List of integer sequences with links to LODA programs.

  • A050024 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the smallest integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.
  • A050025 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.
  • A050026 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.
  • A050027 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.
  • A050028 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 2.
  • A050029 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 2.
  • A050030 (program): a(n) = a(n-1) + a(m) for n >= 3, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1.
  • A050031 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 2.
  • A050032 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique value such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 3.
  • A050033 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 3.
  • A050034 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 3.
  • A050035 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 3.
  • A050036 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 4.
  • A050037 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 4.
  • A050038 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(3) = 4.
  • A050039 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = 1 and a(2) = 4.
  • A050040 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.
  • A050041 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.
  • A050042 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.
  • A050043 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 1.
  • A050044 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 2.
  • A050045 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 2.
  • A050046 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 2.
  • A050047 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 2.
  • A050048 (program): a(n) = a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 2.
  • A050049 (program): a(n) = a(n-1) + a(m) for n >= 3, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 2.
  • A050050 (program): a(n) = a(n-1)+a(m), where m=n-1-2^p and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050051 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 3.
  • A050052 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 4.
  • A050053 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 4.
  • A050054 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 4.
  • A050055 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 2, and a(3) = 4.
  • A050056 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 1.
  • A050057 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 1.
  • A050058 (program): a(n) = a(n-1)+a(m), where m=n-1-2^p and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050059 (program): a(n)=a(n-1)+a(m), where m=2n-2-2^(p+1) and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050060 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 2.
  • A050061 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 2.
  • A050062 (program): a(n) = a(n-1)+a(m), where m=n-1-2^p and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050063 (program): a(n)=a(n-1)+a(m), where m=2n-2-2^(p+1) and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050064 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 3.
  • A050065 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 3.
  • A050066 (program): a(n) = a(n-1) + a(m) for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = a(3) = 3.
  • A050067 (program): a(n)=a(n-1)+a(m), where m=2n-2-2^(p+1) and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050068 (program): a(n) = a(n-1) + a(m) for n >= 3, where m = 2*n - 3 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 3.
  • A050069 (program): a(n) = a(n-1) + a(m) for n >= 3, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 3.
  • A050070 (program): a(n) = a(n-1)+a(m), where m=n-1-2^p and 2^p<n-1<=2^(p+1), for m >= 4.
  • A050071 (program): a(n) = a(n-1)+a(m), where m=2n-2-2^(p+1) and 2^p<n-1<=2^(p+1), for n >= 4.
  • A050072 (program): a(n) = |a(n-1) - a(m)| for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.
  • A050073 (program): a(n) = |a(n-1) - a(m)| for n >= 4, where m = n - 1 - 2^p and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = a(2) = a(3) = 1.
  • A050140 (program): a(n) = 2*floor(n*phi)-n, where phi = (1+sqrt(5))/2.
  • A050141 (program): a(n) = 2*floor((n+1)*phi) - 2*floor(n*phi) - 1 where phi = (1 + sqrt(5))/2 is the golden ratio.
  • A050146 (program): a(n) = T(n,n), array T as in A050143.
  • A050147 (program): a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.
  • A050148 (program): a(n) = T(n,n-2), array T as in A050143.
  • A050149 (program): a(n) = T(n,n-3), array T as in A050143.
  • A050150 (program): Odd numbers with prime number of divisors.
  • A050151 (program): a(n) = T(n,n+2), array T as in A050143.
  • A050152 (program): a(n) = T(n,n+3), array T as in A050143.
  • A050155 (program): Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).
  • A050156 (program): T(n,k)=M(2n+3,n+3,k+3), 0<=k<=n, n >= 0, array M as in A050144.
  • A050157 (program): T(n, k) = S(2n, n, k) for 0<=k<=n and n>=0, where S(p, q, r) is the number of upright paths from (0, 0) to (p, p-q) that do not rise above the line y = x-r.
  • A050158 (program): T(n, k) = S(2*n + 1, n, k + 1) for 0<=k<=n and n >= 0, array S as in A050157.
  • A050163 (program): T(n, k) = S(2n+2, n+2, k+2) for 0<=k<=n and n >= 0, array S as in A050157.
  • A050164 (program): T(n,k)=S(2n+3,n+3,k+3), 0<=k<=n, n >= 0, array S as in A050157.
  • A050165 (program): Triangle read by rows: T(n,k) = M(2n+1,k,-1), 0 <= k <= n, n >= 0, array M as in A050144.
  • A050166 (program): Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.
  • A050168 (program): a(0) = 1; for n > 0, a(n) = binomial(n, floor(n/2)) + binomial(n-1, floor(n/2)).
  • A050169 (program): Triangle read by rows: T(n,k) = gcd(C(n,k), C(n,k-1)), n >= 1, 1 <= k <= n.
  • A050174 (program): T(n,k) = S(n,k,k-2), 1<=k<=n-2, n >= 3, array S as in A050157.
  • A050176 (program): T(n,k) = M0(n+1,k,f(n,k)), where M0(p,q,r) is the number of upright paths from (0,0) to (1,0) to (p,p-q) that meet the line y = x-r and do not rise above it and f(n,k) is the least t such that M0(n+1,k,f) is not 0.
  • A050181 (program): T(2n+3, n), array T as in A051168; a count of Lyndon words.
  • A050182 (program): a(n) = T(2*n+4, n), array T as in A051168 (a count of Lyndon words).
  • A050185 (program): T(2n+7,n), array T as in A051168; a count of Lyndon words.
  • A050187 (program): a(n) = n * floor((n-1)/2).
  • A050188 (program): T(n,3), array T as in A050186; a count of aperiodic binary words.
  • A050189 (program): T(n,4), array T as in A050186; a count of aperiodic binary words.
  • A050190 (program): T(n,5), array T as in A050186; a count of aperiodic binary words.
  • A050192 (program): a(n)=a(n-1)+a(n-2)-d, where d=a(n/2) if n is even, else d=0; 2 initial terms.
  • A050206 (program): Triangle read by rows: smallest denominator of the expansion of k/n using the greedy algorithm, 1<=k<=n-1.
  • A050214 (program): Product((d+n/d): d divides n and d^2<=n); a(1)=1.
  • A050228 (program): a(n) is the number of subsequences {s(k)} of {1,2,3,…n} such that s(k+1)-s(k) is 1 or 3.
  • A050229 (program): Numbers k such that for any x in 1..k-1 there exists a y in 0..k-2 such that x^2 == 2^y (mod k).
  • A050231 (program): a(n) is the number of n-tosses having a run of 3 or more heads for a fair coin (i.e., probability is a(n)/2^n).
  • A050232 (program): a(n) is the number of n-tosses having a run of 4 or more heads for a fair coin (i.e., probability is a(n)/2^n).
  • A050233 (program): a(n) is the number of n-tosses having a run of 5 or more heads for a fair coin (i.e., probability is a(n)/2^n).
  • A050250 (program): Number of nonzero palindromes less than 10^n.
  • A050253 (program): G.f.: ( 1 - x^2 - sqrt( 1 - 2*x^2 - 4*x^3 - 3*x^4 ) ) / ( 2*x^3 ).
  • A050265 (program): Primes of the form 2*n^2 + 11.
  • A050268 (program): Primes of the form 36*n^2 - 810*n + 2753, listed in order of increasing parameter n >= 0.
  • A050270 (program): Largest value b for Diophantine 1-doubles (a,b) ordered by smallest b.
  • A050271 (program): Numbers k such that k = floor(sqrt(k)*ceiling(sqrt(k))).
  • A050275 (program): Largest value c for Diophantine 1-triples (a,b,c) ordered by smallest c,b.
  • A050278 (program): Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once.
  • A050289 (program): Zeroless pandigital numbers: numbers containing the digits 1-9 (each appearing at least once) and no 0’s.
  • A050291 (program): Number of double-free subsets of {1, 2, …, n}.
  • A050292 (program): a(2n) = 2n - a(n), a(2n+1) = 2n + 1 - a(n) (for n >= 0).
  • A050294 (program): Maximum cardinality of a 3-fold-free subset of {1, 2, …, n}.
  • A050296 (program): Maximum cardinality of a strongly triple-free subset of {1, 2, …, n}.
  • A050297 (program): Number of scalars which can be constructed from the Riemann tensor and metric tensor in n dimensions.
  • A050315 (program): Main diagonal of A050314.
  • A050320 (program): Number of ways n is a product of squarefree numbers > 1.
  • A050321 (program): k such that A050292(k) is different from A004396(k).
  • A050328 (program): Number of ordered factorizations of n into squarefree numbers > 1.
  • A050351 (program): Number of 3-level labeled linear rooted trees with n leaves.
  • A050352 (program): Number of 4-level labeled linear rooted trees with n leaves.
  • A050353 (program): Number of 5-level labeled linear rooted trees with n leaves.
  • A050354 (program): Number of ordered factorizations of n with one level of parentheses.
  • A050356 (program): Number of ordered factorizations of n with 2 levels of parentheses.
  • A050358 (program): Number of ordered factorizations of n with 3 levels of parentheses.
  • A050361 (program): Number of factorizations into distinct prime powers greater than 1.
  • A050369 (program): Number of ordered factorizations of n into 2 kinds of 2, 3 kinds of 3, …
  • A050376 (program): “Fermi-Dirac primes”: numbers of the form p^(2^k) where p is prime and k >= 0.
  • A050377 (program): Number of ways to factor n into “Fermi-Dirac primes” (members of A050376).
  • A050399 (program): Least k such that n = A009195(k) (= gcd(phi(k), k)).
  • A050403 (program): Partial sums of A051877.
  • A050404 (program): Partial sums of A051878.
  • A050405 (program): Partial sums of A051879.
  • A050406 (program): Partial sums of A051880.
  • A050407 (program): a(n) = n*(n^2 - 6*n + 11)/6.
  • A050408 (program): a(n) = (117*n^2 - 99*n + 2)/2.
  • A050409 (program): Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.
  • A050410 (program): Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2.
  • A050435 (program): a(n) = composite(composite(n)), where composite = A002808, composite numbers.
  • A050436 (program): Third-order composites.
  • A050438 (program): Fourth-order composites.
  • A050439 (program): Fifth-order composites.
  • A050440 (program): Sixth-order composites.
  • A050441 (program): Partial sums of A051865.
  • A050442 (program): Octahedral torus number: a(n) = n^2 + 2*(Sum_{k=1..n-1} k^2) - 2*(floor((n+1)/2)^2 + 2*(Sum_{k=1..floor((n+1)/2)-1} k^2)) + (1 - (-1)^n)/2.
  • A050443 (program): a(0)=4, a(1)=0, a(2)=0, a(3)=3; thereafter a(n) = a(n-3) + a(n-4).
  • A050448 (program): a(n) = Sum_{d|n, d==1 (mod 4)} d^4.
  • A050449 (program): a(n) = Sum_{d|n, d==1 (mod 4)} d.
  • A050450 (program): Sum_{d|n, d=1 mod 4} d^2.
  • A050451 (program): a(n) = Sum_{d|n, d=1 mod 4} d^3.
  • A050452 (program): a(n) = Sum_{d|n, d=3 mod 4} d.
  • A050453 (program): Sum_{d|n, d=3 mod 4} d^2.
  • A050454 (program): a(n) = Sum_{d|n, d=3 mod 4} d^3.
  • A050455 (program): a(n) = Sum_{d|n, d=3 mod 4} d^4.
  • A050456 (program): a(n) = Sum_{d|n, d==1 mod 4} d^4 - Sum_{d|n, d==3 mod 4} d^4.
  • A050457 (program): a(n) = Sum_{ d divides n, d==1 mod 4} d - Sum_{ d divides n, d==3 mod 4} d.
  • A050458 (program): Difference between Sum_{d|n, d == 1 mod 4} d^2 and Sum_{d|n, d == 3 mod 4} d^2.
  • A050459 (program): a(n) = Sum_{d|n, d==1 mod 4} d^3 - Sum_{d|n, d==3 mod 4} d^3.
  • A050460 (program): a(n) = Sum_{ d divides n, n/d=1 mod 4} d.
  • A050461 (program): a(n) = Sum_{d|n, n/d=1 mod 4} d^2.
  • A050462 (program): a(n) = Sum_{d|n, n/d=1 mod 4} d^3.
  • A050463 (program): a(n) = Sum_{d|n, n/d=1 mod 4} d^4.
  • A050464 (program): a(n) = Sum_{d divides n, n/d=3 mod 4} d.
  • A050465 (program): Sum_{d|n, n/d=3 mod 4} d^2.
  • A050466 (program): a(n) = Sum_{d|n, n/d=3 mod 4} d^3.
  • A050467 (program): Sum_{d|n, n/d=3 mod 4} d^4.
  • A050468 (program): Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.
  • A050469 (program): a(n) = Sum_{ d divides n, n/d=1 mod 4} d - Sum_{ d divides n, n/d=3 mod 4} d.
  • A050470 (program): a(n) = Sum_{d|n, n/d == 1 (mod 4)} d^2 - Sum_{d|n, n/d == 3 (mod 4)} d^2.
  • A050471 (program): a(n) = sum_{d|n, n/d=1 mod 4} d^3 - sum_{d|n, n/d=3 mod 4} d^3.
  • A050476 (program): a(n) = C(n)*(6n+1) where C(n)=Catalan numbers (A000108).
  • A050477 (program): a(n) = C(n)*(7n+1) where C(n)=Catalan numbers (A000108).
  • A050478 (program): a(n) = C(n)*(8n+1) where C(n)=Catalan numbers (A000108).
  • A050479 (program): a(n) = C(n)*(9n+1) where C(n)=Catalan numbers (A000108).
  • A050482 (program): Sum of remainders when n-th prime is divided by all preceding integers.
  • A050483 (program): Partial sums of A051947.
  • A050484 (program): Partial sums of A051946.
  • A050486 (program): a(n) = binomial(n+6,6)*(2n+7)/7.
  • A050487 (program): Geometric Connell sequence: start with 1; then next two numbers == 2 mod 3; next four == 3 mod 3; next eight == 1 mod 3; etc.
  • A050488 (program): a(n) = 3*(2^n-1) - 2*n.
  • A050489 (program): a(n) = C(n)*(10n+1) where C(n)=Catalan numbers (A000108).
  • A050490 (program): a(n) = C(n)*(11n+1) where C(n)=Catalan numbers (A000108).
  • A050491 (program): a(n) = C(n)*(12n+1) where C(n)=Catalan numbers (A000108).
  • A050492 (program): Thickened cube numbers: a(n) = n*(n^2 + (n-1)^2) + (n-1)*2*n*(n-1).
  • A050493 (program): a(n) = sum of binary digits of n-th triangular number.
  • A050494 (program): Partial sums of A051923.
  • A050506 (program): Nearest integer to log(Fibonacci(n)).
  • A050508 (program): Golden rectangle numbers: n * A007067(n).
  • A050509 (program): House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.
  • A050510 (program): Golden rectangular box numbers: a(n) = n*A007067(n)*A007067(A007067(n)).
  • A050511 (program): a(n) = (-1)^n * Sum_{i=0..n} binomial(n+1,i+1)*Catalan(i).
  • A050512 (program): a(n) = (a(n-1)*a(n-3) - a(n-2)^2) / a(n-4), with a(0) = 0, a(1) = a(2) = a(3) = 1, a(4) = -1.
  • A050514 (program): Cards left over after dealing evenly to n people.
  • A050519 (program): Increments of arithmetic progression of at least 6 terms having the same value of phi in A050518.
  • A050520 (program): Values of phi in arithmetic progression of at least 6 terms having the same value of phi in A050518.
  • A050530 (program): Numbers k such that k - phi(k) is prime.
  • A050531 (program): Number of multigraphs with loops on 3 nodes with n edges.
  • A050533 (program): Thickened pyramidal numbers: a(n) = 2*(n+1)*n + Sum_{i=1..n} (4*i*(i-1) + 1).
  • A050534 (program): Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n*(n+1)*(n-1)*(n-2)/8.
  • A050603 (program): A001511 with every term repeated.
  • A050604 (program): Column 3 of A050600: a(n) = add1c(n,3).
  • A050605 (program): Column/row 2 of A050602: a(n) = add3c(n,2).
  • A050606 (program): Column/row 3 of A050602: a(n) = add3c(n,3).
  • A050611 (program): Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+1) = FL(n+1)Product(L(2^i)^bit(n,i),i=0..).
  • A050612 (program): Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+3) = FL(n+3)Product(L(2^i)^bit(n,i),i=0..).
  • A050613 (program): Products of distinct terms of 1 and rest from A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^i)^bit(n,i).
  • A050614 (program): Products of distinct terms of A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^(i+1))^bit(n,i).
  • A050615 (program): Products of distinct terms of Fibonacci(2^(i+2)): a(n) = Product_{i=0..floor(log_2(n+1))} F(2^(i+2))^bit(n,i).
  • A050620 (program): Quotients arising from sequence A035014.
  • A050621 (program): Smallest n-digit number divisible by 2^n.
  • A050622 (program): Numbers m that are divisible by 2^k, where k is the digit length of m.
  • A050623 (program): Smallest n-digit number divisible by 3^n.
  • A050624 (program): Let b(n) = A050623(n) = smallest n-digit number divisible by 3^n; sequence gives b(n)/3^n.
  • A050626 (program): Product of digits of n is a nonzero square.
  • A050674 (program): Inserting a digit ‘0’ between adjacent digits of n makes a prime.
  • A050683 (program): Number of nonzero palindromes of length n.
  • A050684 (program): Number of nonzero palindromes < 10^n and containing at least one digit ‘1’.
  • A050685 (program): Number of nonzero palindromes < 10^n and containing at least one digit ‘0’.
  • A050686 (program): Number of palindromes of length n and containing the digit 1 (or any other fixed nonzero digit).
  • A050703 (program): Numbers that when added to the sum of their prime factors (with multiplicity) become prime.
  • A050720 (program): Number of nonzero palindromes of length n containing the digit ‘0’.
  • A050726 (program): Decimal expansion of 5^n contains no pair of consecutive equal digits (probably finite).
  • A050727 (program): Decimal expansion of 6^n contains no pair of consecutive equal digits (probably finite).
  • A050728 (program): Decimal expansion of 7^n contains no pair of consecutive equal digits (probably finite).
  • A050735 (program): Numbers of form 5^k (values of k see A050726) containing no pair of consecutive equal digits (probably finite).
  • A050736 (program): Numbers of form 6^k (values of k see A050727) containing no pair of consecutive equal digits (probably finite).
  • A050737 (program): Numbers of form 7^k (values of k see A050728) containing no pair of consecutive equal digits (probably finite).
  • A050763 (program): Numbers k such that the decimal expansion of k^k contains no pair of consecutive equal digits (probably finite).
  • A050795 (program): Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.
  • A050796 (program): Numbers n such that n^2 + 1 is expressible as the sum of two nonzero squares in at least one way (the trivial solution n^2 + 1 = n^2 + 1^2 is not counted).
  • A050799 (program): Values of n^2 - 1 resulting from A050795.
  • A050800 (program): Values of n^2 + 1 resulting from A050796.
  • A050803 (program): Cubes expressible as the sum of two nonzero squares in at least one way.
  • A050804 (program): Numbers n such that n^3 is the sum of two nonzero squares in exactly one way.
  • A050815 (program): Number of positive Fibonacci numbers with n decimal digits.
  • A050820 (program): Odd numbers in the sequence generated by a(n)=|a(n-1)+2a(n-2)-n|.
  • A050821 (program): Even numbers in the sequence generated by a(n)=|a(n-1)+2a(n-2)-n| where a(0)=a(1)=1.
  • A050871 (program): Row sums of even numbered rows of array T in A050870 (periodic binary words).
  • A050872 (program): a(n) = (1/2)*A050871 (row sums of array T in A050870, periodic binary words).
  • A050873 (program): Triangular array T read by rows: T(n,k) = gcd(n,k).
  • A050874 (program): Binary numbers d(1)…d(j) such that d(i) = d(j+1-i) for all but 6 values of i.
  • A050875 (program): Binary numbers d(1)…d(j) such that d(i) != d(j+1-i) for all but 6 values of i.
  • A050914 (program): a(n) = n*3^n + 1.
  • A050915 (program): a(n) = n*4^n + 1.
  • A050916 (program): a(n) = n*5^n + 1.
  • A050917 (program): a(n) = n*6^n + 1.
  • A050919 (program): a(n) = n*7^n + 1.
  • A050925 (program): Numerator of (n+1)*Bernoulli(n).
  • A050926 (program): Binary representation of A007908(n).
  • A050928 (program): Sum of digits of A050926(n).
  • A050931 (program): Numbers having a prime factor congruent to 1 mod 6.
  • A050932 (program): Denominator of (n+1)*Bernoulli(n).
  • A050935 (program): a(1)=0, a(2)=0, a(3)=1, a(n+1) = a(n) - a(n-2).
  • A050939 (program): Numbers that are not the sum of consecutive Fibonacci numbers.
  • A050941 (program): Numbers that are not the sum of consecutive triangular numbers.
  • A050946 (program): “Stirling-Bernoulli transform” of Fibonacci numbers.
  • A050970 (program): Numerator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} (4k+1)^(-n).
  • A050971 (program): 4*Denominator of S(n)/Pi^n, where S(n) = Sum_{k=-inf..+inf} ((4k+1)^(-n)).
  • A050975 (program): Haupt-exponents of 3 modulo integers relatively prime to 3.
  • A050976 (program): Haupt-exponents of 4 modulo integers relatively prime to 4.
  • A050977 (program): Haupt-exponents of 5 modulo integers relatively prime to 5.
  • A050978 (program): Haupt-exponents of 6 modulo integers relatively prime to 6.
  • A050979 (program): Haupt-exponents of 7 modulo integers relatively prime to 7.
  • A050980 (program): Haupt-exponents of 8 modulo integers relatively prime to 8.
  • A050981 (program): Haupt-exponents of 9 modulo integers relatively prime to 9.
  • A050982 (program): 5-idempotent numbers.
  • A050983 (program): de Bruijn’s S(4,n).
  • A050984 (program): de Bruijn’s S(5,n).
  • A050985 (program): Cubefree part of n.
  • A050988 (program): 6-idempotent numbers.
  • A050989 (program): 7-idempotent numbers.
  • A050995 (program): Reduced denominators of series expansion for integrand in Renyi’s parking constant.
  • A050997 (program): Fifth powers of primes.
  • A050999 (program): Sum of squares of odd divisors of n.
  • A051000 (program): Sum of cubes of odd divisors of n.
  • A051001 (program): Sum of 4th powers of odd divisors of n.
  • A051002 (program): Sum of 5th powers of odd divisors of n.
  • A051022 (program): Interpolate 0’s between each pair of digits of n.
  • A051023 (program): Middle column of rule-30 1-D cellular automaton, from a lone 1 cell.
  • A051027 (program): a(n) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.
  • A051032 (program): Summatory Rudin-Shapiro sequence for 2^(n-1).
  • A051033 (program): a(n) = binomial(n, floor(n/3)).
  • A051034 (program): Minimal number of primes needed to sum to n.
  • A051035 (program): Composite numbers which can be represented as the sum of two primes (i.e., A002808 excluding A025583).
  • A051036 (program): a(n) = binomial(n, floor(n/4)).
  • A051037 (program): 5-smooth numbers, i.e., numbers whose prime divisors are all <= 5.
  • A051038 (program): 11-smooth numbers: numbers whose prime divisors are all <= 11.
  • A051039 (program): 4-Stohr sequence.
  • A051040 (program): 5-Stohr sequence.
  • A051047 (program): For n > 5, a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); initial terms are 1, 3, 8, 120, 1680.
  • A051048 (program): Sqrt[a(n)a(n+1)+1] of A051047.
  • A051049 (program): Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.
  • A051052 (program): a(n) = binomial(n, floor(n/5)).
  • A051053 (program): a(n) = binomial(n, floor(n/6)).
  • A051054 (program): a(n) = Sum_{k=1..n} C(n, floor(n/k)).
  • A051062 (program): a(n) = 16*n + 8.
  • A051063 (program): 27*n+9 or 27*n+18.
  • A051064 (program): 3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.
  • A051065 (program): a(n) = A004128(n) mod 2.
  • A051066 (program): Partial sums of A051065.
  • A051067 (program): A051066 read mod 2.
  • A051068 (program): Partial sums of A014578.
  • A051069 (program): A051068 read mod 2.
  • A051100 (program): Primes p such that x^62 = -2 has a solution mod p.
  • A051102 (program): Floor of exp(n-th prime).
  • A051109 (program): Hyperinflation sequence for banknotes.
  • A051111 (program): Expansion of x/(x^4-3*x^3+4*x^2-2*x+1).
  • A051119 (program): n/p^k, where p = largest prime dividing n and p^k = highest power of p dividing n.
  • A051122 (program): a(n) = Fibonacci(n) AND Fibonacci(n+1).
  • A051123 (program): a(n) = Fibonacci(n) OR Fibonacci(n+1).
  • A051124 (program): a(n) = Fibonacci(n) XOR Fibonacci(n+1).
  • A051125 (program): Table T(n,k) = max{n,k} read by antidiagonals (n >= 1, k >= 1).
  • A051126 (program): Table T(n,k) = n mod k read by antidiagonals (n >= 1, k >= 1).
  • A051127 (program): Table T(n,k) = k mod n read by antidiagonals (n >= 1, k >= 1).
  • A051128 (program): Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).
  • A051129 (program): Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).
  • A051132 (program): Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.
  • A051133 (program): a(n) = binomial(2n,n)*n*(2n+1)/2.
  • A051135 (program): a(n) = number of times n appears in the Hofstadter-Conway $10000 sequence A004001.
  • A051138 (program): Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).
  • A051139 (program): a(n) = A000994(n+2) - A000995(n+2).
  • A051140 (program): a(n) = (A000110(n) - A000994(n+2))/2.
  • A051144 (program): Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.
  • A051158 (program): Decimal expansion of Sum_{n >= 0} 1/(2^2^n+1).
  • A051159 (program): Triangular array made of three copies of Pascal’s triangle.
  • A051160 (program): Coefficients in expansion of (1-x)^floor(n/2)(1+x)^ceiling(n/2).
  • A051162 (program): Triangle T(n,k) = n+k, n >= 0, 0 <= k <= n.
  • A051163 (program): Sequence is defined by property that (a0,a1,a2,a3,…) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,…).
  • A051170 (program): T(n,5), array T as in A051168; a count of Lyndon words; aperiodic necklaces with 5 black beads and n-5 white beads.
  • A051172 (program): T(n,7), array T as in A051168; a count of Lyndon words; aperiodic necklaces with 7 black beads and n-7 white beads.
  • A051173 (program): Triangular array T read by rows: T(u,v) = lcm(u,v).
  • A051176 (program): If n mod 3 = 0 then n/3 else n.
  • A051178 (program): Numbers k such that k divides number of divisors of k!.
  • A051188 (program): Sept-factorial numbers.
  • A051189 (program): Octo-factorial numbers.
  • A051190 (program): a(n) = Product_{k=1..n-1} gcd(k,n).
  • A051192 (program): T(n,7), array T as in A050186; a count of aperiodic binary words.
  • A051193 (program): a(n) = Sum_{k=1..n} lcm(n,k).
  • A051194 (program): Triangular array T read by rows: T(n,k) = number of positive integers that divide both n and k.
  • A051195 (program): T(2n+2,n), array T as in A050186; a count of aperiodic binary words.
  • A051196 (program): T(2n+3,n), array T as in A050186; a count of aperiodic binary words.
  • A051197 (program): T(2n+4,n), array T as in A050186; a count of aperiodic binary words.
  • A051201 (program): Sum of elements of the set { [ n/k ] : 1 <= k <= n }.
  • A051232 (program): 9-factorial numbers.
  • A051234 (program): Possible orders of central factor groups of groups.
  • A051236 (program): Largest integer a(n) for which the integer interval [ 0,a(n) ] is a subset of the set of determinants of all n X n 0-1 matrices.
  • A051244 (program): Binary numbers d(1)…d(j) such that d(i) = d(j+1-i) for all but two values of i.
  • A051245 (program): Binary numbers d(1)…d(j) such that d(i) != d(j+1-i) for all but two values of i.
  • A051246 (program): Binary numbers d(1)…d(j) such that d(i) != d(j+1-k) for all but 4 values of i.
  • A051247 (program): Binary numbers d(1)…d(j) such that d(i) = d(j+1-k) for all but 4 values of i.
  • A051250 (program): Numbers whose reduced residue system consists of 1 and prime powers only.
  • A051253 (program): Weights of rotation-symmetric functions in n variables.
  • A051256 (program): Numbers formed from binomial coefficients (mod 2) interpreted as digits in factorial base.
  • A051257 (program): Numbers formed from binomial coefficients (mod 2+k) interpreted as digits in factorial base.
  • A051262 (program): 10-factorial numbers.
  • A051263 (program): Expansion of 1/((1-x)*(1-x^3)^2*(1-x^5)).
  • A051270 (program): Numbers that are divisible by exactly 5 different primes.
  • A051271 (program): Number of numbers either relatively prime to or divisors of primorial number(n).
  • A051272 (program): Number of numbers neither relatively prime to nor divisors of primorial number(n).
  • A051274 (program): Expansion of (1+x^4)/((1-x^2)*(1-x^3)).
  • A051275 (program): Expansion of (1+x^2)/((1-x^2)*(1-x^3)).
  • A051276 (program): Nonzero coefficients in one of the 5-adic expansions of sqrt(-1).
  • A051277 (program): Coefficients in 7-adic expansion of sqrt(2).
  • A051283 (program): Numbers k such that if one writes k = Product p_i^e_i (p_i primes) and P = max p_i^e_i, then k/P > P.
  • A051286 (program): Whitney number of level n of the lattice of the ideals of the fence of order 2n.
  • A051287 (program): Triangular array T read by rows: T(n,k)=P(n,k,|n-2k|), where P(n,k,c)=number of vectors (x(1),x(2,),…,x(n)) of k 1’s and n-k 0’s such that x(i)=x(n+1-i) for exactly c values of i.
  • A051289 (program): Triangular array T read by rows: T(n,k)=P(2n+1,n,2k+1), where P(n,k,c)=number of vectors (x(1),x(2,),…,x(n)) of k 1’s and n-k 0’s such that x(i)=x(n+1-i) for exactly c values of i.
  • A051290 (program): Triangular array T read by rows: T(n,k)=P(2n+3,n,2k+3), where P(n,k,c)=number of vectors (x(1),x(2,),…,x(n)) of k 1’s and n-k 0’s such that x(i)=x(n+1-i) for exactly c values of i.
  • A051291 (program): Whitney number of level n of the lattice of the ideals of the fence of order 2 n + 1.
  • A051292 (program): Whitney number of level n of the lattice of the ideals of the crown of size 2 n.
  • A051293 (program): Number of nonempty subsets of {1,2,3,…,n} whose elements have an integer average.
  • A051294 (program): a(n) = F(n^2)/F(n), where F(n) = A000045(n) is the n-th Fibonacci number.
  • A051295 (program): a(0)=1; thereafter, a(m+1) = Sum_{k=0..m} k!*a(m-k).
  • A051296 (program): INVERT transform of factorial numbers.
  • A051297 (program): (Terms in A028266)/2.
  • A051298 (program): (Terms in A028273)/2.
  • A051329 (program): A generalized Thue-Morse sequence.
  • A051336 (program): Number of arithmetic progressions in {1,2,3,…,n}, including trivial arithmetic progressions of lengths 1 and 2.
  • A051340 (program): A simple 2-dimensional array, read by antidiagonals: T[i,j] = 1 for j>0, T[i,0] = i+1; i,j = 0,1,2,3,…
  • A051343 (program): Number of ways of writing n as a sum of 3 nonnegative cubes (counted naively).
  • A051344 (program): Number of ways of writing n as a sum of 3 positive cubes (counted naively).
  • A051348 (program): a(n) = F(n) / Product_{p|n} F(p), where F(k) is k-th Fibonacci number and the p’s in product are the distinct primes dividing n.
  • A051349 (program): Sum of first n nonprimes.
  • A051350 (program): Sum of digit-sums of first n nonprimes.
  • A051351 (program): a(n) = a(n-1) + sum of digits of n-th prime.
  • A051352 (program): a(0) = 0; for n>0, a(n) = a(n-1) + n if n not prime else a(n-1) - n.
  • A051357 (program): Chernoff sequence A006939 divided by 2.
  • A051358 (program): (Terms in A028279)/2.
  • A051359 (program): (Terms in A028286)/2.
  • A051361 (program): A class of Boolean functions of n variables and rank 3.
  • A051369 (program): a(n+1) = a(n) + sum of digits of a(n)^2.
  • A051370 (program): a(n+1) = a(n) + sum of digits of a(n)^2.
  • A051371 (program): a(n+1) = a(n) + sum of digits of a(n)^2.
  • A051373 (program): a(n+1) = a(n) + sum of digits of (a(n)^3).
  • A051375 (program): Number of Boolean functions of n variables and rank 3 from Post class F(5,inf).
  • A051376 (program): Number of Boolean functions of n variables and rank 4 from Post class F(5,inf).
  • A051382 (program): Numbers k whose base 3 expansion matches (0|1)*(02)?(0|1)* (no more than one “02” allowed in midst of 0’s and 1’s).
  • A051396 (program): a(n) = (2*n-2)*(2*n-3)*a(n-1)+1.
  • A051397 (program): a(n) = (2*n-2)*(2*n-1)*a(n-1)+1.
  • A051398 (program): a(n) = -(n-3)*a(n-1) + 2*(n-2)^2.
  • A051399 (program): a(n) = (n-1)!*a(n-1)+1.
  • A051403 (program): a(n) = (n+2)*(a(n-1) - a(n-2)).
  • A051405 (program): a(n) = (3^n+1)*(3^(n+1)+1)/4.
  • A051406 (program): a(n) = (3^n+1) * (3^(n+1)+1) / 8.
  • A051407 (program): a(n) = 3^n*(3^(n+1)+1)/2.
  • A051408 (program): a(n+1) = a(n) + sum of digits of (a(n)^3).
  • A051409 (program): a(n+1) = a(n) + sum of digits of (a(n)^3).
  • A051410 (program): a(n+1) = a(n) + sum of digits of a(n)^3.
  • A051411 (program): a(n+1) = a(n) + sum of digits of a(n)^3.
  • A051412 (program): a(n+1) = a(n) + sum of digits of a(n)^3.
  • A051413 (program): a(n+1) = a(n) + sum of digits of (a(n)^3).
  • A051414 (program): a(n+1) = a(n) + sum of digits of a(n)^3.
  • A051417 (program): Quotients of consecutive values of lcm {1, 3, 5 …,2n-1} or A025547(n+1)/A025547(n).
  • A051418 (program): Square of LCM of {1, 2, …, n}.
  • A051425 (program): (Terms in A029665)/2.
  • A051426 (program): Least common multiple of {2, 4, 6, …, 2n}.
  • A051427 (program): Number of strictly Deza graphs with n nodes.
  • A051428 (program): (Terms in A029658)/2.
  • A051429 (program): (Terms in A029659)/2.
  • A051430 (program): (Terms in A029661)/2.
  • A051431 (program): a(n) = (n+10)!/10!.
  • A051432 (program): (Terms in A029617)/2.
  • A051433 (program): (Terms in A029605)/2.
  • A051434 (program): (Terms in A029607)/2.
  • A051435 (program): (Terms in A029613)/2.
  • A051437 (program): Number of undirected walks of length n+1 on an oriented triangle, visiting n+2 vertices, with n “corners”; the symmetry group is C3. Walks are not self-avoiding.
  • A051442 (program): a(n) = n^(n+1)+(n+1)^n.
  • A051443 (program): a(n) = n^(n+1)*(n+1)^n.
  • A051448 (program): Sum of prime divisors of n (with multiplicity) is a square.
  • A051449 (program): Number of fibered rational knots with n crossings.
  • A051450 (program): Number of positive rational knots with 2n+1 crossings.
  • A051451 (program): a(n) = lcm{ 1,2,…,x } where x is the n-th prime power (A000961).
  • A051452 (program): a(n) = 1 + lcm(1..k) where k is the n-th prime power A000961(n).
  • A051455 (program): (Terms in A029623)/2.
  • A051456 (program): (Terms in A029625)/2.
  • A051457 (program): (Terms in A029627)/2.
  • A051458 (program): (Terms in A029631)/2.
  • A051462 (program): Molien series for group G_{1,2}^{8} of order 1536.
  • A051467 (program): (Terms in A029640)/2.
  • A051468 (program): (Terms in A029641)/2.
  • A051469 (program): (Terms in A029643)/2.
  • A051471 (program): (Terms in A029647)/2.
  • A051472 (program): (A028317)/2.
  • A051473 (program): (Terms in A028321)/2.
  • A051474 (program): (Terms in A014450)/2.
  • A051476 (program): (Terms in A014733)/4.
  • A051478 (program): a(n) is the number of values k satisfying phi(k) = 4*n+2, n>0.
  • A051489 (program): a(n) = n^(n+2) + (n+2)^n.
  • A051490 (program): a(n) = n^(n+2)*(n+2)^n.
  • A051494 (program): Expansion of (1 - x + x^2 + x^3)/(1 - x^2)^3.
  • A051495 (program): Expansion of (1 + x + 2*x^2 - x^3 + x^4)/(1 - 3*x^3 + 3*x^6 - x^9).
  • A051497 (program): (Terms in A014476)/2.
  • A051500 (program): a(n) = (3^n + 1)^2/4.
  • A051503 (program): a(n) = min { n, floor(100/n) }.
  • A051513 (program): a(n) = min { 2^n, floor(100/n) }.
  • A051514 (program): (Terms in A014762)/4.
  • A051515 (program): (Terms in A014738)/4.
  • A051524 (program): Second unsigned column of triangle A051338.
  • A051525 (program): Third unsigned column of triangle A051338.
  • A051533 (program): Numbers that are the sum of two positive triangular numbers.
  • A051536 (program): a(n) = least common multiple of {1, 4, 7, 10, 13 …, 3n+1} (A016777).
  • A051537 (program): Triangle read by rows: T(i,j) = lcm(i,j)/gcd(i,j) for 1 <= j <= i.
  • A051538 (program): Least common multiple of {b(1),…,b(n)}, where b(k) = k(k+1)(2k+1)/6 = A000330(k).
  • A051539 (program): a(n) is the least common multiple of {1, 5, 9, 13, 17, …, 4n+1} (A016813).
  • A051540 (program): Least common multiple of {2, 5, 8, 11, 14, …, 3n+2} (A016789).
  • A051541 (program): Quotients of consecutive values of LCM {1, 5, 9, 13, 17, …, (4n+1)}.
  • A051542 (program): Quotients of consecutive values of LCM {b(1),…,b(n)}, b() = A000330.
  • A051543 (program): Quotients of consecutive values of lcm of first n triangular numbers (A000217).
  • A051544 (program): Quotients of consecutive values of lcm {1, 4, 7, 10, 13 …,(3n+1)} (A016777).
  • A051545 (program): Second unsigned column of triangle A051339.
  • A051546 (program): Third unsigned column of triangle A051339.
  • A051552 (program): Quotients of consecutive values of LCM {b(0), b(1) …,b(n)}, b() = A016789.
  • A051560 (program): Second unsigned column of triangle A051379.
  • A051561 (program): Third unsigned column of triangle A051379.
  • A051562 (program): Second unsigned column of triangle A051380.
  • A051563 (program): Third unsigned column of triangle A051380.
  • A051564 (program): Second unsigned column of triangle A051523.
  • A051565 (program): Third unsigned column of triangle A051523.
  • A051575 (program): a(n) = LCM { Catalan(0), …, Catalan(n) }.
  • A051576 (program): Order of Burnside group B(3,n) of exponent 3 and rank n.
  • A051577 (program): a(n) = (2*n + 3)!!/3 = A001147(n+2)/3.
  • A051578 (program): a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).
  • A051579 (program): a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).
  • A051580 (program): a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).
  • A051581 (program): a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).
  • A051582 (program): a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).
  • A051583 (program): a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).
  • A051588 (program): Number of 3 X n binary matrices such that any 2 rows have a common 1.
  • A051596 (program): Numerical values or Gematriot of Hebrew letters {aleph, bet, …, tav}.
  • A051597 (program): Rows of triangle formed using Pascal’s rule except begin and end n-th row with n+1.
  • A051601 (program): Rows of triangle formed using Pascal’s rule except we begin and end the n-th row with n.
  • A051602 (program): a(n) is the maximal number of squares that can be formed from n points in the plane.
  • A051604 (program): a(n) = (3*n+4)!!!/4!!!
  • A051605 (program): a(n) = (3*n+5)!!!/5!!!.
  • A051606 (program): a(n) = (3*n+6)!!!/6!!!, related to A032031 ((3*n)!!! triple factorials).
  • A051607 (program): a(n) = (3*n+7)!!!/7!!!, related to A007559(n+1) ((3*n+1)!!! triple factorials).
  • A051608 (program): a(n) = (3*n+8)!!!/8!!!, related to A008544(n+1) ((3*n+2)!!! triple factorials).
  • A051609 (program): a(n) = (3*n+9)!!!/9!!!, related to A032031 ((3*n)!!! triple factorials).
  • A051611 (program): Numbers that are not the sum of 2 nonzero triangular numbers.
  • A051612 (program): a(n) = sigma(n) - phi(n).
  • A051617 (program): a(n) = (4*n+5)(!^4)/5(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).
  • A051618 (program): a(n) = (4*n+6)(!^4)/6(!^4).
  • A051619 (program): a(n) = (4*n+7)(!^4)/7(!^4), related to A034176(n+1) ((4*n+3)(!^4) quartic, or 4-factorials).
  • A051620 (program): a(n) = (4*n+8)(!^4)/8(!^4), related to A034177(n+1) ((4*n+4)(!^4) quartic, or 4-factorials).
  • A051621 (program): a(n) = (4*n+9)(!^4)/9(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).
  • A051622 (program): a(n) = (4*n+10)(!^4)/10(!^4), related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).
  • A051623 (program): a(n) = Sum_{x=1+floor(sqrt(n))..floor(sqrt(2n))} (x^2 - n).
  • A051624 (program): 12-gonal (or dodecagonal) numbers: a(n) = n*(5*n-4).
  • A051628 (program): Number of digits in decimal expansion of 1/n before the periodic part begins.
  • A051630 (program): Poincaré series [or Poincare series] (or Molien series) for Gamma_2(1,2)_(2).
  • A051631 (program): Triangle formed using Pascal’s rule except begin and end n-th row with n-1.
  • A051632 (program): Rows of triangle formed using Pascal’s rule except we begin and end the n-th row with n-2.
  • A051633 (program): a(n) = 5*2^n - 2.
  • A051634 (program): Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2.
  • A051635 (program): Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.
  • A051638 (program): a(n) = sum_{k=0..n} (C(n,k) mod 3).
  • A051639 (program): Concatenation of 3^k, k = 0,..,n.
  • A051644 (program): Primes of the form 6*p + 1 where p is also prime.
  • A051645 (program): Primes p such that 30*p+1 is also prime.
  • A051646 (program): Primes of the form 30*p + 1 where p is also prime.
  • A051648 (program): Primes of form 210*p + 1 where p is a prime.
  • A051656 (program): Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).
  • A051662 (program): House numbers: a(n) = (n+1)^3 + Sum_{i=1..n} i^2.
  • A051666 (program): Rows of triangle formed using Pascal’s rule except begin and end n-th row with n^2.
  • A051667 (program): a(n) = 6*2^n - 4*n - 6.
  • A051668 (program): Rows of triangle formed using Pascal’s rule except begin and end n-th row with (n+1)^2.
  • A051669 (program): 11*2^n - 4*n - 10.
  • A051672 (program): Triangle of up-down sums of k-th powers: a(n,k)=sum(i^k,i=1..n)+sum((n-i)^k,i=1..n-1), n,k>0.
  • A051673 (program): Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.
  • A051674 (program): a(n) = prime(n)^prime(n).
  • A051677 (program): Tetrahedron-tree numbers: a(n)=sum(b(m),m=1..n), b(m)=1, 1,3, 1,3,6, 1,3,6,10,…, 1,2,…,i*(i+1)2.
  • A051678 (program): Square-pyramid-tree numbers: a(n) = sum(b(m),m=1..n), b(m) = 1^2, 1^2,2^2, 1^2,2^2,3^2,.. = (A002260)^2.
  • A051679 (program): Total number of even entries in first n rows of Pascal’s triangle (the zeroth and first rows being 1; 1,1).
  • A051682 (program): 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.
  • A051683 (program): Triangle read by rows: T(n,k) = n!*k.
  • A051684 (program): Auxiliary sequence for calculation of number of even permutations of degree n and order exactly 2.
  • A051687 (program): a(n) = (5*n+6)(!^5)/6, related to A008548 ((5*n+1)(!^5) quintic, or 5-factorials).
  • A051688 (program): a(n) = (5*n+7)(!^5)/7(!^5), related to A034323 ((5*n+2)(!^5) quintic, or 5-factorials).
  • A051689 (program): a(n) = (5*n+8)(!^5)/8(!^5), related to A034300 ((5*n+3)(!^5) quintic, or 5-factorials).
  • A051690 (program): a(n) = (5*n+9)(!^5)/9(!^5), related to A034301 ((5*n+2)(!^5) quintic, or 5-factorials).
  • A051691 (program): a(n) = (5*n+10)(!^5)/10(!^5), related to A052562 ((5*n)(!^5) quintic, or 5-factorials).
  • A051694 (program): Smallest Fibonacci number that is divisible by n-th prime.
  • A051696 (program): Greatest common divisor of n! and n^n.
  • A051697 (program): Closest prime to n (break ties by taking the smaller prime).
  • A051699 (program): Distance from n to closest prime.
  • A051708 (program): Number of ways to move a chess rook from the lower left corner to square (n,n), with the rook moving only up or right.
  • A051709 (program): a(n) = sigma(n) + phi(n) - 2n.
  • A051711 (program): a(0) = 1; for n > 0, a(n) = n!*4^n/2.
  • A051712 (program): Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.
  • A051713 (program): Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.
  • A051724 (program): Numerator of n/12.
  • A051725 (program): Denominator of n/12.
  • A051726 (program): Numerator of n(n-1)(n-2)/720.
  • A051727 (program): Denominator of n(n-1)(n-2)/720.
  • A051731 (program): Triangle read by rows: T(n,k) = 1 if k divides n, T(n,k) = 0 otherwise (for n >= 1 and 1 <= k <= n).
  • A051733 (program): Numbers n such that A051732(n) = n-1.
  • A051736 (program): Number of 3 X n (0,1)-matrices with no consecutive 1’s in any row or column.
  • A051740 (program): Partial sums of A007584.
  • A051743 (program): a(n) = (1/24)*n*(n + 5)*(n^2 + n + 6).
  • A051744 (program): a(n) = n*(n+1)*(n^2+5*n+18)/24.
  • A051745 (program): a(n) = n*(n^4 + 10*n^3 + 35*n^2 + 50*n + 144)/120.
  • A051746 (program): a(n) = n(n+7)(n+1)(n^2+2n+12)/120.
  • A051747 (program): a(n) = n*(n+1)*(n+2)*(n^2+7*n+32)/120.
  • A051750 (program): Primes whose cubes lack zeros.
  • A051751 (program): Cubes arising in A051750.
  • A051754 (program): Consider problem of placing N queens on an n X n board so that each queen attacks precisely 1 other. Sequence gives maximal number of queens.
  • A051755 (program): Consider problem of placing N queens on an n X n board so that each queen attacks precisely 2 others. Sequence gives maximal number of queens.
  • A051764 (program): Number of torus knots with n crossings.
  • A051777 (program): Triangle read by rows, where row (n) = n mod n, n mod (n-1), n mod (n-2), …n mod 1.
  • A051778 (program): Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), …n mod 2.
  • A051786 (program): Propp’s cubic recurrence: a(0)=a(1)=a(2)=a(3)=1; for n>3, a(n)=(1+a(n-1)*a(n-2)*a(n-3))/a(n-4).
  • A051789 (program): C(n)*(C(n)-1)/2, where C(n) are the Catalan numbers (A000108).
  • A051790 (program): a(n) = C(n)*(C(n)-1)*(C(n)-2)/6, where C(n) are the Catalan numbers (A000108).
  • A051792 (program): a(n) = (-1)^(n-1)*a(n-1)+(-1)^(n-2)*a(n-2), a(1)=1, a(2)=1.
  • A051797 (program): Partial sums of A007585.
  • A051798 (program): a(n) = (n+1)*(n+2)*(n+3)*(9n+4)/24.
  • A051799 (program): Partial sums of A007587.
  • A051801 (program): Product of the nonzero digits of n.
  • A051802 (program): Nonzero multiplicative digital root of n.
  • A051803 (program): Numbers with nonzero multiplicative digital root 1.
  • A051804 (program): Numbers with nonzero multiplicative digital root 2.
  • A051805 (program): Numbers with nonzero multiplicative digital root 3.
  • A051806 (program): Numbers with nonzero multiplicative digital root 4.
  • A051807 (program): Numbers with nonzero multiplicative digital root 5.
  • A051808 (program): Numbers with nonzero multiplicative digital root 6.
  • A051809 (program): Numbers with nonzero multiplicative digital root 7.
  • A051810 (program): Numbers with nonzero multiplicative digital root 8.
  • A051811 (program): Numbers with nonzero multiplicative digital root 9.
  • A051830 (program): Fibonacci(Pn+1) mod Pn, where Pn is the n-th prime.
  • A051831 (program): a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.
  • A051834 (program): Fibonacci(Pn-1) mod Pn, where Pn is the n-th prime.
  • A051836 (program): a(n) = n*(n+1)*(n+2)*(n+3)*(3*n+2)/120.
  • A051843 (program): Partial sums of A002419.
  • A051844 (program): a(n) = LCM_{k=0..n} (2^k + 1).
  • A051846 (program): Digits 1..n in strict descending order n..1 interpreted in base n+1.
  • A051847 (program): Bisection of A051846, divided by the term position.
  • A051848 (program): Bisection of A023811, divided by the term position.
  • A051865 (program): 13-gonal (or tridecagonal) numbers: a(n) = n*(11*n - 9)/2.
  • A051866 (program): 14-gonal (or tetradecagonal) numbers: a(n) = n*(6*n-5).
  • A051867 (program): 15-gonal (or pentadecagonal) numbers: n(13n-11)/2.
  • A051868 (program): 16-gonal (or hexadecagonal) numbers: a(n) = n*(7*n-6).
  • A051869 (program): 17-gonal (or heptadecagonal) numbers: n*(15*n-13)/2.
  • A051870 (program): 18-gonal (or octadecagonal) numbers: a(n) = n*(8*n-7).
  • A051871 (program): 19-gonal (or enneadecagonal) numbers: n(17n-15)/2.
  • A051872 (program): 20-gonal (or icosagonal) numbers: a(n) = n*(9*n-8).
  • A051873 (program): 21-gonal numbers: a(n) = n*(19n - 17)/2.
  • A051874 (program): 22-gonal numbers: a(n) = n*(10*n-9).
  • A051875 (program): 23-gonal numbers: a(n) = n(21n-19)/2.
  • A051876 (program): 24-gonal numbers: a(n) = n*(11*n-10).
  • A051877 (program): Partial sums of A051740.
  • A051878 (program): Partial sums of A051797.
  • A051879 (program): Partial sums of A051798.
  • A051880 (program): a(n) = binomial(n+4,4)*(2*n+1).
  • A051885 (program): Smallest number whose sum of digits is n.
  • A051890 (program): a(n) = 2*(n^2 - n + 1).
  • A051893 (program): a(n) = Sum_{i=1..n-1} i^2*a(i), a(1) = 1.
  • A051895 (program): Partial sums of second pentagonal numbers with even index (A049453).
  • A051903 (program): Maximal exponent in prime factorization of n.
  • A051904 (program): Minimal exponent in prime factorization of n.
  • A051913 (program): Numbers n such that phi(n)/phi(phi(n)) = 3.
  • A051920 (program): a(n) = binomial(n, floor(n/2)) + 1.
  • A051923 (program): Partial sums of A051836.
  • A051924 (program): a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).
  • A051925 (program): a(n) = n*(2*n+5)*(n-1)/6.
  • A051926 (program): Number of independent sets of nodes in graph C_4 X P_n (n>2).
  • A051927 (program): Number of independent vertex sets in the n-prism graph Y_n = K_2 X C_n (n > 2).
  • A051928 (program): Number of independent sets of vertices in graph K_3 X C_n (n > 2).
  • A051929 (program): Number of independent sets of vertices in graph K_4 X C_n (n > 2).
  • A051930 (program): Number of independent sets of vertices in graph K_5 X C_n (n > 2).
  • A051931 (program): Number of independent sets of nodes in graph K_6 X C_n (n > 2).
  • A051932 (program): Number of independent sets of nodes in graph K_7 X C_n (n > 2).
  • A051933 (program): Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.
  • A051935 (program): a(n) = smallest number > a(n-1) such that a(1) + a(2) + … + a(n) is a prime.
  • A051936 (program): Truncated triangular numbers: a(n) = n*(n+1)/2 - 9.
  • A051937 (program): Truncated triangular pyramid numbers: a(n) = Sum_{k=4..n} (k*(k+1)/2 - 9).
  • A051938 (program): Truncated triangular numbers: a(n) = n*(n+1)/2 - 18.
  • A051939 (program): Truncated triangular pyramid numbers: a(n) = (n-5)*(n^2 + 8*n - 66)/6.
  • A051940 (program): Truncated triangular numbers: n*(n+1)/2 - 3*t*(t+1)/2 with t=4.
  • A051941 (program): Truncated triangular pyramid numbers: a(n) = (n-7)*(n^2 + 10*n - 108)/6, n >= 8.
  • A051942 (program): a(n) = n*(n+1)/2 - 45.
  • A051943 (program): Truncated triangular pyramid numbers: a(n) = Sum_{k=9..n} (k*(k+1)/2 - 45).
  • A051944 (program): a(n) = C(n)*(4n+1) where C(n) = Catalan numbers (A000108).
  • A051945 (program): a(n) = C(n)*(5n+1) where C(n) = Catalan numbers (A000108).
  • A051946 (program): Expansion of g.f.: (1+4*x)/(1-x)^7.
  • A051947 (program): Partial sums of A034263.
  • A051949 (program): Differences of two factorial numbers.
  • A051950 (program): Differences between values of tau(n) (A000005): a(n) = tau(n)-tau(n-1).
  • A051951 (program): Second differences of tau(n).
  • A051952 (program): Numbers that are not a sum of 3 positive squares nor are of the form 4^a*(8b+7) and which are not multiples of 4.
  • A051953 (program): Cototient(n) := n - phi(n).
  • A051958 (program): a(n) = 2 a(n-1) + 24 a(n-2), a(0)=0, a(1)=1.
  • A051959 (program): Expansion of (1+6x)/( (1-2x-x^2)(1-x)^2).
  • A051960 (program): a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108.
  • A052008 (program): a(n) = ‘n with digits sorted in ascending order’ + ‘n with digits sorted in descending order’.
  • A052036 (program): Smallest number that must be added to n to make or keep n palindromic.
  • A052038 (program): First nonzero digit in expansion of 1/n.
  • A052040 (program): Numbers whose square is zeroless.
  • A052041 (program): Squares lacking the digit zero in their decimal expansion.
  • A052042 (program): Primes that lack the digit zero in the decimal expansion of their squares.
  • A052043 (program): Squares of primes lacking the digit zero in their decimal expansion.
  • A052044 (program): Numbers k such that k^3 lacks the digit zero in its decimal expansion.
  • A052045 (program): Cubes lacking the digit zero in their decimal expansion.
  • A052060 (program): Numbers n such that the digits of 2^n occur with the same frequency.
  • A052089 (program): Primes formed by concatenating k with k-1.
  • A052100 (program): a(n) = lcm(n, phi(n), n - phi(n)).
  • A052101 (program): One of the three sequences associated with the polynomial x^3 - 2.
  • A052102 (program): The second of the three sequences associated with the polynomial x^3 - 2.
  • A052103 (program): The third of the three sequences associated with the polynomial x^3 - 2.
  • A052106 (program): a(n) = lcm(n, n - phi(n)).
  • A052115 (program): Number of nonnegative integer pairs (i,j) with binomial(i+r,r) + binomial(j+r,r) <= binomial(n+r,r), r=2.
  • A052119 (program): Decimal expansion of number with continued fraction expansion 0, 1, 2, 3, 4, 5, 6, …
  • A052124 (program): E.g.f.: exp(-2x)/(1-x)^3.
  • A052125 (program): a(n) = n/A034684(n).
  • A052126 (program): a(1) = 1; for n>1, a(n)=n/(largest prime dividing n).
  • A052127 (program): Sum a(n) x^n / n!^2 = exp(-2x)/(1-x)^3.
  • A052128 (program): a(1) = 1; for n > 1, a(n) is the largest divisor of n that is coprime to a larger divisor of n.
  • A052133 (program): CONTINUANT transform of 0, 1, 1, 2, 1, 3, 2, 3, … (A002487).
  • A052140 (program): 4^n*n!^2*Sum_{k=0..n} 1/k!.
  • A052141 (program): Number of paths from (0,0) to (n,n) that always move closer to (n,n) (and do not pass (n,n) and backtrack).
  • A052142 (program): E.g.f.: exp(x/(1-4*x)^(1/2)).
  • A052143 (program): E.g.f.: exp(x)/sqrt(1-4*x).
  • A052144 (program): A000172(n)^2.
  • A052145 (program): a(n) = (2n-1)*(2n-1)!/n.
  • A052146 (program): a(n) = floor((sqrt(1+8*n)-3)/2).
  • A052147 (program): a(n) = prime(n) + 2.
  • A052149 (program): Number of nonsquare rectangles on an n X n board.
  • A052150 (program): Partial sums of A000340, second partial sums of A003462.
  • A052153 (program): Rhombi (in 3 different orientations) in a rhombus with 60-degree acute angles.
  • A052156 (program): Number of compositions of n into 2*j-1 kinds of j’s for all j>=1.
  • A052158 (program): Lower prime of a difference of 6 (G-minor-6 primes) between consecutive primes of 6k+1 form.
  • A052159 (program): Lower prime of a difference of 6 (G-minor-6 primes) between consecutive primes of 6k+5 form.
  • A052161 (program): Partial sums of A014825, second partial sums of A002450.
  • A052169 (program): Equivalent of the Kurepa hypothesis for left factorial.
  • A052173 (program): Another version of the Catalan triangle A008315.
  • A052174 (program): Triangle of numbers arising in enumeration of walks on square lattice.
  • A052177 (program): Number of walks on simple cubic lattice (starting on the xy plane, never going below it and finishing a height 1 above it).
  • A052178 (program): Number of walks of length n on the simple cubic lattice terminating at height 2 above the (x,y)-plane.
  • A052179 (program): Triangle of numbers arising in enumeration of walks on cubic lattice.
  • A052181 (program): Partial sums of A050483.
  • A052182 (program): Determinant of n X n matrix whose rows are cyclic permutations of 1..n.
  • A052183 (program): A second-order recursive sequence.
  • A052186 (program): Number of permutations of [n] with no strong fixed points.
  • A052200 (program): Number of n-state, 2-symbol, d+ in {LEFT, RIGHT}, 5-tuple (q, s, q+, s+, d+) (halting or not) Turing machines.
  • A052201 (program): Equivalent of the Kurepa hypothesis for left factorial.
  • A052203 (program): a(n) = (4n+1)*binomial(4n,n)/(3n+1).
  • A052204 (program): a(n) = (5n+1)*C(4n,n)/(3n+1).
  • A052206 (program): Partial sums of A050405.
  • A052207 (program): Number of sequences {s(i): i=0..n} such that |s(i)-s(i-1)|=1, i=1..n and s(i)=0 at four values of i, one of which is i=0.
  • A052208 (program): a(n) = Pell(n)*Pell(2*n)/2.
  • A052209 (program): b(n)*b(2*n), b(n) = A001353(n+1).
  • A052216 (program): Sum of two powers of 10.
  • A052217 (program): Numbers whose sum of digits is 3.
  • A052218 (program): Numbers whose sum of digits is 4.
  • A052219 (program): Numbers whose sum of digits is 5.
  • A052220 (program): Numbers whose sum of digits is 6.
  • A052221 (program): Numbers whose sum of digits is 7.
  • A052222 (program): Numbers whose sum of digits is 8.
  • A052223 (program): Numbers whose sum of digits is 9.
  • A052224 (program): Numbers whose sum of digits is 10.
  • A052225 (program): (n+1)!*(n+3)-3.
  • A052226 (program): Partial sums of A050404.
  • A052227 (program): a(n) = (4*n+1)*binomial(3*n,n)/(2*n+1).
  • A052244 (program): Partial sums of A014827.
  • A052245 (program): Expansion of 10*x / ((1 - x) * (1 - 10*x)^2) in powers of x.
  • A052246 (program): Concatenation of integers from n down to 0.
  • A052248 (program): Greatest prime divisor of all composite numbers between p and next prime.
  • A052254 (program): Partial sums of A050406.
  • A052255 (program): Partial sums of A050484.
  • A052262 (program): Partial sums of A014824.
  • A052267 (program): Number of 2 X n matrices over GF(3) under row and column permutations.
  • A052268 (program): First differences of 10^n (A011557).
  • A052276 (program): Nonnegative numbers of the form n^3 (+/-) 3, n >= 0.
  • A052277 (program): a(n) = (4n+2)!/2^(2n+1).
  • A052278 (program): a(n) = (4n+3)!/4^n.
  • A052282 (program): Number of 3 X 3 stochastic matrices under row and column permutations.
  • A052284 (program): Number of compositions of n into nonprime numbers.
  • A052288 (program): First differences of the average of two consecutive primes (A024675).
  • A052293 (program): Numbers of the form 11 + x^2 + x or 11 + 2*x^2.
  • A052294 (program): Pernicious numbers: numbers with a prime number of 1’s in their binary expansion.
  • A052295 (program): a(n) = (n*(n+1)/2)!.
  • A052297 (program): Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.
  • A052316 (program): Number of labeled rooted trees with n nodes and 2-colored internal (non-leaf) nodes.
  • A052317 (program): Number of labeled trees with n nodes and 2-colored internal (non-leaf) nodes.
  • A052332 (program): Number of labeled digraphs where every node has indegree 0 or outdegree 0 and no isolated nodes.
  • A052333 (program): Riesel problem: start with n; repeatedly double and add 1 until reach a prime. Sequence gives prime reached, or 0 if no prime is ever reached.
  • A052338 (program): a(n) = A050443(n-th prime)/(n-th prime).
  • A052341 (program): Shifts left two places under BIN1 transform.
  • A052343 (program): Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).
  • A052344 (program): Number of ways to write n as the unordered sum of two nonzero triangular numbers.
  • A052368 (program): Primes p such that p+7! is also prime.
  • A052369 (program): Largest prime factor of n, where n runs through composite numbers.
  • A052379 (program): Number of integers from 1 to 10^(n+1)-1 that lack 0 and 1 as a digit.
  • A052380 (program): a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.
  • A052382 (program): Numbers without 0 as a digit, a.k.a. zeroless numbers.
  • A052383 (program): Numbers without 1 as a digit.
  • A052386 (program): Number of integers from 1 to 10^n-1 that lack 0 as a digit.
  • A052387 (program): Number of 3 X n binary matrices such that any 2 rows have a common 1, up to column permutations.
  • A052392 (program): T(2n+1,n), array T as in A054120.
  • A052404 (program): Numbers without 2 as a digit.
  • A052405 (program): Numbers without 3 as a digit.
  • A052406 (program): Numbers without 4 as a digit.
  • A052409 (program): a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).
  • A052410 (program): Write n = m^k with m, k integers, k >= 1, then a(n) is the smallest possible choice for m.
  • A052413 (program): Numbers without 5 as a digit.
  • A052414 (program): Numbers without 6 as a digit.
  • A052419 (program): Numbers without 7 as a digit.
  • A052421 (program): Numbers without 8 as a digit.
  • A052422 (program): Number of n-crossing hyperbolic knots having symmetry group D8.
  • A052423 (program): Highest common factor of nonzero digits of n.
  • A052424 (program): Numbers with no single-digit factors (apart from 1 and n).
  • A052449 (program): a(n) = 1 + Product_{k=1..n} Fibonacci(k).
  • A052453 (program): Number of nonisomorphic (3,n) cage graphs.
  • A052454 (program): Positive integer values of n such that 10n^2 - 9 is a square.
  • A052459 (program): a(n) = n*(2*n^2 + 1)*(n^2 + 1)/6.
  • A052460 (program): 3-magic series constant.
  • A052461 (program): 4-magic series constant.
  • A052462 (program): a(n) is the minimal positive integral solution k to 24*k == 1 (mod 5^n).
  • A052463 (program): a(n) is the smallest nonnegative solution k to 24*k == 1 (mod 7^(2*n-2)).
  • A052465 (program): a(n) is the smallest positive integral solution k to 24*k == 1 (mod 11^n).
  • A052466 (program): a(n) is the smallest positive solution k to 24*k == 1 (mod 13^n).
  • A052467 (program): Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.
  • A052468 (program): Numerators in the Taylor series for arccosh(x) - log(2*x).
  • A052469 (program): Denominators in the Taylor series for arccosh(x) - log(2*x).
  • A052472 (program): Number of independent components for a Weyl tensor in n dimensions.
  • A052473 (program): a(n) = binomial(2*n-5,n-2) + 2.
  • A052477 (program): Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(3)).
  • A052481 (program): a(n) = 2^n*(binomial(n,2) + 1).
  • A052482 (program): a(n) = 2^(n-2)*binomial(n+1,2).
  • A052485 (program): Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.
  • A052488 (program): a(n) = floor(n*H(n)) where H(n) is the n-th harmonic number, Sum_{k=1..n} 1/k (A001008/A002805).
  • A052492 (program): Initial pile sizes that guarantee a win for player 2 in a variant of Fibonacci NIM where the players may not take one stone.
  • A052499 (program): If n is in the sequence then so are 2n and 4n-1.
  • A052501 (program): Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5.
  • A052502 (program): Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.
  • A052503 (program): Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.
  • A052504 (program): Number of permutations sigma of [5n] without fixed points such that sigma^5 = Id.
  • A052505 (program): Number of labeled 3-constrained functional graphs.
  • A052506 (program): Expansion of e.g.f. exp(x*exp(x)-x).
  • A052509 (program): Knights-move Pascal triangle: T(n,k), n >= 0, 0 <= k <= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,…,n-1, n >= 2.
  • A052510 (program): Number of labeled planar binary trees with 2n-1 elements (external nodes or internal nodes).
  • A052511 (program): Prime(n) - 1 - A006218(n).
  • A052512 (program): Number of rooted labeled trees of height at most 2.
  • A052515 (program): Number of ordered pairs of complementary subsets of an n-set with both subsets of cardinality at least 2.
  • A052516 (program): Number of pairs of sets of cardinality at least 3.
  • A052517 (program): Number of ordered pairs of cycles over all n-permutations having two cycles.
  • A052518 (program): Number of pairs of cycles of cardinality at least 2.
  • A052519 (program): Number of pairs of cycles of cardinality at least 3.
  • A052520 (program): Number of pairs of sequences of cardinality at least 2.
  • A052521 (program): Number of pairs of sequences of cardinality at least 3.
  • A052524 (program): Number of ordered labeled rooted trees on n nodes with non-leaf nodes having more than two children.
  • A052527 (program): Expansion of (1-x)/(1-x-x^2-x^3+x^4).
  • A052528 (program): Expansion of (1 - x)/(1 - 2*x - 2*x^2 + 2*x^3).
  • A052529 (program): Expansion of (1-x)^3/(1 - 4*x + 3*x^2 - x^3).
  • A052530 (program): a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 2.
  • A052531 (program): If n is even then 2^n+1 otherwise 2^n.
  • A052532 (program): Expansion of (1-x)/(1-x-x^3-x^4+x^5).
  • A052533 (program): Expansion of (1-x)/(1-x-3*x^2).
  • A052534 (program): Expansion of (1-x)*(1+x)/(1-2*x-x^2+x^3).
  • A052535 (program): Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).
  • A052536 (program): Number of compositions of n when parts 1 and 2 are of two kinds.
  • A052537 (program): Expansion of (1-x)/(1-x-2*x^3).
  • A052538 (program): Expansion of (1-x)/(1-2*x-3*x^2+3*x^3).
  • A052539 (program): a(n) = 4^n + 1.
  • A052540 (program): Expansion of (1-x)/(1-2*x-x^3+x^4).
  • A052541 (program): Expansion of 1/(1-3*x-x^3).
  • A052542 (program): a(n) = 2*a(n-1) + a(n-2), with a(0) = 1, a(1) = 2, a(2) = 4.
  • A052543 (program): Expansion of (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).
  • A052544 (program): Expansion of (1-x)^2/(1 - 4*x + 3*x^2 - x^3).
  • A052545 (program): Expansion of (1-x)^2/(1-3*x+x^3).
  • A052546 (program): Expansion of (1-x)/(1-x-x^2-2*x^3+2*x^4).
  • A052547 (program): Expansion of (1-x)/(1-x-2*x^2+x^3).
  • A052548 (program): a(n) = 2^n + 2.
  • A052549 (program): a(n) = 5*2^(n-1) - 1, n>0, with a(0)=1.
  • A052550 (program): Expansion of (1-2*x)/(1 - 3*x - x^2 + 2*x^3).
  • A052551 (program): Expansion of 1/((1 - x)*(1 - 2*x^2)).
  • A052552 (program): a(2*n+1) = 1, a(2*n) = 2*a(2*n-2) - 1.
  • A052553 (program): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by antidiagonals.
  • A052554 (program): Expansion of e.g.f.: (1-x)/(1 - x - x^2).
  • A052555 (program): Expansion of e.g.f. 1/(1-2*x-x^2).
  • A052556 (program): Expansion of e.g.f. 1/(1-x-x^3).
  • A052557 (program): Expansion of e.g.f. (1-x)/(1-x-x^3).
  • A052558 (program): a(n) = n! *((-1)^n + 2*n + 3)/4.
  • A052559 (program): Expansion of e.g.f. (1-x)/(1 - 2*x - x^2 + x^3).
  • A052560 (program): a(n) = 3*n!.
  • A052561 (program): a(n) = (1 + 2^n) * n!.
  • A052562 (program): a(n) = 5^n * n!.
  • A052563 (program): E.g.f.: (1-x)/(1-3*x).
  • A052564 (program): Expansion of e.g.f. x*(1-x)/(1-2*x).
  • A052565 (program): E.g.f. (1+x^3-x^4)/(1-x).
  • A052566 (program): Expansion of e.g.f. (2 + x)/(1 - x^2).
  • A052567 (program): E.g.f.: (1-x)^2/(1-3*x+x^2).
  • A052568 (program): E.g.f.: (1-x)/(1-3*x+x^2).
  • A052569 (program): E.g.f. 1/((1-x)(1-x^3)).
  • A052570 (program): E.g.f.: x/(1-4*x).
  • A052571 (program): E.g.f. x^3/(1-x)^2.
  • A052572 (program): E.g.f. (1+2x-2x^2)/(1-x)^2.
  • A052573 (program): (1+3^n)*n!.
  • A052574 (program): Expansion of e.g.f. (1-2x)/(1-3x+x^2).
  • A052575 (program): Expansion of e.g.f. (1-x)/(1-2*x-2*x^2+2*x^3).
  • A052576 (program): E.g.f. (1+x^2-2x^3)/(1-2x).
  • A052577 (program): a(n) = (3^(n+1)-1)*n!/2.
  • A052578 (program): a(0) = 0, a(n) = 4*n! for n > 0.
  • A052579 (program): E.g.f. (2+x+x^2)/((1-x)(1+x+x^2)).
  • A052580 (program): E.g.f. (1-2x)/(1-2x-x^2).
  • A052581 (program): E.g.f. (1-x)/(1-x-x^4).
  • A052582 (program): a(n) = 2*n*n!.
  • A052583 (program): E.g.f. x(1-x)/(1-x-x^2).
  • A052584 (program): E.g.f. (2-4x+x^2)/((1-x)(1-2x)).
  • A052585 (program): E.g.f. 1/(1-x-2*x^2).
  • A052586 (program): Expansion of e.g.f.: (1-x^3)/(1-x-x^3).
  • A052587 (program): E.g.f. x^2(1-x)/(1-2x).
  • A052588 (program): E.g.f. (1-x)/(1-x-x^2-x^3+x^4).
  • A052589 (program): a(n) = (2^n - 1)*n!.
  • A052590 (program): E.g.f. (1-x)/(1-4x+2x^2).
  • A052591 (program): E.g.f. x/((1-x)(1-x^2)).
  • A052592 (program): E.g.f. (1-x)/(1-4x).
  • A052593 (program): E.g.f. 1/(1-x-x^4).
  • A052594 (program): E.g.f. x(1+x-2x^2)/(1-2x).
  • A052595 (program): E.g.f. 1/(1-3x-x^2).
  • A052596 (program): E.g.f. (1+x^4-x^5)/(1-x).
  • A052597 (program): E.g.f. 1/(1-x^2-x^3).
  • A052598 (program): E.g.f. (1-x)/(1-x-2x^2).
  • A052599 (program): Expansion of e.g.f.: 1/(1-2x-x^3).
  • A052600 (program): E.g.f. 1/((1-2*x)*(1-x^2)).
  • A052601 (program): E.g.f. (1-x)/(1-x-2x^3).
  • A052602 (program): E.g.f. x^2*(1-x)/(1-x-x^2).
  • A052603 (program): E.g.f. (1-x)^3/(1-4x+3x^2-x^3).
  • A052604 (program): E.g.f. (1-x)/(1-2x-x^3+x^4).
  • A052605 (program): Expansion of E.g.f. x*(1-x)/(1-x-x^3).
  • A052606 (program): E.g.f. (1-x)^2/(1-4x+x^2).
  • A052607 (program): E.g.f. (1-x^3)/(1-x^2-x^3).
  • A052608 (program): E.g.f. (1-x)/(1-2x-x^2).
  • A052609 (program): a(n) = (2*n - 2)*n!.
  • A052610 (program): E.g.f. 1/(1-x-2x^3).
  • A052611 (program): E.g.f. 1/(1-2x-2x^2).
  • A052612 (program): Expansion of e.g.f. x*(2+x)/(1-x^2).
  • A052613 (program): E.g.f. (1-2x)/(1-2x-x^2+x^3).
  • A052614 (program): E.g.f. 1/((1-x)(1-x^4)).
  • A052615 (program): E.g.f. x^3*(1+2x-2x^2)/(1-x).
  • A052616 (program): E.g.f. (3+2x)/(1-x^2).
  • A052617 (program): E.g.f. (1+x-x^2)/((1-x)(1-2x)).
  • A052618 (program): Expansion of e.g.f. 1/((1-x)^2*(1-x^2)).
  • A052619 (program): E.g.f. 3x^3/(1-x).
  • A052620 (program): E.g.f. (1-x)^2/(1-3x+x^3).
  • A052621 (program): E.g.f. (2+x+x^2+x^3)/(1-x^4).
  • A052622 (program): E.g.f. (1-x^2)/(1-2x-x^2).
  • A052623 (program): E.g.f. x(1-x)^2/(1-3x+x^2).
  • A052624 (program): E.g.f. (1+x^2-2x^3+x^4)/(1-x)^2.
  • A052625 (program): E.g.f. (1-x)^2/(1-2x+x^2-x^3).
  • A052626 (program): (2^n+2)*n!.
  • A052627 (program): E.g.f. (1-x)/(1-x-x^5).
  • A052628 (program): E.g.f. (2+x^3-x^4)/(1-x).
  • A052629 (program): E.g.f. (1-x)/(1-5x+3x^2).
  • A052630 (program): E.g.f. 1/(1-4x-x^2).
  • A052631 (program): a(n) = n!*Pell(n) (or n!*A000129(n)).
  • A052632 (program): E.g.f. 1/(1-x-x^5).
  • A052633 (program): E.g.f. x^2*(1+x-x^2)/(1-x)^2.
  • A052634 (program): Expansion of e.g.f. 1/((1-2*x^2)*(1-x)).
  • A052635 (program): E.g.f. (1-3x)/(1-3x-x^2).
  • A052636 (program): Expansion of e.g.f. (2-x-2x^2)/((1-x)(1-2x^2)).
  • A052637 (program): E.g.f. 3x(1+x-x^2)/(1-x).
  • A052638 (program): E.g.f. x^2*(1+x-2x^2)/(1-2x).
  • A052639 (program): E.g.f. (1-2x)/(1-2x-x^3).
  • A052640 (program): E.g.f. x*(1-x)/(1-2*x-x^2+x^3).
  • A052641 (program): Expansion of e.g.f. (1-x)/(1-3*x-x^2+x^3).
  • A052642 (program): E.g.f. x^2*(2+x-x^2)/(1-x).
  • A052643 (program): E.g.f. (1+x-x^2)^2/(1-x)^2.
  • A052644 (program): E.g.f. (1+3x-3x^2)/(1-x)^2.
  • A052645 (program): E.g.f. 2*x^2*(1+x-x^2)/(1-x).
  • A052646 (program): E.g.f. 1/((1-x)(1-x-x^2)).
  • A052647 (program): E.g.f. (2-2x-x^2)/((1-2x)(1-x^2)).
  • A052648 (program): Expansion of e.g.f. 5*x/(1-x).
  • A052649 (program): E.g.f. (2+x-x^2)/(1-x)^2.
  • A052650 (program): E.g.f. 1/((1-2x)(1-x)^2).
  • A052651 (program): E.g.f. (1-x)/(1-x-x^3-x^4+x^5).
  • A052652 (program): E.g.f. x^4/(1-2x).
  • A052653 (program): E.g.f. (1-2x^2)/(1-x-2x^2).
  • A052654 (program): E.g.f. 1/((1-x)(1-4x)).
  • A052655 (program): a(2) = 6, otherwise a(n) = n*n!.
  • A052656 (program): E.g.f. x*(1+2x-4x^2)/(1-2x).
  • A052657 (program): E.g.f. x^2/((1-x)^2*(1+x)).
  • A052658 (program): E.g.f. (1-x^2)*(1-x)/(1-2x-x^2+x^3).
  • A052659 (program): Expansion of e.g.f. (1-2x)(1-x)/(1-4x+2x^2).
  • A052660 (program): E.g.f. (2-2x-x^2)/((1-x)(1-x-x^2)).
  • A052661 (program): E.g.f. (2-3x)/((1-x)(1-x-x^2)).
  • A052662 (program): E.g.f. (1-x^2)/(1-2x-x^2+x^3).
  • A052663 (program): E.g.f. x^4*(1+x-x^2)/(1-x).
  • A052664 (program): E.g.f. (1-x)/(1-2x-3x^2+3x^3).
  • A052665 (program): a(0)=0, for n >= 1, a(n) = ((2^(n-1)-1)*n!.
  • A052666 (program): E.g.f. 1/(1-x-3x^2).
  • A052667 (program): E.g.f. 1/(1-2x-x^4).
  • A052668 (program): Expansion of e.g.f. 1/(1 - 3*x - x^3).
  • A052669 (program): Expansion of e.g.f. (1-2*x)/(1-3*x-x^2+2*x^3).
  • A052670 (program): Expansion of e.g.f. x^2/(1-4*x).
  • A052671 (program): Expansion of e.g.f. x^3*(1-x)/(1-2*x).
  • A052672 (program): Expansion of e.g.f. (1-x)/(1-x-2*x^2+x^3).
  • A052673 (program): a(n) = 3*n*n!.
  • A052674 (program): Expansion of e.g.f. (1-x)/(1-3*x-2*x^2+2*x^3).
  • A052675 (program): Expansion of e.g.f. (1-x)/(1-5*x).
  • A052676 (program): Expansion of e.g.f. 3*x/(1 - 2*x).
  • A052677 (program): Expansion of e.g.f. (1-x)/(1-4*x+x^2).
  • A052678 (program): Expansion of e.g.f. x^3/(1-3*x).
  • A052679 (program): Expansion of e.g.f. (1-x^2)/(1-x^2-x^3).
  • A052680 (program): Expansion of e.g.f. (1-2*x)/(1-4*x+2*x^2).
  • A052681 (program): Expansion of e.g.f. (1-x)/(1 - x - x^2 - 2*x^3 + 2*x^4).
  • A052682 (program): Expansion of e.g.f. (1-x)/(1-x-3*x^2).
  • A052683 (program): Expansion of e.g.f. 2*x^4/(1-x).
  • A052684 (program): Expansion of e.g.f. 1/(1-2*x^2-x^3).
  • A052685 (program): Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).
  • A052686 (program): Expansion of e.g.f. x^2*(1+3*x-3*x^2)/(1-x).
  • A052687 (program): Expansion of e.g.f. (1+x-x^3)/((1-x)*(1-x^2)).
  • A052688 (program): Expansion of e.g.f. x/((1-x)*(1-x^3)).
  • A052689 (program): Expansion of e.g.f. (1+x-x^2)/((1-x)*(1-x^2)).
  • A052690 (program): Expansion of e.g.f. x*(1+x-3*x^2)/(1-3*x).
  • A052691 (program): Expansion of e.g.f. (1-x)/(1-2*x+x^2-x^3).
  • A052692 (program): Expansion of e.g.f. (1-x^4)/(1-x-x^4).
  • A052693 (program): Expansion of e.g.f. (1-x)/(1-3*x+x^3).
  • A052694 (program): Expansion of e.g.f. (1 + x^3 - 2*x^4)/(1-2*x).
  • A052695 (program): Expansion of e.g.f. (2-5*x)/((1-x)*(1-4*x)).
  • A052696 (program): Expansion of e.g.f. (1-x)^2/(1-4*x+3*x^2-x^3).
  • A052697 (program): Expansion of e.g.f. 1/(1-x^3-x^4).
  • A052698 (program): Expansion of e.g.f. x/((1-x)*(1-3*x)).
  • A052699 (program): Expansion of e.g.f. (1+x^5-x^6)/(1-x).
  • A052700 (program): Expansion of e.g.f. x*(1-x)/(1-3*x).
  • A052701 (program): a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.
  • A052702 (program): A simple context-free grammar.
  • A052703 (program): A simple context-free grammar.
  • A052704 (program): Apart from the leading term, a(n) = Catalan(n-1)*4^(n-1).
  • A052705 (program): Expansion of 2*x^2/(1 - 2*x - 2*x^2 + sqrt(1 - 4*x - 4*x^2)).
  • A052706 (program): A simple context-free grammar.
  • A052707 (program): Odd powers of 2 multiplied by Catalan numbers.
  • A052709 (program): Expansion of (1-sqrt(1-4*x-4*x^2))/(2*(1+x)).
  • A052711 (program): Expansion of e.g.f. x*(1 - 2*x - sqrt(1-4*x))/2.
  • A052712 (program): Expansion of e.g.f. (1 + 4*x - sqrt(1-8*x))/8.
  • A052713 (program): Expansion of e.g.f. (1-sqrt(1-8*x))/2.
  • A052714 (program): a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.
  • A052715 (program): Expansion of e.g.f. (1-2*x-sqrt(1-4*x))/2 - x*(1-2*x-sqrt(1-4*x)) - x^2.
  • A052716 (program): Expansion of e.g.f. (x + 1 - sqrt(1-6*x+x^2))/2.
  • A052717 (program): Expansion of e.g.f. x*(1 - sqrt(1 - 4*x))/2.
  • A052718 (program): E.g.f. 1 - x - sqrt(1-4*x).
  • A052719 (program): Expansion of e.g.f. (1 - 2*x*sqrt(1-4*x))*(1 - sqrt(1-4*x))/4.
  • A052720 (program): Expansion of e.g.f.: (1 - 4*x + 3*x^2)*(1 - 2*x - sqrt(1-4*x))/2 - x^2 + 2*x^3.
  • A052721 (program): Expansion of e.g.f. x*(1-2*x)*(1 - 2*x - sqrt(1-4*x))/2 - x^3.
  • A052722 (program): Expansion of e.g.f. (1 - 2*x - sqrt(1-4*x))^2 * (1 - sqrt(1-4*x))/8.
  • A052723 (program): Expansion of e.g.f. (1 - x - sqrt(1-2*x+x^2-4*x^3))/(2*x).
  • A052724 (program): A simple context-free grammar in a labeled universe.
  • A052726 (program): E.g.f. (1-sqrt(1-4*x-4*x^2))/ (2*(1+x)).
  • A052727 (program): A simple context-free grammar in a labeled universe.
  • A052728 (program): A simple context-free grammar in a labeled universe.
  • A052729 (program): A simple context-free grammar in a labeled universe.
  • A052730 (program): A simple context-free grammar in a labeled universe.
  • A052731 (program): E.g.f. [1-x -sqrt(1-2x-3x^2)]/(2x) - [1+x-sqrt(1-2x-3x^2)]/2 .
  • A052732 (program): E.g.f.: (1-2x-sqrt(1-4*x))*x^2/2
  • A052733 (program): E.g.f.: x^2*(1-sqrt(1-4*x))/2.
  • A052734 (program): a(n) = 4^(n-1) * n! * Catalan(n-1) for n > 0, with a(0) = 0.
  • A052735 (program): E.g.f. (1-x -sqrt(1-2*x-3*x^2) )/2.
  • A052736 (program): E.g.f. [1 -3x -sqrt(1-6x+x^2) -x*(1-x-sqrt(1-6x+x^2)) ]/2.
  • A052737 (program): a(n) = ((2*n)!/n!)*2^(2*n+1).
  • A052739 (program): E.g.f. (1-sqrt(1-4x-4x^2))/2 -x*(1+x).
  • A052740 (program): A simple context-free grammar in a labeled universe.
  • A052741 (program): A simple context-free grammar in a labeled universe.
  • A052742 (program): A simple context-free grammar in a labeled universe.
  • A052743 (program): E.g.f. ( 1-x-sqrt(1-2*x+x^2-4*x^3) )/(2*x^2).
  • A052744 (program): E.g.f. x*(1-2*x-2*x^2-sqrt(1-4*x-4*x^2))/ (2*(1+x)^2).
  • A052745 (program): A simple grammar.
  • A052746 (program): a(0) = 0; a(n) = (2*n)^(n-1), n > 0.
  • A052747 (program): a(0) = a(1) = a(2) = 0; a(n) = n!/(n-2) for n > 2.
  • A052748 (program): Expansion of e.g.f.: -(log(1-x))^3.
  • A052749 (program): 2n*S2(n-1,2).
  • A052750 (program): a(n) = (2*n + 1)^(n - 1).
  • A052752 (program): a(n) = (3*n+1)^(n-1).
  • A052753 (program): Expansion of e.g.f.: log(1-x)^4.
  • A052754 (program): Expansion of e.g.f.: (log(1-x))^2*x^2.
  • A052756 (program): E.g.f.: (-1/3)*LambertW(-3*x).
  • A052758 (program): Expansion of e.g.f.: -(log(1-x))^3*x.
  • A052759 (program): E.g.f.: x^3*log(1/(1-x)).
  • A052760 (program): Expansion of e.g.f.: x^2*(exp(x)-1)^2.
  • A052761 (program): a(n) = 3!*n*S(n-1,3), where S denotes the Stirling numbers of second kind.
  • A052762 (program): Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).
  • A052764 (program): E.g.f.: -1/4*LambertW(-4*x).
  • A052765 (program): Expansion of e.g.f.: -x^2*(log(1-x))^3.
  • A052766 (program): Expansion of e.g.f.: (log(1-x))^2*x^3.
  • A052767 (program): Expansion of e.g.f.: -(log(1-x))^5.
  • A052768 (program): A simple grammar.
  • A052769 (program): E.g.f.: x^3*(exp(x)-1)^2.
  • A052770 (program): A simple grammar.
  • A052771 (program): E.g.f.: x^3*exp(x)^2.
  • A052774 (program): a(n) = (4*n+1)^(n-1).
  • A052776 (program): a(n) = 4!*n*Stirling2(n-1,4).
  • A052777 (program): E.g.f.: x^2*(exp(x)-1)^3.
  • A052778 (program): E.g.f.: x^4*log(-1/(-1+x)).
  • A052779 (program): Expansion of e.g.f.: (log(1-x))^6.
  • A052780 (program): Expansion of e.g.f. x^2*exp(4*x).
  • A052782 (program): a(n) = (5*n+1)^(n-1).
  • A052783 (program): A simple grammar.
  • A052784 (program): E.g.f.: x^3*(exp(x)-1)^3.
  • A052785 (program): a(n) = 5!*n*Stirling2(n-1, 5).
  • A052786 (program): Expansion of e.g.f.: -x^3*(log(1-x))^3.
  • A052787 (program): Product of 5 consecutive integers.
  • A052789 (program): Expansion of e.g.f. -(1/5)*LambertW(-5*x).
  • A052790 (program): Expansion of e.g.f.: x^2*log(1-x)^4.
  • A052791 (program): 3^(n-3)*n*(n-1)*(n-2).
  • A052792 (program): Expansion of e.g.f.: x^2*(exp(x)-1)^4.
  • A052793 (program): A simple grammar.
  • A052794 (program): E.g.f.: -x^5*log(1-x).
  • A052795 (program): a(n) = (6*n)!/(5*n+1)!.
  • A052796 (program): E.g.f.: x^4*exp(x)^2.
  • A052799 (program): Expansion of e.g.f.: x^4*(log(1-x))^2.
  • A052800 (program): E.g.f.: x^5*exp(x)-x^5.
  • A052801 (program): A simple grammar: labeled pairs of sequences of cycles.
  • A052804 (program): A simple grammar: cycles of rooted cycles.
  • A052808 (program): E.g.f.: -log(1-x+log(1-x)).
  • A052809 (program): A simple grammar: number of cycles of cycles.
  • A052810 (program): 1 + number of partitions of n, n>0.
  • A052816 (program): G.f.: (1+x)*Product_{m>0} (1 + x^m).
  • A052820 (program): E.g.f.: 1/(1-x+log(1-x)).
  • A052823 (program): A simple grammar: cycles of pairs of sequences.
  • A052825 (program): A simple grammar.
  • A052828 (program): A simple grammar: union of cycles and cycles of cycles.
  • A052830 (program): A simple grammar: sequences of rooted cycles.
  • A052832 (program): A simple grammar.
  • A052834 (program): a(n) = Bell(n+1)-Bell(n)-1, n>0.
  • A052837 (program): Number of partitions of 2n whose Ferrers-Young diagram allows more than one different domino tiling.
  • A052838 (program): Expansion of e.g.f.: (exp(x/(1-x)) - 1)^2.
  • A052839 (program): Number of partitions of n into distinct summands (A000009), plus 1 (apart from the first term).
  • A052840 (program): A simple grammar.
  • A052841 (program): E.g.f.: 1/(exp(x)*(2-exp(x))).
  • A052844 (program): E.g.f.: exp(x*(2-x)/(1-x)).
  • A052845 (program): Expansion of e.g.f.: exp(x^2/(1-x)).
  • A052847 (program): G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).
  • A052848 (program): Number of ordered set partitions with a designated element in each block and no block containing less than two elements.
  • A052849 (program): a(0) = 0; a(n+1) = 2*n! (n >= 0).
  • A052850 (program): E.g.f.: x/(1-x)+log((1-x)/(1-2*x)).
  • A052852 (program): Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).
  • A052854 (program): Number of forests of ordered trees on n total nodes.
  • A052856 (program): E.g.f.: (1-3*exp(x)+exp(2*x))/(exp(x)-2).
  • A052857 (program): A simple grammar.
  • A052858 (program): E.g.f.: log(-1/(-1+x*exp(x)-x)).
  • A052859 (program): Expansion of e.g.f.: exp(exp(2*x) - 2*exp(x) + 1).
  • A052860 (program): A simple grammar: rooted sequences of cycles.
  • A052861 (program): E.g.f.: log((1-x)/(1-2*x))*x/(1-x).
  • A052862 (program): Expansion of e.g.f. log(-1/(-2+exp(x)))*x.
  • A052863 (program): Expansion of e.g.f. log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).
  • A052865 (program): E.g.f.: log(-1/(-1+x))^2 / (-1 + log(-1/(-1+x)))^2.
  • A052866 (program): Expansion of e.g.f. x/(1 - x) + exp(x/(1 - x)).
  • A052867 (program): E.g.f.: log(-(-1+x)^2/(-1+2*x)).
  • A052868 (program): E.g.f.: LambertW(x/(-1+x))/x*(-1+x).
  • A052871 (program): E.g.f.: -LambertW(x/(-1+x)).
  • A052873 (program): E.g.f. satisfies: A(x) = exp(x*A(x)/(1 - x*A(x))).
  • A052874 (program): E.g.f.: -x/(-1+x)*(exp(-x/(-1+x))-1).
  • A052875 (program): E.g.f.: (exp(x)-1)^2/(2-exp(x)).
  • A052876 (program): Expansion of e.g.f. (exp(x)-1)^2/(-2+exp(x))^2.
  • A052877 (program): E.g.f.: exp(x)-1+log(-1/(-2+exp(x))).
  • A052878 (program): E.g.f.: log((1-x)/(1-3*x+x^2)).
  • A052881 (program): E.g.f.: log(1/(1-x))*x/(1-x).
  • A052882 (program): A simple grammar: rooted ordered set partitions.
  • A052883 (program): Expansion of e.g.f.: log((-1+x)/(-1+2*x))^2.
  • A052885 (program): E.g.f. A(x) is inverse to F(x) = x*exp(-x)/(1+x).
  • A052886 (program): Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).
  • A052887 (program): Expansion of e.g.f.: exp(x^2/(1 - x)^2).
  • A052888 (program): E.g.f. is series reversion of log(1+x)*exp(-x).
  • A052889 (program): Number of rooted set partitions.
  • A052892 (program): E.g.f. is series reversion of log(1+x)*(1-x).
  • A052896 (program): E.g.f.: (exp(exp(x)-1)-1)^2.
  • A052897 (program): Expansion of e.g.f.: exp(2*x/(1-x)).
  • A052898 (program): 2*n! + 1.
  • A052899 (program): Expansion of g.f.: ( 1-2*x ) / ((x-1)*(4*x^2+2*x-1)).
  • A052900 (program): Expansion of (1-x)/(1-x-3x^3).
  • A052901 (program): Periodic with period 3: a(3n)=3, a(3n+1)=a(3n+2)=2.
  • A052903 (program): Expansion of (1-x^3)/(1-2x-x^3+x^4).
  • A052904 (program): Expansion of (1-x)/(1-2x-4x^2+4x^3).
  • A052905 (program): a(n) = (n^2 + 7*n + 2)/2.
  • A052906 (program): Expansion of (1-x^2)/(1-3*x-x^2).
  • A052907 (program): Expansion of 1/(1 - 2*x^2 - 2*x^3).
  • A052908 (program): Expansion of 1 + x/(1 - 2*x - x^3 + x^4).
  • A052909 (program): Expansion of (1+x-x^2)/((1-x)*(1-3*x)).
  • A052910 (program): Expansion of 1 + 2/(1-2*x-x^3).
  • A052911 (program): Expansion of (1-x)/(1 - 3*x - x^2 + 2*x^3).
  • A052912 (program): Expansion of 1/(1-2*x-2*x^3).
  • A052913 (program): a(n+2) = 5*a(n+1) - 2*a(n), with a(0) = 1, a(1) = 4.
  • A052914 (program): Expansion of (1-x)/(1 - x - x^3 - 2*x^4 + 2*x^5).
  • A052915 (program): Expansion of (1-x)/(1 - x - x^2 - 3*x^3 + 3*x^4).
  • A052916 (program): Expansion of (1-x)/(1 - x - 2*x^3 + x^4).
  • A052917 (program): Expansion of 1/(1-3*x-x^4).
  • A052918 (program): a(0) = 1, a(1) = 5, a(n+1) = 5*a(n) + a(n-1).
  • A052919 (program): a(n) = 1 + 2*3^(n-1) with a(0)=2.
  • A052920 (program): a(n) = a(n-3) + a(n-5) with initial values 1,0,0,1,0.
  • A052921 (program): Expansion of (1 - x)/(1 - 3*x + 2*x^2 - x^3).
  • A052922 (program): Expansion of 1/(1 - 2*x^3 - x^4).
  • A052923 (program): Expansion of (1-x)/(1 - x - 4*x^2).
  • A052924 (program): Expansion of g.f.: (1-x)/(1 - 3*x - x^2).
  • A052925 (program): Expansion of (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).
  • A052926 (program): Expansion of (1-3*x)/(1 - 4*x - x^2 + 3*x^3).
  • A052927 (program): Expansion of 1/(1-4*x-x^3).
  • A052928 (program): The even numbers repeated.
  • A052929 (program): Expansion of (2-3*x-x^2)/((1-x^2)*(1-3*x)).
  • A052930 (program): Expansion of (1-x)/(1 - x - 2*x^2 - 2*x^3 + 2*x^4).
  • A052931 (program): Expansion of 1/(1 - 3*x^2 - x^3).
  • A052932 (program): Expansion of (1-x)/(1 - 2*x - x^4 + x^5).
  • A052933 (program): Expansion of (1-x^2)/(1 - x - 3*x^2 + 2*x^4).
  • A052934 (program): Expansion of (1-x)/(1-6*x).
  • A052935 (program): Expansion of (2-2*x-x^3)/((1-2*x)*(1-x^3)).
  • A052936 (program): Expansion of (1-x)*(1-2*x)/(1-5*x+5*x^2).
  • A052937 (program): Expansion of (2-3*x-x^2)/((1-x)*(1-2*x-x^2)).
  • A052938 (program): Expansion of (1 + 2*x - 2*x^2)/( (1+x)*(1-x)^2 ).
  • A052939 (program): Expansion of (1-x)*(1+x)/(1-3*x-x^2+2*x^3).
  • A052940 (program): a(0) = 1; a(n) = 3*2^n - 1, for n > 0.
  • A052941 (program): Expansion of (1-x)/(1 - 4*x + x^2 + x^3).
  • A052942 (program): Expansion of 1/((1+x)*(1-2*x+2*x^2-2*x^3)).
  • A052943 (program): Expansion of (1-x^2)/(1-2*x^2-x^3+x^5).
  • A052944 (program): a(n) = 2^n + n - 1.
  • A052945 (program): Number of compositions of n when each odd part can be of two kinds.
  • A052946 (program): Expansion of (1-x)^2/(1-3*x+2*x^3-x^4).
  • A052947 (program): Expansion of 1/(1-x^2-2*x^3).
  • A052948 (program): Expansion of g.f.: (1-2*x)/(1-3*x+2*x^3).
  • A052949 (program): Expansion of (2-4*x+x^3)/((1-x)*(1-2*x-x^2+x^3)).
  • A052950 (program): Expansion of (2-3*x-x^2+x^3)/((1-x)*(1+x)*(1-2*x)).
  • A052951 (program): Expansion of (1 + x - 2*x^2)/(1 - 2*x)^2.
  • A052952 (program): a(n) = Fibonacci(n+2) - (1-(-1)^n)/2.
  • A052953 (program): Expansion of 2*(1-x-x^2)/((1-x)*(1+x)*(1-2*x)).
  • A052954 (program): Expansion of (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)).
  • A052955 (program): a(2n) = 2*2^n - 1, a(2n+1) = 3*2^n - 1.
  • A052956 (program): a(n) = 2^n + Fibonacci(n+1).
  • A052957 (program): Expansion of 2*(1-x-x^2)/((1-2*x)*(1-2*x^2)).
  • A052958 (program): Expansion of g.f.: (1-x)/(1-3*x-2*x^3+2*x^4).
  • A052959 (program): a(2n) = a(2n-1)+a(2n-2), a(2n+1) = a(2n)+a(2n-1)-1, a(0)=2, a(1)=1.
  • A052960 (program): Expansion of ( 1-x-x^2 ) / ( 1-2*x-2*x^2+x^3+x^4 ).
  • A052961 (program): Expansion of (1 - 3*x) / (1 - 5*x + 3*x^2).
  • A052962 (program): Expansion of (1-2x^2)/(1-x-3x^2+2x^4).
  • A052963 (program): a(0)=1, a(1)=2, a(2)=5, a(n) = 3*a(n+2) - a(n+3).
  • A052964 (program): Expansion of (1-x)/((1-2x)(1+x-x^2)).
  • A052965 (program): Expansion of (1-x)/(1-3x-4x^2+4x^3).
  • A052966 (program): Expansion of (1-x)/(1-x-4x^2+2x^3).
  • A052967 (program): Expansion of (1 - x)/(1 - 2*x - x^2 + x^4).
  • A052968 (program): a(n) = 1 + 2^(n-1) + n for n > 0, a(0) = 2.
  • A052969 (program): Expansion of (1-x)/(1-x-2x^2+x^4).
  • A052970 (program): Expansion of (1-2x)/(1-2x-2x^2+2x^3).
  • A052971 (program): Expansion of (1-x)/(1-2x-2x^3+2x^4).
  • A052972 (program): Expansion of (1-x^3)/(1-x-x^2-x^3+x^5).
  • A052973 (program): Expansion of ( 1-x ) / ( 1-x-3*x^2+x^3 ).
  • A052974 (program): Expansion of (1 - 2x)/(1 - 2x - x^2 - x^3 + 2x^4).
  • A052975 (program): Expansion of (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).
  • A052976 (program): Expansion of (1-2x)/(1-3x-x^3+2x^4).
  • A052977 (program): Expansion of (1-x)(1+x)/(1 - x - x^2 - x^3 + x^5).
  • A052978 (program): Expansion of (1-2*x)/(1-4*x-2*x^2+4*x^3).
  • A052979 (program): Expansion of (1-x)(1+x)/(1-2*x-3*x^2+2*x^4).
  • A052980 (program): Expansion of (1 - x)/(1 - 2*x - x^3).
  • A052981 (program): Expansion of ( 1-x ) / ( 1-4*x-3*x^2+3*x^3 ).
  • A052982 (program): Expansion of ( 1-x ) / ( 1-2*x-2*x^2+x^4 ).
  • A052984 (program): a(n) = 5*a(n-1) - 2*a(n-2) for n>1, with a(0) = 1, a(1) = 3.
  • A052985 (program): Expansion of ( 1-x ) / ( 1-3*x+x^2-x^3+x^4 ).
  • A052986 (program): Expansion of ( 1-2*x ) / ( (x-1)*(2*x^2+3*x-1) ).
  • A052987 (program): Expansion of (1-2x^2)/(1-2x-2x^2+2x^3).
  • A052988 (program): Expansion of (1-x^2)/(1-2x-2x^2+x^3+x^4).
  • A052989 (program): Expansion of ( 1-x ) / ( 1-x-x^2-x^4+x^5 ).
  • A052990 (program): Expansion of ( 1-x ) / ( 1-4*x-x^2+2*x^3 ).
  • A052991 (program): Expansion of (1-x-x^2)/(1-3x-x^2).
  • A052992 (program): Expansion of 1/((1 - x)*(1 - 2*x)*(1 + 2*x)).
  • A052993 (program): a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3), with a(0)=a(1)=1, a(2)=4.
  • A052994 (program): Expansion of 2x(1-x)/(1-2x-x^2+x^3).
  • A052995 (program): Expansion of 2*x*(1 - x)/(1 - 3*x + x^2).
  • A052996 (program): G.f.: (1+x^2-x^3)/((1-x)(1-2*x)).
  • A052997 (program): Expansion of (1+x-x^3)/((1-2*x)*(1-x^2)).
  • A053000 (program): a(n) = (smallest prime > n^2) - n^2.
  • A053001 (program): Largest prime < n^2.
  • A053004 (program): Decimal expansion of AGM(1,sqrt(2)).
  • A053005 (program): Denominator of beta(2n+1)/Pi^(2n+1), where beta(m) = Sum_{k=0..inf} (-1)^k/(2k+1)^m.
  • A053006 (program): Values of n for which there exist d(1),…,d(n), each in {0,1}, such that Sum[d(i)d(i+k),i=1,n-k] is odd for all k=0,…,n-1.
  • A053021 (program): Number of divisors function applied twice to n!.
  • A053023 (program): Divisor function applied thrice to n!.
  • A053024 (program): a(n) = n*p where p is the next prime >= n.
  • A053025 (program): Number of iterations of A000005 required to reach 2 when started at n!.
  • A053029 (program): Numbers with 4 zeros in Fibonacci numbers mod m.
  • A053031 (program): Numbers with 1 zero in Fibonacci numbers mod m.
  • A053041 (program): Smallest n-digit number divisible by n.
  • A053042 (program): a(n) = n^n + n!.
  • A053044 (program): a(n) is the number of iterations of the Euler totient function to reach 1, starting at n!.
  • A053057 (program): Squares whose digit sum is also a square.
  • A053059 (program): Squares whose product of digits is also a nonzero square.
  • A053061 (program): a(n) is the decimal concatenation of n and n^2.
  • A053062 (program): Concatenate n, 2n, 3n, … nn.
  • A053067 (program): a(n) is the concatenation of next n numbers (omit leading 0’s).
  • A053079 (program): a(1)=1; a(m+1) = Sum_{k=1..m} gcd(k, a(m+1-k)).
  • A053088 (program): a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.
  • A053089 (program): a(n) = prime(n)^prime(n+1).
  • A053090 (program): Number of F^3-convex polyominoes on honeycomb lattice with given semiperimeter.
  • A053091 (program): F^3-convex polyominoes on the honeycomb lattice by number of cells.
  • A053096 (program): When the Euler phi function is iterated with initial value A002110(n) = primorial, a(n) = number of iterations required to reach the fixed number = 1.
  • A053100 (program): a(n) = ((6*n+7)(!^6))/7, related to A008542 ((6*n+1)(!^6) sextic, or 6-factorials).
  • A053101 (program): a(n) = ((6*n+8)(!^6))/8(!^6), related to A034689 (((6*n+2)(!^6))/2 sextic, or 6-factorials).
  • A053102 (program): a(n) = ((6*n+9)(!^6))/9(!^6), related to A034723 (((6*n+3)(!^6))/3 sextic, or 6-factorials).
  • A053103 (program): a(n) = ((6*n+10)(!^6))/10(!^6), related to A034724 (((6*n+4)(!^6))/4 sextic, or 6-factorials).
  • A053104 (program): a(n) = ((7*n+8)(!^7))/8, related to A045754 ((7*n+1)(!^7) sept-, or 7-factorials).
  • A053105 (program): a(n) = ((7*n+9)(!^7))/9(!^7), related to A034829 (((7*n+2)(!^7))/2 sept-, or 7-factorials).
  • A053106 (program): a(n) = ((7*n+10)(!^7))/10(1^7), related to A034830 (((7*n+3)(!^7))/3 sept-, or 7-factorials).
  • A053107 (program): Expansion of 1/(1-8*x)^8.
  • A053108 (program): Expansion of 1/(1 - 9*x)^9.
  • A053109 (program): Expansion of 1/(1-10*x)^10.
  • A053110 (program): Expansion of (-1 + 1/(1-7*x)^7)/(49*x); related to A036226.
  • A053111 (program): Expansion of (-1 + 1/(1-8*x)^8)/(64*x); related to A053107.
  • A053112 (program): Expansion of (-1 + 1/(1-9*x)^9)/(81*x); related to A053108.
  • A053113 (program): Expansion of (-1 + 1/(1-10*x)^10)/(100*x); related to A053109.
  • A053114 (program): a(n) = ((8*n+9)(!^8))/9, related to A045755 ((8*n+1)(!^8) octo- or 8-factorials).
  • A053115 (program): a(n) = ((8*n+10)(!^8))/20, related to A034908 ((8*n+2)(!^8) octo- or 8-factorials).
  • A053116 (program): a(n) = ((9*n+10)(!^9))/10, related to A045756 ((9*n+1)(!^9) 9-factorials).
  • A053117 (program): Triangle read by rows of coefficients of Chebyshev’s U(n,x) polynomials (exponents in increasing order).
  • A053118 (program): Triangle of coefficients of Chebyshev’s U(n,x) polynomials (exponents in decreasing order).
  • A053119 (program): Triangle of coefficients of Chebyshev’s S(n,x) polynomials (exponents in decreasing order).
  • A053120 (program): Triangle of coefficients of Chebyshev’s T(n,x) polynomials (powers of x in increasing order).
  • A053121 (program): Catalan triangle (with 0’s) read by rows.
  • A053122 (program): Triangle of coefficients of Chebyshev’s S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).
  • A053123 (program): Triangle of coefficients of shifted Chebyshev’s S(n,x-2)= U(n,x/2-1) polynomials (exponents of x in decreasing order).
  • A053124 (program): Triangle of coefficients of Chebyshev’s U(n,2*x-1) polynomials (exponents of x in increasing order).
  • A053125 (program): Triangle of coefficients of Chebyshev’s U(n,2*x-1) polynomials (exponents of x in decreasing order).
  • A053126 (program): Binomial coefficients binomial(2*n-3,4).
  • A053127 (program): Binomial coefficients C(2*n-4,5).
  • A053128 (program): Binomial coefficients C(2*n-5,6).
  • A053129 (program): Binomial coefficients C(2*n-6,7).
  • A053130 (program): Binomial coefficients C(2*n-7,8).
  • A053131 (program): Binomial coefficients C(2*n-8,9).
  • A053132 (program): One half of binomial coefficients C(2*n-4,5).
  • A053133 (program): One half of binomial coefficients binomial(2*n-8,9).
  • A053134 (program): Binomial coefficients C(2*n+4,4).
  • A053135 (program): Binomial coefficients C(2*n+6,6).
  • A053136 (program): Binomial coefficients C(2*n+7,7).
  • A053137 (program): Binomial coefficients C(2*n+8,8).
  • A053138 (program): Binomial coefficients C(2*n+9,9).
  • A053139 (program): a(n) = phi(n) - mu(n).
  • A053141 (program): a(0)=0, a(1)=2 then a(n) = a(n-2) + 2*sqrt(8*a(n-1)^2 + 8*a(n-1) + 1).
  • A053142 (program): a(n) = A053141(n)/2.
  • A053143 (program): Smallest square divisible by n.
  • A053144 (program): Cototient of the n-th primorial number.
  • A053149 (program): Smallest cube divisible by n.
  • A053150 (program): Cube root of largest cube dividing n.
  • A053152 (program): Number of 2-element intersecting families whose union is an n-element set.
  • A053154 (program): Number of 2-element intersecting families (with not necessary distinct sets) of an n-element set.
  • A053156 (program): Number of 2-element intersecting families (with not necessary distinct sets) whose union is an n-element set.
  • A053158 (program): Sum of n and its cototient function value (A051953): a(n) = 2*n - phi(n), where phi is Euler phi.
  • A053164 (program): 4th root of largest 4th power dividing n.
  • A053165 (program): 4th-power-free part of n.
  • A053166 (program): Smallest positive integer for which n divides a(n)^4.
  • A053167 (program): Smallest 4th power divisible by n.
  • A053175 (program): Catalan-Larcombe-French sequence.
  • A053176 (program): Primes p such that 2p+1 is composite.
  • A053177 (program): Odd composite k such that (k-1)/2 is prime.
  • A053178 (program): Numbers ending in 1 which are not prime.
  • A053179 (program): Numbers ending in 3 which are not prime.
  • A053180 (program): Numbers ending in 7 which are not prime.
  • A053181 (program): Composite numbers ending in 9.
  • A053184 (program): Primes p such that p^2+p-1 is prime.
  • A053185 (program): Primes of the form p^2 + p - 1 when p is prime.
  • A053186 (program): Square excess of n: difference between n and largest square <= n.
  • A053187 (program): Square nearest to n.
  • A053188 (program): Distance from n to nearest square.
  • A053191 (program): a(n) = n^2 * phi(n).
  • A053192 (program): a(n) is the cototient of n^3.
  • A053193 (program): Cototient of odd numbers.
  • A053196 (program): Cototients of even numbers.
  • A053197 (program): Number of level partitions of n.
  • A053198 (program): Totients of consecutive pure powers of primes.
  • A053199 (program): Triangular array T: put T(n,0)=n+1 for all n >= 0 and all other T(n,k)=0; then put T(n,k)=Sum{T(i,j): 0<=j<=i-n+k, n-k<=i<=n}.
  • A053200 (program): Binomial coefficients C(n,k) reduced modulo n, read by rows; T(0,0)=0 by convention.
  • A053201 (program): Pascal’s triangle (excluding first, last element of each row) read by rows, row n read mod n.
  • A053202 (program): Pascal’s triangle (excluding first, last two elements of each row) read by rows, row n read mod n.
  • A053203 (program): Pascal’s triangle (excluding first, last three elements of each row) read by rows, row n read mod n.
  • A053204 (program): Row sums of A053200.
  • A053205 (program): Row sums of A053201.
  • A053206 (program): Row sums of A053203.
  • A053208 (program): Row sums of A053207.
  • A053209 (program): Row sums of A051598.
  • A053210 (program): Row sums of A051599.
  • A053211 (program): Cototients of consecutive pure powers of primes.
  • A053214 (program): Central binomial coefficients (A000984) read mod 2n, with a(0)=1.
  • A053218 (program): Triangle read by rows where the first element in row n is n, and for k >= 2 element k in row n is the sum of element k-1 in row n and element k-1 in row n-1.
  • A053219 (program): Reverse of triangle A053218, read by rows.
  • A053220 (program): a(n) = (3*n-1) * 2^(n-2).
  • A053221 (program): Row sums of triangle A053218.
  • A053222 (program): First differences of sigma(n).
  • A053223 (program): Second differences of sigma(n).
  • A053224 (program): Numbers k for which sigma(k) < sigma(k+1).
  • A053226 (program): Numbers k for which sigma(k) > sigma(k+1).
  • A053230 (program): First differences between numbers k for which sigma(k) < sigma(k+1).
  • A053238 (program): First differences between numbers k for which sigma(k) > sigma(k+1).
  • A053246 (program): First differences of chowla(n).
  • A053290 (program): Number of nonsingular n X n matrices over GF(3).
  • A053291 (program): Nonsingular n X n matrices over GF(4).
  • A053293 (program): Number of nonsingular n X n matrices over GF(7).
  • A053295 (program): Partial sums of A053739.
  • A053296 (program): Partial sums of A053295.
  • A053297 (program): Row sums of array T in A053199.
  • A053298 (program): Partial sums of A027964.
  • A053307 (program): Number of nonnegative integer 2 X 2 matrices with sum of elements equal to n, under row and column permutations.
  • A053308 (program): Partial sums of A053296.
  • A053309 (program): Partial sums of A053308.
  • A053310 (program): a(n) = (n+3)*binomial(n+8, 8)/3.
  • A053311 (program): Partial sums of A000285.
  • A053312 (program): a(n) contains n digits (either ‘1’ or ‘2’) and is divisible by 2^n.
  • A053313 (program): a(n) contains n digits (either ‘2’ or ‘9’) and is divisible by 2^n.
  • A053314 (program): a(n) contains n digits (either ‘1’ or ‘4’) and is divisible by 2^n.
  • A053315 (program): a(n) contains n digits (either ‘4’ or ‘5’) and is divisible by 2^n.
  • A053316 (program): a(n) contains n digits (either ‘2’ or ‘3’) and is divisible by 2^n.
  • A053317 (program): a(n) contains n digits (either ‘2’ or ‘5’) and is divisible by 2^n.
  • A053318 (program): a(n) contains n digits (either ‘2’ or ‘7’) and is divisible by 2^n.
  • A053319 (program): Distance between the smaller members of successive twin prime pairs.
  • A053320 (program): Distance between pairs of primes differing by 4.
  • A053321 (program): First differences of A031924.
  • A053322 (program): First differences of A031926.
  • A053323 (program): First differences of A031928.
  • A053324 (program): First differences of A031930.
  • A053325 (program): First differences of A031932.
  • A053326 (program): First differences of A031934.
  • A053327 (program): First differences of A031936.
  • A053331 (program): First differences of A031938.
  • A053332 (program): a(n) contains n digits (either ‘4’ or ‘7’) and is divisible by 2^n.
  • A053333 (program): a(n) contains n digits (either ‘4’ or ‘9’) and is divisible by 2^n.
  • A053334 (program): a(n) contains n digits (either ‘1’ or ‘6’) and is divisible by 2^n.
  • A053335 (program): a(n) contains n digits (either ‘3’ or ‘6’) and is divisible by 2^n.
  • A053336 (program): a(n) contains n digits (either ‘5’ or ‘6’) and is divisible by 2^n.
  • A053337 (program): a(n) contains n digits (either ‘6’ or ‘7’) and is divisible by 2^n.
  • A053338 (program): a(n) contains n digits (either ‘6’ or ‘9’) and is divisible by 2^n.
  • A053344 (program): Minimal number of coins needed to pay n cents using coins of denominations 1, 5, 10, 25 cents.
  • A053347 (program): a(n) = binomial(n+7, 7)*(n+4)/4.
  • A053367 (program): Partial sums of A050494.
  • A053368 (program): a(n) = (5n+2)*C(n) where C(n)=Catalan numbers (A000108).
  • A053369 (program): Linear recursion with Catalan numbers.
  • A053376 (program): a(n) contains n digits (either ‘1’ or ‘8’) and is divisible by 2^n.
  • A053377 (program): a(n) contains n digits (either ‘3’ or ‘8’) and is divisible by 2^n.
  • A053378 (program): a(n) contains n digits (either ‘5’ or ‘8’) and is divisible by 2^n.
  • A053379 (program): a(n) contains n digits (either ‘7’ or ‘8’) and is divisible by 2^n.
  • A053380 (program): a(n) contains n digits (either ‘8’ or ‘9’) and is divisible by 2^n.
  • A053381 (program): Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.
  • A053384 (program): a(n) = A053398(4,n).
  • A053385 (program): A053398(5, n).
  • A053386 (program): A053398(6, n).
  • A053387 (program): A053398(7, n).
  • A053388 (program): A053398(8, n).
  • A053389 (program): A053398(9, n).
  • A053390 (program): a(n) = A053398(10, n).
  • A053398 (program): Nim-values from game of Kopper’s Nim.
  • A053399 (program): A053398(3, n).
  • A053404 (program): Expansion of 1/((1+3*x)*(1-4*x)).
  • A053405 (program): Definition: A kara B = C, where C is the least nonnegative integer such that: C * B >= A and C * (B-1) < A. Sequence gives smallest a such that n kara a is undefined.
  • A053407 (program): Traditional type sizes (in typographic points) for printing.
  • A053408 (program): Numbers n such that A003266(n) + 1 is prime.
  • A053410 (program): a(1) = 0, a(2) = 16, a(2n+1) = 10*a(2n) - a(2n-1), a(2n) = 10*a(2n-1) - a(2n-2) + 16.
  • A053411 (program): Circle numbers (version 1): a(n)= number of points (i,j), i,j integers, contained in a circle of diameter n, centered at the origin.
  • A053412 (program): n-th nonzero Fibonacci numbers arising in A053408.
  • A053413 (program): Primes of the form A003266(n) + 1.
  • A053414 (program): Circle numbers (version 2): a(n) is the number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (0, 1/2).
  • A053415 (program): Circle numbers (version 3): a(n) = number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (1/2, 1/2).
  • A053416 (program): Circle numbers (version 4): a(n)= number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (0,0).
  • A053417 (program): Circle numbers (version 5): a(n) = number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (1/2,0).
  • A053422 (program): n times (n 1’s): a(n) = n*(10^n - 1)/9.
  • A053425 (program): Even numbers n such that the 120 points of the 600-cell exactly integrate homogeneous polynomials of degree n.
  • A053428 (program): a(n) = a(n-1) + 20*a(n-2), n >= 2; a(0)=1, a(1)=1.
  • A053430 (program): a(n) = (6^(n+1) - (-5)^(n+1))/11.
  • A053436 (program): a(n) = n+1 + ceiling(n/2)*(ceiling(n/2)-1)*(ceiling(n/2)+1)/6.
  • A053438 (program): Expansion of (1+x+2*x^3)/((1-x)*(1-x^2)).
  • A053439 (program): Expansion of (1+x+2*x^3)/((1-x)*(1-x^2)^2).
  • A053440 (program): Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, …, n+1}.
  • A053441 (program): Moments of generalized Motzkin paths.
  • A053442 (program): Moments of generalized Motzkin paths.
  • A053445 (program): Second differences of partition numbers A000041.
  • A053446 (program): Multiplicative order of 3 mod m, where gcd(m, 3) = 1.
  • A053447 (program): Multiplicative order of 4 mod 2n+1.
  • A053448 (program): Multiplicative order of 5 mod m, where gcd(m, 5) = 1.
  • A053449 (program): Multiplicative order of 6 mod n, where gcd(n, 6) = 1.
  • A053450 (program): Multiplicative order of 7 mod n, where gcd(n,7) = 1.
  • A053451 (program): Multiplicative order of 8 mod 2n+1.
  • A053452 (program): Multiplicative order of 9 mod n, where gcd(n, 9) = 1.
  • A053453 (program): Duplicate of A002329.
  • A053455 (program): a(n) = ((8^n) - (-6)^n)/14.
  • A053456 (program): Open disk numbers (version 1): a(n) is the number of points (i,j), i,j, integers, contained in an open disk of diameter n, centered at (0,0).
  • A053457 (program): Open disk numbers (version 2): a(n) is the number of points (i,j), i,j, integers, contained in an open disk of diameter n, centered at (0,1/2).
  • A053458 (program): Open disk numbers (version 3): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (0,0).
  • A053459 (program): Open disk numbers (version 4): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (1/2,0).
  • A053464 (program): a(n) = n*5^(n-1).
  • A053469 (program): a(n) = n*6^(n-1).
  • A053470 (program): a(n) is the cototient of n (A051953) iterated twice.
  • A053471 (program): a(n) is the cototient of n (A051953) iterated 3 times.
  • A053472 (program): a(n) is the cototient of n (A051953) iterated 4 times.
  • A053475 (program): 1 + the number of iterations of A051953 (Euler-cototient) function needed to reach 0, starting at n.
  • A053477 (program): Sum of iterates of divisor number function A000005.
  • A053478 (program): Sum of iterates when phi, A000010, is iterated until fixed point 1.
  • A053480 (program): Sum of values when cototient function A051953 is iterated until fixed point is reached.
  • A053481 (program): First differences of A029767.
  • A053482 (program): Binomial transform of A029767.
  • A053484 (program): Numerators in expansion of exp(2x)/(1-x).
  • A053485 (program): Denominators in expansion of exp(2x)/(1-x).
  • A053486 (program): E.g.f.: exp(3x)/(1-x).
  • A053487 (program): E.g.f.: exp(4x)/(1-x).
  • A053493 (program): Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.
  • A053496 (program): Number of degree-n permutations of order dividing 6.
  • A053497 (program): Number of degree-n permutations of order dividing 7.
  • A053498 (program): Number of degree-n permutations of order dividing 8.
  • A053499 (program): Number of degree-n permutations of order dividing 9.
  • A053500 (program): Number of degree-n permutations of order dividing 10.
  • A053502 (program): Number of degree-n permutations of order dividing 12.
  • A053504 (program): Number of degree-n permutations of order dividing 24.
  • A053505 (program): Number of degree-n permutations of order dividing 30.
  • A053506 (program): a(n) = (n-1)*n^(n-2).
  • A053507 (program): a(n) = binomial(n-1,2)*n^(n-3).
  • A053508 (program): a(n) = binomial(n-1,3)*n^(n-4).
  • A053509 (program): a(n) = binomial(n-1,4)*n^(n-5).
  • A053518 (program): Numerators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+…))))))).
  • A053519 (program): Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+…))))))).
  • A053520 (program): Denominators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+…))))))).
  • A053524 (program): a(n) = (6^n - (-2)^n)/8.
  • A053525 (program): Expansion of e.g.f.: (1-x)/(2-exp(x)).
  • A053526 (program): Number of bipartite graphs with 3 edges on nodes {1..n}.
  • A053529 (program): a(n) = n! * number of partitions of n.
  • A053532 (program): Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6).
  • A053535 (program): Expansion of 1/((1+3*x)*(1-9*x)).
  • A053536 (program): Expansion of 1/((1+4*x)*(1-12*x)).
  • A053537 (program): Expansion of 1/((1+5*x)*(1-15*x)).
  • A053538 (program): Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).
  • A053539 (program): a(n) = n * 8^(n-1).
  • A053540 (program): a(n) = n*9^(n-1).
  • A053541 (program): a(n) = n*10^(n-1).
  • A053542 (program): Distance from n-th composite number (A002808) to next prime.
  • A053543 (program): Distance to closest prime in sequence of composites.
  • A053545 (program): Comparisons needed for Batcher’s sorting algorithm applied to 2^n items.
  • A053553 (program): Extreme points of set of n X n symmetric substochastic matrices.
  • A053556 (program): Denominator of Sum_{k=0..n} (-1)^k/k!.
  • A053557 (program): Numerator of Sum_{k=0..n} (-1)^k/k!.
  • A053565 (program): a(n) = 2^(n-1)*(3*n-4).
  • A053566 (program): Expansion of (11*x-2)/(1-3*x)^2.
  • A053569 (program): Sum of divisors of numbers which cannot be sum of divisors of any number.
  • A053570 (program): Sum of totient functions over arguments running through reduced residue system of n.
  • A053573 (program): a(n) = 5*a(n-1) + 14*a(n-2), a(0)=1, a(1)=5.
  • A053574 (program): Exponent of 2 in phi(n) where phi(n) = A000010(n).
  • A053575 (program): Odd part of phi(n): a(n) = A000265(A000010(n)).
  • A053577 (program): Cototient function n - phi(n) is a power of 2.
  • A053581 (program): First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649).
  • A053585 (program): If n = p_1^e_1 * … * p_k^e_k, p_1 < … < p_k primes, then a(n) = p_k^e_k.
  • A053589 (program): Greatest primorial number (A002110) which divides n.
  • A053599 (program): Number of nonempty subsequences {s(k)} of 1..n such that the difference sequence is palindromic.
  • A053601 (program): Number of bases of an n-dimensional vector space over GF(2).
  • A053602 (program): a(n) = a(n-1) - (-1)^n*a(n-2), a(0)=0, a(1)=1.
  • A053603 (program): Number of ways to write n as an ordered sum of two nonzero triangular numbers.
  • A053604 (program): Number of ways to write n as an ordered sum of 3 nonzero triangular numbers.
  • A053606 (program): a(n) = (Fibonacci(6*n+3) - 2)/4.
  • A053610 (program): Number of positive squares needed to sum to n using the greedy algorithm.
  • A053615 (program): Pyramidal sequence: distance to nearest product of two consecutive integers (promic or heteromecic numbers).
  • A053616 (program): Pyramidal sequence: distance to nearest triangular number.
  • A053618 (program): a(n) = ceiling(binomial(n,4)/n).
  • A053625 (program): Product of 6 consecutive integers.
  • A053626 (program): a(n) is the smallest positive integer k such that harmonic mean of n and k is an integer.
  • A053627 (program): Smallest integer which is the harmonic mean of n and an integer.
  • A053631 (program): Pythagorean spiral: a(n-1)+1, a(n) and a(n)+1 are the sides of a right triangle (a primitive Pythagorean triangle).
  • A053634 (program): a(n) = Sum_{ d divides n } phi(d)*2^(n/d)/(2n).
  • A053635 (program): a(n) = Sum_{d|n} phi(d)*2^(n/d).
  • A053636 (program): a(n) = Sum_{odd d|n} phi(d)*2^(n/d).
  • A053637 (program): a(n) = ceiling(2^(n-1)/n).
  • A053638 (program): a(n) = ceiling(2^n/n).
  • A053639 (program): a(n) = ceiling(2^(n+1)/n).
  • A053641 (program): Rotate one binary digit to the right, calculate, then rotate one binary digit to the left.
  • A053642 (program): Rotate one binary digit to the left, calculate, then rotate one binary digit to the right.
  • A053643 (program): a(n) = ceiling(binomial(n,6)/n).
  • A053644 (program): Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.
  • A053645 (program): Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal.
  • A053646 (program): Distance to nearest power of 2.
  • A053650 (program): Cototient function of n^2.
  • A053654 (program): Multiples of 123456789.
  • A053655 (program): a(n) = (10^n - 1)*(10^(2*n-1) - 1)/81.
  • A053656 (program): Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).
  • A053657 (program): a(n) = Product_{p prime} p^{ Sum_{k>= 0} floor[(n-1)/((p-1)p^k)]}.
  • A053661 (program): For n > 1: if n is present, 2n is not.
  • A053666 (program): Product of digits of n-th prime.
  • A053667 (program): Product of digits of n^2.
  • A053668 (program): Product of digits of n^3.
  • A053669 (program): Smallest prime not dividing n.
  • A053670 (program): Least number coprime to n and n+1.
  • A053671 (program): Least number coprime to n, n+1 and n+2.
  • A053672 (program): Least number coprime to n, n+1, n+2 and n+3.
  • A053673 (program): Least number > 1 coprime to n, n+1, n+2, n+3 and n+4.
  • A053674 (program): Least number coprime to n, n+1, n+2, n+3, n+4 and n+5.
  • A053683 (program): Number of nonprimes <= prime(n)^2.
  • A053692 (program): Number of self-conjugate 4-core partitions of n.
  • A053694 (program): Number of self-conjugate 5-core partitions of n.
  • A053698 (program): a(n) = n^3 + n^2 + n + 1.
  • A053699 (program): a(n) = n^4 + n^3 + n^2 + n + 1.
  • A053700 (program): a(n) = 111111 in base n.
  • A053715 (program): a(n) = n-th triangular number (the sum of the first n integers) in base n.
  • A053716 (program): a(n) = 1111111 in base n.
  • A053717 (program): a(n) = n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.
  • A053723 (program): Number of 5-core partitions of n.
  • A053726 (program): “Flag numbers”: number of dots that can be arranged in successive rows of K, K-1, K, K-1, K, …, K-1, K (assuming there is a total of L > 1 rows of size K > 1).
  • A053728 (program): For n=1,2,3,…, compute sum of aliquot divisors of n; if result is prime append this prime to sequence.
  • A053729 (program): Self-convolution of 1,4,27,256,3125,46656,… (cf. A000312).
  • A053730 (program): a(n) = 2^(n-2)*(n^2 - n + 4).
  • A053731 (program): a(n) = ceiling(binomial(n,8)/n).
  • A053733 (program): a(n) = ceiling(binomial(n,9)/n).
  • A053735 (program): Sum of digits of (n written in base 3).
  • A053737 (program): Sum of digits of (n written in base 4).
  • A053738 (program): If k is in sequence then 2*k and 2*k+1 are not (and 1 is in the sequence); numbers with an odd number of digits in binary.
  • A053739 (program): Partial sums of A014166.
  • A053741 (program): Sum of even numbers in range 10*n to 10*n+9.
  • A053742 (program): Sum of odd numbers in range 10*n to 10*n+9.
  • A053743 (program): Sum of numbers in range 10*n to 10*n+9.
  • A053744 (program): Sum of 3 consecutive digits of Pi.
  • A053754 (program): If k is in the sequence then 2*k and 2*k+1 are not (and 0 is in the sequence); when written in binary k has an even number of bits (0 has 0 digits).
  • A053755 (program): a(n) = 4*n^2 + 1.
  • A053763 (program): a(n) = 2^(n^2 - n).
  • A053764 (program): a(n) = 3^(n^2 - n).
  • A053765 (program): a(n) = 4^(n^2 - n).
  • A053767 (program): Sum of first n composite numbers.
  • A053785 (program): Nextprime(n^4) - n^4.
  • A053786 (program): a(n) = next prime after n^4.
  • A053787 (program): Nextprime(n^5) - n^5.
  • A053788 (program): Next prime after n^5.
  • A053789 (program): a(n) = A020639(A053790(n)).
  • A053790 (program): Composite numbers arising as sum of first k primes.
  • A053793 (program): n^2+n modulo 7.
  • A053794 (program): a(n) = (n^2 + n) modulo 8.
  • A053795 (program): Composite numbers ending in 1, 3, 7 or 9.
  • A053796 (program): a(n) = (n^2+n) modulo 5.
  • A053797 (program): Lengths of successive gaps between squarefree numbers.
  • A053798 (program): Number of basis partitions of n+16 with Durfee square size 4.
  • A053799 (program): Number of basis partitions of n+9 with Durfee square size 3.
  • A053800 (program): Number of basis partitions of n+25 with Durfee square size 5.
  • A053805 (program): Expansion of (1 + x)^12 / (1 - x)^13.
  • A053806 (program): Numbers where a gap begins in the sequence of squarefree numbers (A005117).
  • A053807 (program): a(n) = Sum_{k=1..n, n mod k = 1} k^2.
  • A053808 (program): Partial sums of A001891.
  • A053809 (program): Second partial sums of A001891.
  • A053813 (program): Numbers which are an integral multiple of the sum of their proper divisors: prime and perfect numbers.
  • A053814 (program): a(n) = n modulo (sum of proper divisors of n).
  • A053815 (program): Floor(n / (sum of proper divisors of n)).
  • A053817 (program): a(0)=1, a(n) = n*(a(n-1) + n).
  • A053818 (program): a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2.
  • A053819 (program): a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3.
  • A053820 (program): a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^4.
  • A053821 (program): A discrete approximation to log(n): Sum_{ d divides n } A029833(d).
  • A053822 (program): Dirichlet inverse of sigma_2 function (A001157).
  • A053824 (program): Sum of digits of (n written in base 5).
  • A053825 (program): Dirichlet inverse of sigma_3 function (A001158).
  • A053826 (program): Dirichlet inverse of sigma_4 function (A001159).
  • A053827 (program): Sum of digits of (n written in base 6).
  • A053828 (program): Sum of digits of (n written in base 7).
  • A053829 (program): Sum of digits of (n written in base 8).
  • A053830 (program): Sum of digits of (n written in base 9).
  • A053831 (program): Sum of digits of n written in base 11.
  • A053832 (program): Sum of digits of n written in base 12.
  • A053833 (program): Sum of digits of n written in base 13.
  • A053836 (program): Sum of digits of n written in base 16.
  • A053837 (program): Sum of digits of n modulo 10.
  • A053838 (program): a(n) = (sum of digits of n written in base 3) modulo 3.
  • A053839 (program): a(n) = (sum of digits of n written in base 4) modulo 4.
  • A053840 (program): (Sum of digits of n written in base 5) modulo 5.
  • A053841 (program): (Sum of digits of n written in base 6) modulo 6.
  • A053842 (program): (Sum of digits of n written in base 7) modulo 7.
  • A053843 (program): (Sum of digits of n written in base 8) modulo 8.
  • A053844 (program): (Sum of digits of n written in base 9) modulo 9.
  • A053845 (program): Primes of form prime(1) + … + prime(k) + 1.
  • A053850 (program): Odd numbers divisible by a square > 1.
  • A053858 (program): Squarefree even composite numbers with an odd number of prime factors.
  • A053864 (program): A second order generalization of the Mobius function of n.
  • A053866 (program): Parity of A000203(n), the sum of the divisors of n; a(n) = 1 when n is a square or twice a square, 0 otherwise.
  • A053867 (program): Parity of sum of divisors of n less than n.
  • A053868 (program): Numbers whose sum of proper divisors is odd.
  • A053869 (program): Sum of divisors of n less than n is even.
  • A053871 (program): a(0)=1; a(1)=0; a(n) = 2*(n-1)*(a(n-1) + a(n-2)).
  • A053879 (program): a(n) = n^2 mod 7.
  • A053983 (program): a(n) = (2*n-1)*a(n-1) - a(n-2), a(0)=a(1)=1.
  • A053984 (program): a(n) = (2*n-1)*a(n-1) - a(n-2), a(0) = 0, a(1) = 1.
  • A053985 (program): Replace 2^k with (-2)^k in binary expansion of n.
  • A053987 (program): Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-…))))))).
  • A053988 (program): Denominators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-…))))))).
  • A053989 (program): Smallest k such that nk-1 is prime.
  • A053995 (program): Bases of n-dimensional vector space over GF(3).
  • A053996 (program): Number of bases of n-dimensional vector space over GF(4).
  • A053997 (program): Sum of primes in n-th shell of prime spiral.
  • A053998 (program): Smallest prime in n-th shell of prime spiral.
  • A054000 (program): a(n) = 2*n^2 - 2.
  • A054008 (program): n read modulo (number of divisors of n).
  • A054009 (program): n read modulo (number of proper divisors of n).
  • A054010 (program): Numbers n with property that n is divisible by the number of its proper divisors.
  • A054011 (program): n is not divisible by the number of its proper divisors.
  • A054012 (program): Nonzero values of n read modulo (number of proper divisors of n).
  • A054013 (program): Chowla function of n read modulo n.
  • A054014 (program): Chowla function of n read modulo (the number of divisors of n).
  • A054015 (program): a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.
  • A054020 (program): Chowla’s function of n is not divisible by the number of proper divisors of n.
  • A054021 (program): Numbers n such that Chowla’s function of n is divisible by the number of proper divisors of n.
  • A054022 (program): Chowla function of n is divisible by the number of divisors of n.
  • A054023 (program): Chowla function of n is not divisible by the number of divisors of n.
  • A054024 (program): Sum of the divisors of n reduced modulo n.
  • A054025 (program): Sum of divisors of n read modulo (number of divisors of n).
  • A054026 (program): a(n) is the number of sets of natural numbers [a,b,c,d,e] that can be produced with the numbers [0..n] such that the values of all the distinct parenthesized expressions of a-b-c-d-e are different.
  • A054027 (program): Numbers that do not divide their sum of divisors.
  • A054042 (program): Decimal expansion of 1 - 1/sqrt(10).
  • A054047 (program): Numbers which can be read as (possibly different) numbers when rotated by 180 degrees *final zeros allowed).
  • A054054 (program): Smallest digit of n.
  • A054055 (program): Largest digit of n.
  • A054056 (program): Numbers of form 29+n^2+n or 29+2*n^2.
  • A054057 (program): Numbers of form 41+n^2+n or 41+2*n^2.
  • A054058 (program): Inverse Moebius transform of A000031 (starting at term 0).
  • A054060 (program): Least k for which the integers floor(k*(Pi/2 - arctan(m))) for m=1,2,…,n are distinct.
  • A054066 (program): Position of n-th 1 in A054065.
  • A054067 (program): Position of first appearance of n in A054065.
  • A054071 (program): Position of 1 in the permutation of 1,2,…,n obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,…,n.
  • A054072 (program): Position of n in the permutation of 1,2,…,n obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,…,n.
  • A054074 (program): Position of n-th 1 in A054073.
  • A054075 (program): Position of first appearance of n in A054073.
  • A054080 (program): Inverse Moebius transform of A001037 (starting at term 0).
  • A054082 (program): Permutation of N: a(1)=2, a(2)=1 and for each k >= 2, let p(k)=least natural number not already an a(i), q(k)=p(k)+k-1, a(2k-1)=q(k), a(2k)=p(k).
  • A054084 (program): Permutation of N: for each k >= 1, let p(k)=least natural number not already an a(i), q(k)=p(k)+k, a(2k-1)=q(k), a(2k)=p(k).
  • A054087 (program): s(3n-2), s=A054086; also a bisection of A003511.
  • A054088 (program): a(n) = A054086(3n); also a bisection of A003511.
  • A054089 (program): For k >= 1, let p(k)=least h in N not already an a(i), q(k)=p(k)+k, a(2k)=q(k), a(2k+1)=p(k).
  • A054091 (program): Row sums of A054090.
  • A054092 (program): T(n,n), array T as in A054090.
  • A054093 (program): T(n,n-1), array T as in A054090.
  • A054094 (program): T(n,n-2), array T as in A054090.
  • A054095 (program): T(n,n-3), array T as in A054090.
  • A054096 (program): T(n,2), array T as in A054090.
  • A054097 (program): T(n,3), array T as in A054090.
  • A054099 (program): Sum{T(n,k): k=0,1,…,n}, array T as in A054098.
  • A054100 (program): T(n,n), array T as in A054098.
  • A054101 (program): T(n,n-1), array T as in A054098.
  • A054102 (program): T(n,n-2), array T as in A054098.
  • A054103 (program): T(n,n-3), array T as in A054098.
  • A054104 (program): T(n,2), array T as in A054098.
  • A054105 (program): T(n,3), array T as in A054098.
  • A054107 (program): T(n,n-3), array T as in A054106.
  • A054108 (program): a(n) = (-1)^(n+1)*Sum_{k=0..n+1}(-1)^k*binomial(2*k,k).
  • A054109 (program): a(n) = T(2*n+1, n), array T as in A054106.
  • A054111 (program): Row sums of array T as in A054110.
  • A054112 (program): T(n,n-3), array T as in A054110.
  • A054113 (program): T(2n,n), array T as in A054110.
  • A054114 (program): T(2n+1,n), array T as in A054110.
  • A054115 (program): Triangular array generated by its row sums: T(n,0)=1 for n >= 1, T(n,1)=r(n-1), T(n,k)=T(n,k-1)+r(n-k) for k=2,3,…,n, n >= 2, r(h)=sum of the numbers in row h of T.
  • A054116 (program): T(n,n-1), array T as in A054115.
  • A054117 (program): T(n,n-2), array T as in A054115.
  • A054118 (program): Subdiagonal T(n,n-3), array T as in A054115.
  • A054119 (program): a(n) = n! + (n-1)! + (n-2)!.
  • A054121 (program): T(n,n-3), array T as in A054120.
  • A054122 (program): T(2n,n), array T as in A054120.
  • A054123 (program): Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.
  • A054124 (program): Left Fibonacci row-sum array, n >= 0, 0<=k<=n.
  • A054127 (program): a(1) = 2; a(n) = 9*2^(n-2) - n - 2, n>1.
  • A054128 (program): T(n,2), array T as in A054126.
  • A054129 (program): T(n,3), array T as in A054126.
  • A054130 (program): T(n,4), array T as in A054126.
  • A054131 (program): T(2n,n), array T as in A054126.
  • A054132 (program): T(2n+1,n), array T as in A054126.
  • A054133 (program): T(2n-1,n) where T is the array in A054126.
  • A054135 (program): a(n) = T(n,1), array T as in A054134.
  • A054136 (program): T(n,2), array T as in A054134.
  • A054137 (program): T(n,3), array T as in A054134.
  • A054138 (program): T(n,4), array T as in A054134.
  • A054139 (program): T(2n,n), array T as in A054134.
  • A054140 (program): T(2n+1,n), array T as in A054134.
  • A054141 (program): T(2n-1,n), array T as in A054134.
  • A054142 (program): Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
  • A054143 (program): Triangular array T given by T(n,k) = Sum_{0 <= j <= i-n+k, n-k <= i <= n} C(i,j) for n >= 0 and 0 <= k <= n.
  • A054144 (program): Triangular array T: put T(n,0)=n for all n >= 0 and all other T(n,k)=0; then put T(n,k)=Sum{T(i,j): 0<=j<=i-n+k, n-k<=i<=n}.
  • A054145 (program): Row sums of array T as in A054144.
  • A054146 (program): a(n) = A054145(n)/2.
  • A054147 (program): a(n) = T(2n,n), array T as in A054144.
  • A054148 (program): T(2n+1,n), array T as in A054144.
  • A054149 (program): T(2n-1,n), array T as in A054144.
  • A054155 (program): Inverse Moebius transform of A000029 (starting at term 0).
  • A054168 (program): Inverse Moebius transform of A000013 (starting at term 0).
  • A054172 (program): Inverse Moebius transform of A000048 (starting at term 0).
  • A054181 (program): Inverse Moebius transform of A000011 (starting at term 0).
  • A054185 (program): Binomial transform of A000031.
  • A054190 (program): Binomial transform of A001037.
  • A054192 (program): Binomial transform of A000029.
  • A054196 (program): Binomial transform of A000013.
  • A054197 (program): Binomial transform of A000048.
  • A054198 (program): Binomial transform of A000011.
  • A054201 (program): a(n) = (n-1)! * Sum_{k=1..n} k^k/k!.
  • A054204 (program): Integers expressible as sums of distinct even-subscripted Fibonacci numbers.
  • A054211 (program): Numbers n such that n concatenated with n-1 is prime.
  • A054243 (program): Number of partitions of n into distinct positive parts <= n, where parts are combined by XOR.
  • A054246 (program): Non-Cayley-isomorphic circulant p^2-tournaments, indexed by odd primes p.
  • A054248 (program): Binary entropy: a(n) = n + min { a(k)+a(n-k) : 1 <= k <= n-1 }.
  • A054249 (program): Alternately subtract and add 1 to digits in decimal expansion of Pi.
  • A054254 (program): a(n) is n plus the minimum of the a(i)*a(n-i) of the previous i=1..n-1.
  • A054265 (program): Sum of composite numbers between successive primes.
  • A054270 (program): Largest prime below prime(n)^2 (A001248).
  • A054271 (program): Difference between prime(n)^2 and the previous prime.
  • A054272 (program): Number of primes in the interval [prime(n), prime(n)^2].
  • A054275 (program): Susceptibility series H_2 for 2-dimensional Ising model (divided by 2).
  • A054318 (program): a(n)-th star number (A003154) is a square.
  • A054320 (program): Expansion of g.f.: (1 + x)/(1 - 10*x + x^2).
  • A054322 (program): Fourth unsigned column of Lanczos triangle A053125 (decreasing powers).
  • A054323 (program): Fifth column of Lanczos triangle A053125 (decreasing powers).
  • A054324 (program): Sixth unsigned column of Lanczos triangle A053125 (decreasing powers).
  • A054325 (program): Seventh column of Lanczos triangle A053125 (decreasing powers).
  • A054326 (program): Eighth unsigned column of Lanczos triangle A053125 (decreasing powers).
  • A054327 (program): Ninth column of Lanczos triangle A053125 (decreasing powers).
  • A054328 (program): Tenth unsigned column of Lanczos triangle A053125 (decreasing powers).
  • A054329 (program): One quarter of fourth unsigned column of Lanczos’ triangle A053125.
  • A054330 (program): One half of sixth unsigned column of Lanczos’ triangle A053125.
  • A054331 (program): One eighth of eighth unsigned column of Lanczos’ triangle A053125.
  • A054332 (program): One half of tenth unsigned column of Lanczos triangle A053125 (decreasing powers).
  • A054333 (program): 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
  • A054334 (program): 1/512 of 11th unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
  • A054335 (program): A convolution triangle of numbers based on A000984 (central binomial coefficients of even order).
  • A054336 (program): A convolution triangle of numbers based on A001405 (central binomial coefficients).
  • A054337 (program): 7-fold convolution of A000302 (powers of 4).
  • A054338 (program): 8-fold convolution of A000302 (powers of 4).
  • A054339 (program): 9-fold convolution of A000302 (powers of 4).
  • A054340 (program): 10-fold convolution of A000302 (powers of 4).
  • A054341 (program): Row sums of triangle A054336 (central binomial convolutions).
  • A054347 (program): Partial sums of A000201.
  • A054353 (program): Partial sums of Kolakoski sequence A000002.
  • A054354 (program): First differences of Kolakoski sequence A000002.
  • A054355 (program): Binomial transform of Kolakoski sequence A000002.
  • A054385 (program): Beatty sequence for e/(e-1); complement of A022843.
  • A054386 (program): Beatty sequence for Pi/(Pi-1); complement of A022844.
  • A054388 (program): Denominators of coefficients of 1/2^(2n+1) in Newton’s series for Pi.
  • A054390 (program): Number of ways of writing n as a sum of powers of 3, each power being used at most three times.
  • A054391 (program): Number of permutations with certain forbidden subsequences.
  • A054392 (program): Number of permutations with certain forbidden subsequences.
  • A054393 (program): Number of permutations with certain forbidden subsequences.
  • A054394 (program): Number of permutations with certain forbidden subsequences.
  • A054398 (program): Define a sequence of 2^n X 2^n squares as follows: S_0 = [1], S_1 = [1,2; 3,4]; S_2 = [1,2,5,6; 3,4,7,8; 9,10,13,14; 11,12,15,16], etc.; sequence gives triangular array whose n-th row gives differences between successive columns of n-th square.
  • A054401 (program): 5^n-4^n-1.
  • A054403 (program): Result of third stage of sieve of Eratosthenes (after eliminating multiples of 2, 3 and 5).
  • A054404 (program): Number of daughters to wait before picking in sultan’s dowry problem with n daughters.
  • A054405 (program): Row sums of array T as in A055215.
  • A054406 (program): Beatty sequence for (3+sqrt 3)/2; complement of A022838.
  • A054410 (program): Susceptibility series H_3 for 2-dimensional Ising model (divided by 2).
  • A054413 (program): a(n) = 7*a(n-1) + a(n-2), with a(0)=1 and a(1)=7.
  • A054414 (program): a(n) = 1 + floor(n/(1-log(2)/log(3))).
  • A054417 (program): Number of connected 3 X n binary matrices with rightmost column [1,0,0]’.
  • A054418 (program): Number of connected 3 X n binary matrices with rightmost column [1,1,1]’, divided by 4.
  • A054419 (program): Number of connected 3 X n binary matrices (divided by 2).
  • A054420 (program): Number of connectable 3 X n binary matrices.
  • A054421 (program): Number of disconnected 3 X n binary matrices.
  • A054429 (program): Simple self-inverse permutation of natural numbers: List each block of 2^n numbers (from 2^n to 2^(n+1) - 1) in reverse order.
  • A054431 (program): Array read by antidiagonals: T(x, y) tells whether (x, y) are coprime (1) or not (0).
  • A054432 (program): a(n) = Sum_{1<=k<=n, gcd(k,n)=1} 2^(k-1).
  • A054433 (program): Numbers formed by interpreting the reduced residue set of every even number as a Zeckendorf Expansion.
  • A054436 (program): Smallest area of a Pythagorean triangle with n as length of a leg.
  • A054438 (program): Third derivative of n.
  • A054441 (program): Convolution of (shifted) A026671 with A000984 (central binomial coefficients of even order).
  • A054442 (program): Second convolution of A001405 (central binomial numbers).
  • A054443 (program): Third convolution of A001405 (central binomial numbers).
  • A054444 (program): Even-indexed terms of A001629(n), n >= 2, (Fibonacci convolution).
  • A054445 (program): Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle).
  • A054447 (program): Row sums of triangle A054446 (partial row sums triangle of Fibonacci convolution triangle).
  • A054449 (program): Row sums of triangle A054448 (second member of partial row sums triangle family of Fibonacci convolution triangle).
  • A054450 (program): Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).
  • A054451 (program): Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).
  • A054452 (program): Partial sums of A027941(n-1) with a(-1) = 0.
  • A054454 (program): Third column of triangle A054453.
  • A054455 (program): Row sums of triangle A054453.
  • A054456 (program): Convolution triangle of A000129(n) (Pell numbers).
  • A054457 (program): Pell numbers A000129(n+1) (without P(0)) convoluted twice with itself.
  • A054458 (program): Convolution triangle based on A001333(n), n >= 1.
  • A054459 (program): A001333(n), n >= 1, convolved with itself.
  • A054460 (program): A001333(n), n >= 1, convolved twice with itself.
  • A054469 (program): A second-order recursive sequence.
  • A054470 (program): Partial sums of A054469.
  • A054475 (program): Numbers not divisible by any of their digits when written in base 4.
  • A054477 (program): A Pellian-related sequence.
  • A054479 (program): Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.
  • A054480 (program): Number of different positive braids with n crossings of 4 strands.
  • A054485 (program): Expansion of (1+3*x)/(1-4*x+x^2).
  • A054486 (program): Expansion of (1+2*x)/(1-3*x+x^2).
  • A054487 (program): a(n) = (3*n+4)*binomial(n+7, 7)/4.
  • A054488 (program): Expansion of (1+2*x)/(1-6*x+x^2).
  • A054489 (program): Expansion of (1+4*x)/(1-6*x+x^2).
  • A054490 (program): Expansion of (1+5*x)/(1-6*x+x^2).
  • A054491 (program): a(n) = 4*a(n-1) - a(n-2), a(0)=1, a(1)=6.
  • A054492 (program): a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6.
  • A054493 (program): A Pellian-related recursive sequence.
  • A054494 (program): Largest Fibonacci factor of n.
  • A054495 (program): Smallest k such that n/k is a Fibonacci number.
  • A054496 (program): If n = p_1^e_1 *p_2^e_2 *p_3^e_3…, p’s = distinct primes, e’s = positive integers, then a(n) = p_1^(e_1^2) *p_2^(e_2^2) *p_3^(e_3^2) … .
  • A054497 (program): Number of symmetric nonnegative integer 7 X 7 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.
  • A054498 (program): Number of symmetric nonnegative integer 8 X 8 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.
  • A054514 (program): Number of ways to place non-crossing diagonals in convex (n+4)-gon so as to create no triangles or quadrilaterals.
  • A054515 (program): Number of ways to place non-intersecting diagonals in convex (n+2)-gon so as to create no quadrilaterals.
  • A054516 (program): Equivalent of the Kurepa hypothesis for left factorial.
  • A054519 (program): Number of increasing arithmetic progressions of nonnegative integers ending in n, including those of length 1 or 2.
  • A054521 (program): Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).
  • A054522 (program): Triangle T(n,k): T(n,k) = phi(k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n). T(n,k) = number of elements of order k in cyclic group of order n.
  • A054523 (program): Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).
  • A054524 (program): Triangle T(n,k): T(n,k) = mu(k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n).
  • A054525 (program): Triangle T(n,k): T(n,k) = mu(n/k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).
  • A054526 (program): Triangle T(n,k): T(n,k) = phi(k) (n >= 1, 1 <= k <= n).
  • A054527 (program): Triangle read by rows: T(n,k) = Moebius mu(k) (n >= 1, 1 <= k <= n).
  • A054531 (program): Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).
  • A054536 (program): Maximal size of binary code of length n and asymmetric distance 4.
  • A054538 (program): A000013 / 2.
  • A054539 (program): A000016 / 2.
  • A054541 (program): Sum of first n terms equals n-th prime.
  • A054542 (program): A Catalan-like sequence.
  • A054545 (program): Number of labeled digraphs on n unisolated nodes (inverse binomial transform of A053763).
  • A054546 (program): First differences of nonprimes (including 0 and 1, A002808).
  • A054549 (program): Number of symmetric nonnegative integer 9 X 9 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.
  • A054552 (program): a(n) = 4*n^2 - 3*n + 1.
  • A054554 (program): a(n) = 4n^2 - 10n + 7.
  • A054556 (program): a(n) = 4*n^2 - 9*n + 6.
  • A054557 (program): Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 10 1-simplexes.
  • A054558 (program): Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 9 1-simplexes.
  • A054559 (program): Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.
  • A054563 (program): a(n) = n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72.
  • A054564 (program): Prime number spiral (clockwise, Southeast spoke).
  • A054565 (program): Numbers n such that 3 is the first digit of 3^n.
  • A054567 (program): a(n) = 4*n^2 - 7*n + 4.
  • A054569 (program): a(n) = 4*n^2 - 6*n + 3.
  • A054571 (program): a(n) = phi(n - phi(n)), a(1) = 0.
  • A054576 (program): Largest proper factor of the largest proper factor of n.
  • A054577 (program): A Catalan-like sequence.
  • A054579 (program): n^2+n modulo 17.
  • A054580 (program): n^2 modulo 17.
  • A054582 (program): Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.
  • A054584 (program): Number of subgroups of the group generated by a^n=1, b^3=1 and ab=ba.
  • A054585 (program): Sum_{d=1..n} phi(d)*mu(d).
  • A054586 (program): Sum_{d|2n+1} phi(d)*mu(d).
  • A054598 (program): a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).
  • A054599 (program): a(n) = Sum_{d|n} d*2^(n/d - 1).
  • A054600 (program): Sum_{d|n, d odd} d*2^(n/d).
  • A054601 (program): a(n) = Sum_{d|n, d odd} d*2^(n/d - 1), a(0)=0.
  • A054602 (program): a(n) = Sum_{d|3} phi(d)*n^(3/d).
  • A054603 (program): a(n) = Sum_{d|4} phi(d)*n^(4/d).
  • A054604 (program): a(n) = Sum_{d|5} phi(d)*n^(5/d).
  • A054605 (program): a(n) = Sum_{d|6} phi(d)*n^(6/d).
  • A054606 (program): a(n) = Sum_{d|7} phi(d)*n^(7/d).
  • A054607 (program): a(n) = Sum_{d|8} phi(d)*n^(8/d).
  • A054608 (program): a(n) = Sum_{d|9} phi(d)*n^(9/d).
  • A054609 (program): a(n) = Sum_{d|10} phi(d)*n^(10/d).
  • A054610 (program): a(n) = Sum_{d|n} phi(d)*3^(n/d).
  • A054611 (program): a(n) = Sum_{d|n} phi(d)*4^(n/d).
  • A054612 (program): a(n) = Sum_{d|n} phi(d)*5^(n/d).
  • A054613 (program): a(n) = Sum_{d|n} phi(d)*6^(n/d).
  • A054614 (program): a(n) = Sum_{d|n} phi(d)*7^(n/d).
  • A054615 (program): a(n) = Sum_{d|n} phi(d)*8^(n/d).
  • A054616 (program): a(n) = Sum_{d|n} phi(d)*9^(n/d).
  • A054617 (program): a(n) = Sum_{d|n} phi(d)*10^(n/d).
  • A054620 (program): Number of ways to color vertices of a pentagon using <= n colors, allowing only rotations.
  • A054621 (program): Number of ways to color vertices of a heptagon using <= n colors, allowing only rotations.
  • A054622 (program): Number of ways to color vertices of an octagon using <= n colors, allowing only rotations.
  • A054623 (program): Number of ways to color vertices of a 9-gon using <= n colors, allowing only rotations.
  • A054624 (program): Number of ways to color vertices of a 10-gon using <= n colors, allowing only rotations.
  • A054625 (program): Number of n-bead necklaces with 6 colors.
  • A054626 (program): Number of n-bead necklaces with 7 colors.
  • A054627 (program): Number of n-bead necklaces with 8 colors.
  • A054628 (program): Number of n-bead necklaces with 9 colors.
  • A054629 (program): Number of n-bead necklaces with 10 colors.
  • A054638 (program): 0 if pronounced name of n-th letter of English alphabet begin with a vowel sound, otherwise 1. Different from A074322.
  • A054639 (program): Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, …, n} -> {n, 1, n-1, 2, n-2, 3, …} is of order n.
  • A054640 (program): a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).
  • A054641 (program): GCD of divisor-sum of primorials and primorials itself: a(n) = gcd(A002110(n), A000203(A002110(n))).
  • A054644 (program): Number of labeled pure 2-complexes on n nodes with 3 2-simplexes.
  • A054650 (program): Nearest integer to 2^(n-1)/n.
  • A054652 (program): Acyclic orientations of the Hamming graph (K_2) x (K_n).
  • A054668 (program): Number of distinct non-extendable sequences X={x(1),x(2),…,x(k)} where x(1)=1, the x(i)’s are distinct elements of {1,…,n} with |x(i)-x(i+1)|=1 or 2, for i=1,2,…,k.
  • A054683 (program): Numbers whose sum of digits is even.
  • A054684 (program): Numbers whose sum of digits is odd.
  • A054685 (program): Number of partitions of n into distinct prime powers (1 not considered a power).
  • A054686 (program): Multiset consisting of squares and triangular numbers.
  • A054688 (program): Number of nonnegative integer n X n matrices with sum of elements equal to n; polynomial symmetric functions of matrix of order n.
  • A054703 (program): Number of distinct powers of 2 modulo n.
  • A054705 (program): Number of powers of 4 modulo n.
  • A054711 (program): Multiplicative order of 11 mod n, where gcd(n, 11) = 1.
  • A054718 (program): Number of ternary sequences with primitive period n.
  • A054719 (program): Number of 4-ary sequences with primitive period n.
  • A054720 (program): Number of 5-ary sequences with primitive period n.
  • A054721 (program): Number of 6-ary sequences with primitive period n.
  • A054725 (program): a(1)=1; a(n) = Sum_{p | n} e * a(p-1), where sum is over all primes p that divide n, and e is the multiplicity of p in n.
  • A054726 (program): Number of graphs with n nodes on a circle without crossing edges.
  • A054727 (program): Number of forests of rooted trees with n nodes on a circle without crossing edges.
  • A054735 (program): Sums of twin prime pairs.
  • A054740 (program): Cototient(n)/totient(n) when this is an integer.
  • A054741 (program): Numbers m such that totient(m) < cototient(m).
  • A054753 (program): Numbers which are the product of a prime and the square of a different prime (p^2 * q).
  • A054763 (program): Residues of consecutive prime differences modulo 6.
  • A054765 (program): a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 0, a(1) = 1.
  • A054766 (program): a(n+2) = (2*n + 3)*a(n+1) + (n + 1)^2*a(n), a(0) = 1, a(1) = 0.
  • A054768 (program): a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n).
  • A054770 (program): Numbers that are not the sum of distinct Lucas numbers 1,3,4,7,11, … (A000204).
  • A054775 (program): Positive multiples of 6 which are not the midpoint of a pair of twin primes.
  • A054776 (program): a(n) = 3*n*(3*n-1)*(3*n-2).
  • A054777 (program): a(n) = 4*n*(4*n-1)*(4*n-2)*(4*n-3).
  • A054778 (program): 5n*(5n-1)*(5n-2)*(5n-3)*(5n-4).
  • A054779 (program): 6n*(6n-1)*(6n-2)*(6n-3)*(6n-4)*(6n-5).
  • A054780 (program): Number of n-covers of a labeled n-set.
  • A054783 (program): (n^2)-th Fibonacci number.
  • A054785 (program): a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.
  • A054794 (program): Numbers of form 5+n^2+n or 5+2*n^2.
  • A054795 (program): Numbers of form 23+n^2+n or 23+2*n^2.
  • A054796 (program): Numbers of form 17+n^2+n or 17+2*n^2.
  • A054841 (program): If n = 2^a * 3^b * 5^c * 7^d * … then a(n) = a + 10 * b + 100 * c + 1000 * d + … .
  • A054843 (program): Number of sequences of consecutive nonnegative integers (including sequences of length 1) that sum to n.
  • A054844 (program): Number of ways to write n as the sum of any number of consecutive integers (including the trivial one-term sum n = n).
  • A054846 (program): Nearest integer to n^(2/3).
  • A054849 (program): a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.
  • A054850 (program): Binary logarithm of n-th primorial, rounded down to an integer.
  • A054851 (program): a(n) = 2^(n-7)*binomial(n,7). Number of 7D hypercubes in an n-dimensional hypercube.
  • A054854 (program): Number of ways to tile a 4 X n region with 1 X 1 and 2 X 2 tiles.
  • A054855 (program): Number of ways to tile a 5 X n area with 1 X 1 and 2 X 2 tiles.
  • A054856 (program): Number of ways to tile a 4 X n region with 1 X 1, 2 X 2, 3 X 3 and 4 X 4 tiles.
  • A054861 (program): Highest power of 3 dividing n!.
  • A054868 (program): Sum of bits of sum of bits of n: a(n) = wt(wt(n)).
  • A054869 (program): Digits of an idempotent 6-adic number.
  • A054872 (program): Number of (12345, 13245, 21345, 23145, 31245, 32145)-avoiding permutations.
  • A054877 (program): Closed walks of length n along the edges of a pentagon based at a vertex.
  • A054878 (program): Number of closed walks of length n along the edges of a tetrahedron based at a vertex.
  • A054879 (program): Closed walks of length 2n along the edges of a cube based at a vertex.
  • A054880 (program): a(n) = 3*(9^n - 1)/4.
  • A054881 (program): Number of walks of length n along the edges of an octahedron starting and ending at a vertex and also ( with a(0)=0 ) between two opposite vertices.
  • A054883 (program): Number of walks of length n along the edges of a dodecahedron between two opposite vertices.
  • A054884 (program): Number of closed walks of length n along the edges of an icosahedron based at a vertex.
  • A054885 (program): Number of walks of length n along the edges of an icosahedron between two opposite vertices.
  • A054886 (program): Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation).
  • A054888 (program): Layer counting sequence for hyperbolic tessellation by regular pentagons of angle Pi/2.
  • A054889 (program): Layer counting sequence for hyperbolic tessellation by regular pentagons of angle 2*Pi/5.
  • A054890 (program): Layer counting sequence for hyperbolic tessellation by regular heptagons of angle Pi/3.
  • A054893 (program): a(n) = floor(n/4) + floor(n/16) + floor(n/64) + floor(n/256) + ….
  • A054894 (program): a(n+1) = 4*a(n) + 4*a(n-1) - 4*a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=11, a(4)=48.
  • A054895 (program): a(n) = Sum_{k>0} floor(n/6^k).
  • A054896 (program): a(n) = Sum_{k>0} floor(n/7^k).
  • A054897 (program): a(n) = Sum_{k>0} floor(n/8^k).
  • A054898 (program): a(n) = Sum_{k>0} floor(n/9^k).
  • A054899 (program): a(n) = Sum_{k>0} floor(n/10^k).
  • A054900 (program): (n) = floor(n/16) + floor(n/256) + floor(n/4096) + floor(n/65536) + ….
  • A054912 (program): Expansion of e.g.f.: sqrt(exp(5*x)/(2-exp(x))).
  • A054925 (program): a(n) = ceiling(n*(n-1)/4).
  • A054961 (program): Maximal number of binary vectors of length n such that the unions (or bitwise ORs) of any 2 distinct vectors are all distinct.
  • A054963 (program): Number of cells in the first column of all directed column-convex polyominoes of area n+1.
  • A054964 (program): Numbers whose divisors have the form m^k + 1, k>1.
  • A054965 (program): Beatty sequence for log_3(10), i.e., for 1/log_10(3); so largest exponent of 3 which produces an n-digit decimal number.
  • A054966 (program): Numbers that are congruent to {0, 1, 8} mod 9.
  • A054967 (program): Numbers that are congruent to {0, 1, 9} mod 10.
  • A054968 (program): 3*Fibonacci(n) - 11.
  • A054972 (program): Product of (sum of first n primes) and (product of first n primes).
  • A054973 (program): Number of numbers whose divisors sum to n.
  • A054974 (program): Number of nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.
  • A054977 (program): a(0)=2, a(n)=1, n >= 1.
  • A054995 (program): A version of Josephus problem: a(n) is the surviving integer under the following elimination process. Arrange 1,2,3,…,n in a circle, increasing clockwise. Starting with i=1, delete the integer two places clockwise from i. Repeat, counting two places from the next undeleted integer, until only one integer remains.
  • A055003 (program): a(n) = prime(prime(n)-1).
  • A055004 (program): Boris Stechkin’s function.
  • A055010 (program): a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.
  • A055012 (program): Sum of cubes of the digits of n written in base 10.
  • A055013 (program): Sum of 4th powers of digits of n.
  • A055014 (program): Sum of 5th powers of digits of n.
  • A055015 (program): Sum of 6th powers of digits of n.
  • A055017 (program): Difference between sums of alternate digits of n starting with the last, i.e., (Sum of ultimate digit of n, antepenultimate digit of n,…)-(sum of penultimate digit of n, preantepenultimate digit of n,…).
  • A055022 (program): Number of 1-punctured staircase polygons (by perimeter) with a hole of perimeter 4.
  • A055023 (program): a(n) = n/A055032(n).
  • A055025 (program): Norms of Gaussian primes.
  • A055026 (program): Number of Gaussian primes of successive norms (indexed by A055025).
  • A055027 (program): Number of inequivalent Gaussian primes of successive norms (indexed by A055025).
  • A055028 (program): Number of Gaussian primes of norm n.
  • A055029 (program): Number of inequivalent Gaussian primes of norm n.
  • A055030 (program): (Sum(m^(p-1),m=1..p-1)+1)/p as p runs through the primes.
  • A055032 (program): Denominator of (Sum(m^(n-1),m=1..n-1)+1)/n.
  • A055033 (program): usigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448).
  • A055034 (program): a(1) = 1, a(n) = phi(2*n)/2 for n>1.
  • A055035 (program): Degree of minimal polynomial of sin(Pi/n) over the rationals.
  • A055037 (program): Number of numbers <= n with an even number of prime factors (counted with multiplicity).
  • A055038 (program): Number of numbers <= n with an odd number of prime factors (counted with multiplicity).
  • A055039 (program): Numbers of the form 2^(2i+1)*(8j+7).
  • A055040 (program): Numbers of the form 3^(2i+1)*(3*j+2).
  • A055041 (program): Numbers of the form 3^(2i+1)*(3*j+1).
  • A055042 (program): Numbers of the form 2^(2i+1)*(8*j+5).
  • A055043 (program): Numbers of the form 2^(2i+1)*(8*j+3).
  • A055044 (program): Numbers of the form 2^(2i+1)*(8*j+1).
  • A055045 (program): Numbers of the form 4^i*(8*j+5).
  • A055046 (program): Numbers of the form 4^i*(8*j+3).
  • A055047 (program): Numbers of the form 9^i*(3*j+1).
  • A055048 (program): Numbers of the form 9^i*(3*j+2).
  • A055050 (program): Numbers of the form 4^i*(8*j+3) or 4^i*(8*j+7).
  • A055052 (program): Numbers of the form 4^i*(8j+7) or 4^i*(8j+5).
  • A055067 (program): Product of numbers < n which do not divide n (or 1 if no such numbers exist).
  • A055070 (program): Third column of triangle A055864.
  • A055071 (program): Largest square dividing n!.
  • A055076 (program): Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.
  • A055077 (program): Multiplicity of Max{gcd(d, n!/d)} when d runs over the divisors of n!.
  • A055081 (program): Number of positive integers whose harmonic mean with n is a positive integer.
  • A055086 (program): n appears 1+[n/2] times.
  • A055087 (program): Integers 0..n then 0..n then 0..n+1 then 0..n+1 etc.
  • A055096 (program): Triangle read by rows, sums of 2 distinct nonzero squares: T(n,k) = k^2+n^2, (n>=2, 1 <= k <= n-1)
  • A055097 (program): Number of divisors for each term in the triangle A055096. It is 2 for primes (all of the form 4k+1).
  • A055099 (program): Expansion of g.f.: (1 + x)/(1 - 3*x - 2*x^2).
  • A055101 (program): Expansion of square of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/… )))).
  • A055102 (program): Expansion of cube of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/… )))).
  • A055103 (program): Expansion of 4th power of continued fraction 1/ ( 1+q/ ( 1+q^2/ ( 1+q^3/ ( 1+q^4/… )))).
  • A055104 (program): Expansion of 1 + q/((1-q)*(1-q^2)) + q^2/((1-q)*(1-q^2)*(1-q^3)*(1-q^4)).
  • A055112 (program): a(n) = n*(n+1)*(2*n+1).
  • A055113 (program): Number of bracketings of 0^0^0^…^0, with n 0’s, giving the result 0, with conventions that 0^0 = 1^0 = 1^1 = 1, 0^1 = 0.
  • A055115 (program): Base-5 complement of n (write n in base 5, then replace each digit with its base-5 negative).
  • A055116 (program): Base-6 complement of n (write n in base 6, then replace each digit with its base-6 negative).
  • A055117 (program): Base-7 complement of n (write n in base 7, then replace each digit with its base-7 negative).
  • A055118 (program): Base-8 complement of n (write n in base 8, then replace each digit with its base-8 negative).
  • A055119 (program): Base-9 complement of n (write n in base 9, then replace each digit with its base-9 negative).
  • A055120 (program): Digital complement of n (replace each nonzero digit d with 10-d).
  • A055121 (program): Base-11 complement of n (write n in base 11, then replace each digit with its base-11 negative).
  • A055122 (program): Base-12 complement of n (write n in base 12, then replace each digit with its base-12 negative).
  • A055123 (program): Base-13 complement of n (write n in base 13, then replace each digit with its base-13 negative).
  • A055124 (program): Base-14 complement of n (write n in base 14, then replace each digit with its base-14 negative).
  • A055125 (program): Base-15 complement of n (write n in base 15, then replace each digit with its base-15 negative).
  • A055126 (program): Base-16 complement of n (write n in base 16, then replace each digit with its base-16 negative).
  • A055129 (program): Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1’s.
  • A055131 (program): Those composite s for which A055095[s] = 2.
  • A055132 (program): Moebius function (A008683) applied to each term in the triangle A055096.
  • A055134 (program): Triangle read by rows: T(n,k) = number of labeled endofunctions on n points with k fixed points.
  • A055135 (program): Matrix inverse of triangle A055134.
  • A055136 (program): Triangle: a(n,k) = A055135(n,k)/C(n,k).
  • A055137 (program): Regard triangle of rencontres numbers (see A008290) as infinite matrix, compute inverse, read by rows.
  • A055140 (program): Triangle: matchings of 2n people with partners (of either sex) such that exactly k couples are left together.
  • A055141 (program): Matrix inverse of triangle A055140.
  • A055142 (program): E.g.f.: exp(x)*sqrt(1-2x).
  • A055151 (program): Triangular array of Motzkin polynomial coefficients.
  • A055155 (program): a(n) = Sum_{d|n} gcd(d, n/d).
  • A055156 (program): Powers of 3 which are not powers of 3^3.
  • A055197 (program): Numbers k such that A005728(k) is not prime.
  • A055203 (program): Number of different relations between n intervals on a line.
  • A055204 (program): Squarefree part of n!: n! divided by its largest square divisor.
  • A055205 (program): Number of nonsquare divisors of n^2.
  • A055208 (program): Table read by ascending antidiagonals: T(n,k) (n >= 1, k >= 1) is the sum of k-th powers of digits of n.
  • A055209 (program): a(n) = Product_{i=0..n} i!^2.
  • A055210 (program): Sum of totients of square divisors of n.
  • A055212 (program): Number of composite divisors of n.
  • A055214 (program): a(0) = 1; a(n) = 2*n*a(n-1) - 1 for n >= 1.
  • A055216 (program): Triangle T(n,k) by rows, n >= 0, 0<=k<=n: T(n,k) = Sum_{i=0..n-k} binomial(n-k,i) *Sum_{j=0..k-i} binomial(i,j).
  • A055217 (program): a(n) = sum of the first n coefficients of (1+x+x^2)^n.
  • A055218 (program): a(n) = T(2*n+2,n), array T as in A055216.
  • A055219 (program): T(2n+3,n), array T as in A055216.
  • A055220 (program): T(2n+4,n), array T as in A055216.
  • A055221 (program): T(2n+5,n), array T as in A055216.
  • A055222 (program): T(2n+6,n), array T as in A055216.
  • A055223 (program): One-fourth the digital sum of base 5 representations of 2^n.
  • A055225 (program): a(n) = Sum_{k divides n} (n/k)^k.
  • A055226 (program): a(n) = floor(sqrt(n!)).
  • A055227 (program): Nearest integer to sqrt( n! ).
  • A055228 (program): a(n) = ceiling(sqrt(n!)).
  • A055229 (program): Greatest common divisor of largest square dividing n and squarefree part of n.
  • A055230 (program): Greatest common divisor of largest square dividing n! and squarefree part of n!.
  • A055231 (program): Powerfree part of n: product of primes that divide n only once.
  • A055232 (program): Expansion of (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).
  • A055235 (program): Sums of two powers of 3.
  • A055236 (program): Sums of two powers of 4.
  • A055237 (program): Sums of two powers of 5.
  • A055243 (program): First differences of A001628 (Fibonacci convolution).
  • A055244 (program): Number of certain stackings of n+1 squares on a double staircase.
  • A055245 (program): Numerator sequence of mean length of certain stackings of n+1 squares on a double staircase.
  • A055246 (program): At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).
  • A055247 (program): Related to A055246 and A005836. Used for boundaries of open intervals which have to be erased in the Cantor middle third set construction.
  • A055248 (program): Triangle of partial row sums of triangle A007318(n,m) (Pascal’s triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)).
  • A055249 (program): Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal’s triangle A007318).
  • A055250 (program): Seventh column of triangle A055249.
  • A055251 (program): Eighth column of triangle A055249.
  • A055252 (program): Triangle of partial row sums (prs) of triangle A055249.
  • A055253 (program): Number of even digits in 2^n.
  • A055254 (program): Number of odd digits in 2^n.
  • A055255 (program): Number of even digits in 3^n.
  • A055256 (program): Number of odd digits in 3^n.
  • A055257 (program): Sums of two powers of 6.
  • A055258 (program): Sums of two powers of 7.
  • A055259 (program): Sums of two powers of 8.
  • A055260 (program): Sums of two powers of 9.
  • A055261 (program): Sums of two powers of 16.
  • A055262 (program): n + sum of digits of a(n-1).
  • A055263 (program): a(n) = Sum of digits of (n + a(n-1)).
  • A055264 (program): Possible values of A055263; numbers equal to 0, 1, 3 or 6 modulo 9.
  • A055267 (program): a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=7.
  • A055268 (program): a(n) = (11*n + 4)*C(n+3, 3)/4.
  • A055269 (program): a(n) = 4*a(n-1) - a(n-2) + 3 with a(0)=1, a(1)=7.
  • A055270 (program): a(n) = 7*a(n-1) + (-1)^n * binomial(2,2-n) with a(-1)=0.
  • A055271 (program): a(n) = 5*a(n-1) - a(n-2) with a(0)=1, a(1)=7.
  • A055272 (program): First differences of 7^n (A000420).
  • A055273 (program): a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 8.
  • A055274 (program): First differences of 8^n (A001018).
  • A055275 (program): First differences of 9^n (A001019).
  • A055276 (program): First differences of 11^n (A001020).
  • A055278 (program): Number of rooted trees with n nodes and 3 leaves.
  • A055303 (program): Number of labeled rooted trees with n nodes and 2 leaves.
  • A055304 (program): Number of labeled rooted trees with n nodes and 3 leaves.
  • A055305 (program): Number of labeled rooted trees with n nodes and 4 leaves.
  • A055306 (program): Number of labeled rooted trees with n nodes and 5 leaves.
  • A055307 (program): Number of labeled rooted trees with n nodes and 6 leaves.
  • A055308 (program): Number of labeled rooted trees with n nodes and 7 leaves.
  • A055315 (program): Number of labeled trees with n nodes and 3 leaves.
  • A055316 (program): Number of labeled trees with n nodes and 4 leaves.
  • A055317 (program): Number of labeled trees with n nodes and 5 leaves.
  • A055318 (program): Number of labeled trees with n nodes and 6 leaves.
  • A055319 (program): Number of labeled trees with n nodes and 7 leaves.
  • A055320 (program): Number of labeled trees with n nodes and 8 leaves.
  • A055328 (program): Number of rooted identity trees with n nodes and 3 leaves.
  • A055341 (program): Number of mobiles (circular rooted trees) with n nodes and 3 leaves.
  • A055350 (program): Number of labeled mobiles (circular rooted trees) with n nodes and 3 leaves.
  • A055351 (program): Number of labeled mobiles (circular rooted trees) with n nodes and 4 leaves.
  • A055352 (program): Number of labeled mobiles (circular rooted trees) with n nodes and 5 leaves.
  • A055357 (program): Number of increasing mobiles (circular rooted trees) with n nodes and 3 leaves.
  • A055364 (program): Number of asymmetric mobiles (circular rooted trees) with n nodes and 3 leaves.
  • A055372 (program): Invert transform of Pascal’s triangle A007318.
  • A055373 (program): Invert transform applied twice to Pascal’s triangle A007318.
  • A055374 (program): Invert transform applied three times to Pascal’s triangle A007318.
  • A055377 (program): a(n) = largest prime <= n/2.
  • A055388 (program): Number of riffle shuffles of 2n cards required to return the deck to initial state.
  • A055389 (program): a(0) = 1, then twice the Fibonacci sequence.
  • A055392 (program): Number of bracketings of 0#0#0#…#0 giving result 0, where 0#0 = 1, 0#1 = 1#0 = 1#1 = 0.
  • A055394 (program): Numbers that are the sum of a positive square and a positive cube.
  • A055396 (program): Smallest prime dividing n is a(n)-th prime (a(1)=0).
  • A055398 (program): Result of fourth stage of sieve of Eratosthenes (after eliminating multiples of 2, 3, 5, 7).
  • A055400 (program): Cube excess: difference between n and largest cube <= n.
  • A055401 (program): Number of positive cubes needed to sum to n using the greedy algorithm.
  • A055410 (program): Number of points in Z^4 of norm <= n.
  • A055417 (program): Number of points in N^n of norm <= 2.
  • A055426 (program): Number of points in Z^n of norm <= 2.
  • A055436 (program): a(n) = concatenation of n^2 and n.
  • A055437 (program): a(n) = 10*n^2+n.
  • A055438 (program): a(n) = 100*n^2 + n.
  • A055443 (program): Base 3 distribution of first digit of mantissa following Benford’s Law, to minimize chi-squared statistic.
  • A055457 (program): 5^a(n) exactly divides 5n. Or, 5-adic valuation of 5n.
  • A055460 (program): Number of primes with odd exponents in the prime power factorization of n!.
  • A055461 (program): Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.
  • A055462 (program): Superduperfactorials: product of first n superfactorials.
  • A055463 (program): a(n) = a(n-1)*2*a(n-2)-3*a(n-3).
  • A055469 (program): Primes of the form k(k+1)/2+1 (i.e., central polygonal numbers, or one more than triangular numbers).
  • A055472 (program): Primes of the form k(k+1)/2+2 (i.e., two more than a triangular number).
  • A055473 (program): Powers of ten written in base 2.
  • A055474 (program): Powers of ten written in base 3.
  • A055475 (program): Powers of ten written in base 4.
  • A055476 (program): Powers of ten written in base 5.
  • A055477 (program): Powers of ten written in base 6.
  • A055478 (program): Powers of ten written in base 7.
  • A055479 (program): Powers of ten written in base 9.
  • A055483 (program): a(n) is the GCD of n and the reverse of n.
  • A055491 (program): Smallest square divisible by n divided by largest square which divides n.
  • A055494 (program): Numbers k such that k^2 - k + 1 is prime.
  • A055495 (program): Numbers n such that there exists a pair of mutually orthogonal Latin squares of order n.
  • A055497 (program): a(-1) = 4, a(0) = 5; thereafter a(n) = 4 + (Product_{k=1..n} prime(k))^2.
  • A055502 (program): a(0)=0, a(1)=2, a(n) = smallest prime > a(n-1)+a(n-2).
  • A055503 (program): Take n points in general position in the plane; draw all the (infinite) straight lines joining them; sequence gives number of connected regions formed.
  • A055504 (program): n*(n-1)*(n-2)*(n-3)*(n-4)*(2*n-1)/72.
  • A055507 (program): a(n) = Sum_{k=1..n} d(k)*d(n+1-k), where d(k) is number of positive divisors of k.
  • A055518 (program): a_{k+1} = 6*a_k + 11*a_{k-1} - 19*a_{k-2} - 4*a_{k-3} + a_{k-4}, a_1=1, a_2=2, a_3=19, a_4=118, a_5=875.
  • A055519 (program): a(n) = 9*a(n-1) + 33*a(n-2) - 76*a(n-3) - 33*a(n-4) + 9*a(n-5) + a(n-6), a(0)=a(1)=1, a(2)=2, a(3)=35, a(4)=312, a(5)=3779.
  • A055522 (program): Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).
  • A055523 (program): Longest other leg of a Pythagorean triangle with n as length of a leg.
  • A055524 (program): Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).
  • A055526 (program): Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.
  • A055527 (program): Shortest other leg of a Pythagorean triangle with n as length of a leg.
  • A055528 (program): a(n)=10*a(n-1)+n^3, a(0)=0.
  • A055529 (program): Number of bits needed in mantissa to express n! exactly.
  • A055530 (program): The recurrence b(k) = 10*b(k-1) + k^n with b(0)=0 has b(k)/10^k converging to a(n)/9^(n+1).
  • A055531 (program): Number of labeled order relations on n nodes in which longest chain has 2 nodes.
  • A055533 (program): Number of labeled order relations on n nodes in which longest chain has n-1 nodes.
  • A055541 (program): Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.
  • A055546 (program): a(n) = (-1)^(n+1) * 2^n * n!^2.
  • A055554 (program): An arithmetic progression each term of which is followed by at least 4 nonsquarefree consecutive integers.
  • A055555 (program): a(n) = n!*(n!+1)/2.
  • A055562 (program): a(n) = least number greater than a(n-1) not the sum of an earlier pair of consecutive terms, a(0) = 2.
  • A055563 (program): a(n) = least number greater than a(n-1) not the sum of an earlier pair of consecutive terms, a(0) = 3.
  • A055565 (program): Sum of digits of n^4.
  • A055566 (program): Sum of digits of n^5.
  • A055567 (program): Sum of digits of n^6.
  • A055568 (program): Numbers not greater than the sum of digits of their squares.
  • A055569 (program): Sum of digits of a(n)^3 is greater than or equal to a(n).
  • A055579 (program): a(n) = binomial(12*n-1,3*n-1)/((6*n-1)*(12*n-1)).
  • A055580 (program): Björner-Welker sequence: 2^n*(n^2 + n + 2) - 1.
  • A055581 (program): Fifth column of triangle A055252.
  • A055582 (program): Sixth column of triangle A055252.
  • A055583 (program): Seventh column of triangle A055252.
  • A055584 (program): Triangle of partial row sums (prs) of triangle A055252.
  • A055585 (program): Second column of triangle A055584.
  • A055586 (program): Sixth column of triangle A055584.
  • A055587 (program): Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal’s triangle A007318. Essentially A049600 formatted differently.
  • A055588 (program): a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.
  • A055589 (program): Convolution of A049612 with A011782.
  • A055596 (program): Expansion of e.g.f. (2 - x - 2*exp(-x))/(1-x).
  • A055597 (program): Powers of 2 in phi(n!).
  • A055600 (program): Numbers of form 2^i*3^j+1 with i, j >= 0.
  • A055601 (program): Number of n X n binary matrices with no zero rows.
  • A055602 (program): Number of n X n binary matrices with no 0 rows or columns and with n+1 1’s.
  • A055607 (program): a(2n+1) = n^2 - 1 + A002620(n), a(2n) = a(2n-1) + n.
  • A055610 (program): A companion sequence to A011896.
  • A055612 (program): a(n) = Product_{m=1..n} (binomial(n,m)+1).
  • A055613 (program): n!*LaguerreL(n,4,-8).
  • A055615 (program): a(n) = n * mu(n), where mu is the Möbius function A008683.
  • A055620 (program): Digits of an idempotent 6-adic number.
  • A055630 (program): Table T(k,m) = k^2 + m read by antidiagonals.
  • A055631 (program): Sum of Euler’s totient function phi of distinct primes dividing n.
  • A055632 (program): Sum of totient function of primes dividing n is a prime.
  • A055634 (program): 2-adic factorial function.
  • A055636 (program): Partial sums of A144494.
  • A055637 (program): (n-1)!/n or 0 if n does not divide (n-1)!..
  • A055640 (program): Number of nonzero digits in decimal expansion of n.
  • A055641 (program): Number of zero digits in n.
  • A055642 (program): Number of digits in the decimal expansion of n.
  • A055646 (program): Integers in base 15 with each base 15 digit represented by 2 decimal digits.
  • A055647 (program): Integers in base 14 with each base 14 digit represented by 2 decimal digits.
  • A055651 (program): Table T(m,k)=m^k-k^m (with 0^0 taken to be 1) as square array read by antidiagonals.
  • A055652 (program): Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.
  • A055653 (program): Sum of phi(d) [A000010] over all unitary divisors d of n (that is, gcd(d,n/d) = 1).
  • A055654 (program): Difference between n and the result of “Phi-summation” over unitary divisors of n.
  • A055658 (program): Number of (3,n)-partitions of a chain of length n^2.
  • A055659 (program): Number of (2,n)-partitions of a chain of length n^3.
  • A055660 (program): Number of (2,2; n,n)-partitions of a chain of length n^2 + n.
  • A055661 (program): Tower of Hanoi positions (A055662) converted from base 3 to base 10.
  • A055662 (program): Successive positions in Tower of Hanoi (with three pegs {0,1,2}) where xyz means smallest disk is on peg z, second smallest is on peg y, third smallest on peg x, etc. and leading zeros indicate largest disks are all on peg 0.
  • A055663 (program): Number of (3,3; n,n)-partitions of a chain of length n^2 + n.
  • A055664 (program): Norms of Eisenstein-Jacobi primes.
  • A055665 (program): Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).
  • A055666 (program): Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).
  • A055667 (program): Number of Eisenstein-Jacobi primes of norm n.
  • A055668 (program): Number of inequivalent Eisenstein-Jacobi primes of norm n.
  • A055669 (program): Number of prime Hurwitz quaternions of norm prime(n).
  • A055670 (program): a(n) = prime(n) - (-1)^prime(n).
  • A055671 (program): Number of prime Hurwitz quaternions of norm n.
  • A055672 (program): Number of right-inequivalent prime Hurwitz quaternions of norm n.
  • A055673 (program): Absolute values of norms of primes in ring of integers Z[sqrt(2)].
  • A055679 (program): Number of distinct prime factors of phi(n!).
  • A055682 (program): a(n) = floor(n*sqrt(n)) - sigma(n), where sigma(n) is the sum of the divisors of n (A000203).
  • A055684 (program): Number of different n-pointed stars.
  • A055719 (program): d(n)-1 | n and n is not prime.
  • A055720 (program): Numbers k such that d(k)+1 | k.
  • A055734 (program): Number of distinct primes dividing phi(n).
  • A055736 (program): Difference between number of prime factors of n and of phi(n).
  • A055741 (program): Phi(n) has more distinct prime factors than n.
  • A055742 (program): Numbers k such that k and EulerPhi(k) have same number of prime factors, counted without multiplicity.
  • A055743 (program): Phi(n) has fewer distinct prime divisors than n.
  • A055746 (program): Product of first n terms of A003046.
  • A055767 (program): Index (or subscript) of the largest primorial (A002110(k)) which divides EulerPhi of the n-th primorial ((A005867(n)).
  • A055768 (program): Number of distinct primes dividing phi of n-th primorial number.
  • A055769 (program): Largest prime dividing phi of the n-th primorial.
  • A055770 (program): Largest factorial number which divides n.
  • A055772 (program): Square root of largest square dividing n!.
  • A055773 (program): a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.
  • A055774 (program): Least common multiple of n! and n^n.
  • A055775 (program): a(n) = floor(n^n / n!).
  • A055778 (program): Number of 1’s in the base-phi representation of n.
  • A055779 (program): Number of fat trees on n labeled vertices.
  • A055781 (program): Primes q of the form q = 10p + 1, where p is also prime.
  • A055782 (program): Primes q of the form q = 10p + 3, where p is also prime.
  • A055783 (program): Primes q of form q=10p+7, where p is also prime.
  • A055784 (program): Primes q of form q = 10p + 9, where p is also prime.
  • A055786 (program): Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).
  • A055787 (program): a(n) = 2^(4*n-1) - 2^(2*n-1)*binomial(2*n,n).
  • A055789 (program): a(n) = binomial(n, round(sqrt(n))).
  • A055790 (program): a(n) = n*a(n-1) + (n-2)*a(n-2), a(0) = 0, a(1) = 2.
  • A055792 (program): a(n) and floor(a(n)/2) are both squares; i.e., squares which remain squares when written in base 2 and last digit is removed.
  • A055793 (program): Numbers n such that n and floor[n/3] are both squares; i.e., squares which remain squares when written in base 3 and last digit is removed.
  • A055794 (program): Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=1 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j) = T(i-1,j) + T(i-2,j-1) for 1<=j<=i-1.
  • A055795 (program): a(n) = binomial(n,4) + binomial(n,2).
  • A055796 (program): T(2n+3,n), array T as in A055794.
  • A055797 (program): T(2n+4,n), array T as in A055794.
  • A055798 (program): T(2n+5,n), array T as in A055794.
  • A055799 (program): T(2n+6,n), array T as in A055794.
  • A055802 (program): a(n) = T(n,n-2), array T as in A055801.
  • A055803 (program): a(n) = T(n,n-3), array T as in A055801.
  • A055804 (program): a(n) = T(n,n-4), array T as in A055801.
  • A055805 (program): a(n) = T(n,n-5), array T as in A055801.
  • A055806 (program): a(n) = T(n,n-6), array T as in A055801.
  • A055808 (program): a(n) and floor(a(n)/4) are both squares; i.e., squares that remain squares when written in base 4 and last digit is removed.
  • A055809 (program): a(n) = T(n,n-4), array T as in A055807.
  • A055810 (program): a(n) = T(n,n-5), array T as in A055807.
  • A055811 (program): a(n) = T(n,n-6), array T as in A055807.
  • A055812 (program): a(n) and floor(a(n)/5) are both squares; i.e., squares which remain squares when written in base 5 and last digit is removed.
  • A055814 (program): Expansion of e.g.f.: exp(x^3/3 + x^2/2).
  • A055815 (program): a(n) = T(2*n+3,n), array T as in A055807.
  • A055816 (program): a(n) = T(2*n+4,n), array T as in A055807.
  • A055817 (program): a(n) = T(2n+5,n), array T as in A055807.
  • A055819 (program): Row sums of array T in A055818; twice the odd-indexed Fibonacci numbers after initial term.
  • A055820 (program): a(n) = T(n,n-3), array T as in A055818.
  • A055821 (program): a(n) = T(n,n-4), array T as in A055818.
  • A055824 (program): a(n) = T(2*n,n), array T as in A055818.
  • A055825 (program): a(n) = T(2n+1,n), array T as in A055818.
  • A055826 (program): a(n) = T(2n+2,n), array T as in A055818.
  • A055830 (program): Triangle T read by rows: diagonal differences of triangle A037027.
  • A055831 (program): T(n,n-4), where T is the array in A055830.
  • A055832 (program): T(n,n-5), where T is the array in A055830.
  • A055833 (program): T(n,n-6), where T is the array in A055830.
  • A055834 (program): T(2n,n), where T is the array in A055830.
  • A055835 (program): T(2n+1,n), where T is the array in A055830.
  • A055836 (program): T(2n+2, n), where T is the array in A055830.
  • A055837 (program): T(2n+3,n), where T is the array in A055830.
  • A055838 (program): T(2n+4,n), where T is the array in A055830.
  • A055839 (program): T(2n+5,n), where T is the array in A055830.
  • A055840 (program): T(2n+6,n), where T is the array in A055830.
  • A055841 (program): Number of compositions of n into 3*j-1 kinds of j’s for all j >= 1.
  • A055842 (program): Expansion of (1-x)^2/(1-5*x).
  • A055843 (program): Expansion of (1+3*x)/(1-x)^10.
  • A055844 (program): a(n) = (5*n + 9)*binomial(n+8, 8)/9.
  • A055845 (program): a(n) = 4*a(n-1) - a(n-2) with a(0)=1, a(1)=8.
  • A055846 (program): a(n) = 25*6^(n-2), with a(0)=1 and a(1)=4.
  • A055847 (program): a(0)=1, a(1)=6, a(n)=49*8^(n-2) if n>=2.
  • A055848 (program): Expansion of (1+5*x)/(1-x)^10.
  • A055849 (program): a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=9.
  • A055850 (program): a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=10.
  • A055852 (program): Convolution of A055589 with A011782.
  • A055853 (program): Convolution of A055852 with A011782.
  • A055854 (program): Convolution of A055853 with A011782.
  • A055855 (program): Convolution of A055854 with A011782.
  • A055860 (program): a(n) = A000169(n+1) if n > 0; a(0) = 0.
  • A055861 (program): Essentially A053506 but with leading 0 (instead of 1) and offset 0.
  • A055862 (program): Fourth column of triangle A055858.
  • A055863 (program): Fifth column of triangle A055858.
  • A055864 (program): Coefficient triangle for certain polynomials.
  • A055865 (program): Second column of triangle A055864.
  • A055867 (program): Fourth column of triangle A055864.
  • A055868 (program): Fifth column of triangle A055864.
  • A055869 (program): a(n) = (n+1)^n - n^n.
  • A055872 (program): a(n) and floor(a(n)/8) are both squares; i.e., squares that remain squares when written in base 8 and last digit is removed.
  • A055874 (program): a(n) = largest m such that 1, 2, …, m divide n.
  • A055876 (program): a(n) = round( 1 + e^(n-2) ).
  • A055879 (program): Least nondecreasing sequence with a(1) = 1 and Hankel transform {1,1,1,1,…}.
  • A055880 (program): Quotients arising from sequence A053317.
  • A055881 (program): a(n) = largest m such that m! divides n.
  • A055882 (program): a(n) = 2^n*Bell(n). E.g.f.: exp(exp(2x)-1).
  • A055883 (program): Exponential transform of Pascal’s triangle A007318.
  • A055891 (program): CIK (necklace, indistinct, unlabeled) transform of powers of 2.
  • A055895 (program): Inverse Moebius transform of powers of 2.
  • A055897 (program): a(n) = n*(n-1)^(n-1).
  • A055899 (program): Column 3 of triangle A055898.
  • A055908 (program): Column 2 of triangle A055907.
  • A055928 (program): Sum of square divisors of n! = sum of squares of divisors of the square root of largest square dividing n!.
  • A055929 (program): EulerPhi of the factorial of prime(n).
  • A055930 (program): Number of distinct prime factors of totient of (n-th prime)!.
  • A055932 (program): Numbers all of whose prime divisors are consecutive primes starting at 2.
  • A055935 (program): a(0)=1; a(n) = Sum_{j<n, gcd(n,a(j)) = 1} a(j).
  • A055937 (program): a(n) = a(n-1) * a(n-2) - 1.
  • A055938 (program): Integers not generated by b(n) = b(floor(n/2)) + n (cf. A005187).
  • A055941 (program): a(n) = Sum_{j=0..k-1} (i(j) - j) where n = Sum_{j=0..k-1} 2^i(j).
  • A055944 (program): a(n) = n + (reversal of base-2 digits of n) (written in base 10).
  • A055945 (program): a(n) = n - (reversal of base-2 digits of n) (and then the result is written in base 10).
  • A055946 (program): n + reversal of base 3 digits of n (written in base 10).
  • A055947 (program): n - reversal of base 3 digits of n (written in base 10).
  • A055948 (program): n + reversal of base 4 digits of n (written in base 10).
  • A055949 (program): n - reversal of base 4 digits of n (written in base 10).
  • A055950 (program): a(n) = n + reversal of base 5 digits of n (written in base 10).
  • A055951 (program): n - reversal of base 5 digits of n (written in base 10).
  • A055952 (program): n + reversal of base 6 digits of n (written in base 10).
  • A055953 (program): n - reversal of base 6 digits of n (written in base 10).
  • A055954 (program): n + reversal of base 7 digits of n (written in base 10).
  • A055955 (program): a(n) = n - reversal of base 7 digits of n (written in base 10).
  • A055956 (program): n + reversal of base 8 digits of n (written in base 10).
  • A055957 (program): n - reversal of base 8 digits of n (written in base 10).
  • A055958 (program): a(n) = n + reversal of base 9 digits of n (written in base 10).
  • A055959 (program): n - reversal of base 9 digits of n (written in base 10).
  • A055960 (program): n + reversal of base 11 digits of n (written in base 10).
  • A055961 (program): a(n) = n - (reversal of base-11 digits of n) (written in base 10).
  • A055962 (program): n + reversal of base 12 digits of n (written in base 10).
  • A055963 (program): n - reversal of base 12 digits of n (written in base 10).
  • A055964 (program): n + reversal of hexadecimal (base 16) digits of n (written in base 10).
  • A055965 (program): n - reversal of hexadecimal (base 16) digits of n (written in base 10).
  • A055966 (program): n + reversal of base 20 digits of n (written in base 10).
  • A055975 (program): First differences of A003188 (decimal equivalent of the Gray Code).
  • A055976 (program): Remainder when (n-1)! + 1 is divided by n.
  • A055979 (program): Solutions (value of r) of the Diophantine equation 2*x^2 + 3*x + 2 = r^2.
  • A055980 (program): a(n) = floor(Sum_{i=1..n} 1/i).
  • A055981 (program): a(n) = ceiling(n!/d(n!)).
  • A055987 (program): a(n+1) = a(n) converted to base 10 from base 16.
  • A055988 (program): Sequence is its own 4th difference.
  • A055989 (program): a(n) is its own 4th difference.
  • A055990 (program): a(n) is its own 4th difference.
  • A055991 (program): a(n) is its own 4th difference.
  • A055993 (program): Number of square divisors of n!.
  • A055994 (program): Expansion of (1+6x)/(1-x)^10.
  • A055995 (program): a(n) = 64*9^(n-2), a(0)=1, a(1)=7.
  • A055996 (program): a(n) = 81*10^(n-2), a(0)=1, a(1)=8.
  • A055997 (program): Numbers n such that n(n - 1)/2 is a square.
  • A055998 (program): a(n) = n*(n+5)/2.
  • A055999 (program): a(n) = n*(n + 7)/2.
  • A056000 (program): a(n) = n*(n+9)/2.
  • A056001 (program): A second-order recursive sequence.
  • A056002 (program): a(n) = (10^2)*11^(n-2); a(0)=1, a(1)=9.
  • A056003 (program): A second-order recursive sequence.
  • A056005 (program): Number of 3-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 3 labeled nodes and n hyperedges.
  • A056007 (program): Difference between 2^n and largest square strictly less than 2^n.
  • A056010 (program): Number of words of length n in a simple grammar.
  • A056011 (program): Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing.
  • A056014 (program): a(n) = (Fibonacci(2n-1) - Fibonacci(n+1))/2.
  • A056015 (program): A recursive sequence.
  • A056016 (program): a(n) = -2*a(n - 1) -a(n - 2) -a(n - 3), a(0) = a(1) = a(2) = 1.
  • A056020 (program): Numbers that are congruent to +-1 mod 9.
  • A056021 (program): Numbers k such that k^4 == 1 (mod 5^2).
  • A056022 (program): Numbers k such that k^6 == 1 (mod 7^2).
  • A056023 (program): Unique triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are decreasing; (4) even-numbered rows are increasing; and (5) column 1 is increasing.
  • A056026 (program): Numbers k such that k^14 == 1 (mod 15^2).
  • A056038 (program): Largest factorial k! such that (k!)^2 divides n!.
  • A056039 (program): Largest k such that (k!)^2 divides n!.
  • A056040 (program): Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).
  • A056042 (program): a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.
  • A056043 (program): Let k be largest number such that k^2 divides n!; a(n) = k/floor(n/2)!.
  • A056044 (program): Let k be largest number such that k^2 divides n! and let m be largest number such that m! divides k; a(n) = k/m!.
  • A056045 (program): a(n) = Sum_{d|n} binomial(n,d).
  • A056051 (program): a(n) = (n-2)! - 1 (mod n).
  • A056058 (program): Squarefree part of the n-th central binomial coefficient.
  • A056061 (program): Number of square divisors of central binomial coefficients.
  • A056064 (program): The Kubelsky sequence: Jack Benny’s reported age, sampled annually.
  • A056067 (program): Numbers k such that k! is divisible by the square of (f+d)!^2 for d=0 and d=1 (and possibly larger d), where f = floor(k/2).
  • A056074 (program): Number of 3-element ordered antichain covers of an unlabeled n-element set.
  • A056078 (program): Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.
  • A056081 (program): Numbers that are congruent to {1, 26} mod 27.
  • A056082 (program): Numbers k such that k^4 == 1 (mod 5^3).
  • A056084 (program): Numbers k such that k^8 == 1 (mod 9^3).
  • A056096 (program): Maximum value in the distribution by first value of Prufer code of noncrossing spanning trees on a circle of n+2 points; perhaps the number whose Prufer code starts with 2.
  • A056100 (program): Sigma(n)*Phi(n) + 1 (Mod n).
  • A056105 (program): First spoke of a hexagonal spiral.
  • A056106 (program): Second spoke of a hexagonal spiral.
  • A056107 (program): Third spoke of a hexagonal spiral.
  • A056108 (program): Fourth spoke of a hexagonal spiral.
  • A056109 (program): Fifth spoke of a hexagonal spiral.
  • A056112 (program): a(1) = 1; a(m+1) = sum_{k=1 to m} [max(m, a(k))].
  • A056113 (program): Most significant digit of n-th primorial A002110.
  • A056114 (program): Expansion of (1+9*x)/(1-x)^11.
  • A056115 (program): a(n) = n*(n+11)/2.
  • A056116 (program): a(n) = 121*12^(n-2), a(0)=1, a(1)=10.
  • A056117 (program): Expansion of (1+8*x)/(1-x)^9.
  • A056118 (program): a(n) = (11*n+5)*(n+4)*(n+3)*(n+2)*(n+1)/120.
  • A056119 (program): a(n) = n*(n+13)/2.
  • A056120 (program): a(n) = (3^3)*4^(n-3) with a(0)=1, a(1)=1 and a(2)=7.
  • A056121 (program): a(n) = n*(n + 15)/2.
  • A056122 (program): a(n) = (8*n+9)*C(n+8,8)/9.
  • A056123 (program): a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=11.
  • A056124 (program): a(n) = 3*a(n-1) - a(n-2) + 8 with a(0)=1, a(1)=11.
  • A056125 (program): a(n) = (5*n + 4)*binomial(n+7,7)/4.
  • A056126 (program): a(n) = n*(n + 17)/2.
  • A056128 (program): a(n) = (9*n + 11)*binomial(n+10, 10)/11.
  • A056129 (program): Final nonzero digit of n-th primorial.
  • A056133 (program): a(1) = 1, a(m+1) = sum_{k=1 to m}[min(m, a(k))].
  • A056134 (program): Smallest positive integer which is the geometric mean of n and an integer other than n.
  • A056135 (program): Smallest positive integer other than n where geometric mean of n and a(n) is an integer.
  • A056136 (program): Largest positive integer whose harmonic mean with another positive integer is n.
  • A056138 (program): Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.
  • A056140 (program): a(n) = n^2 - previousprime(n)*nextprime(n), for n>2.
  • A056142 (program): Concatenate n, floor[n/10], floor[n/100] … (but do not continue if floor[.]=0).
  • A056143 (program): Concatenate … floor[n/100], floor[n/10], n.
  • A056144 (program): a(1) = 1, a(m+1) = Sum_{k=1..m} gcd(m, a(k)).
  • A056149 (program): a(0) = 1, a(m) = number of a(k), 0 <= k <= m-1, where gcd(m, a(k)) = 1.
  • A056150 (program): Number of combinations for each possible sum when throwing 3 (normal) dice.
  • A056155 (program): Positive integer k, 1 <= k <= n, which maximizes k^(n+1-k).
  • A056158 (program): Equivalent of the Kurepa hypothesis for left factorial.
  • A056159 (program): a(n)=floor[10^(n-1)/n].
  • A056161 (program): Solutions (value of x) of Diophantine equation 2*x^2 + 3*x + 2 = r^2.
  • A056162 (program): a(n) = Sum_{k=0..n} (k!)^(n-k).
  • A056167 (program): Numbers k such that k! is not divisible by the square of (f+1)!, where f = floor(k/2).
  • A056169 (program): Number of unitary prime divisors of n.
  • A056170 (program): Number of non-unitary prime divisors of n.
  • A056171 (program): a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.
  • A056172 (program): Number of non-unitary prime divisors of n!.
  • A056173 (program): Number of unitary prime divisors of central binomial coefficient C(n, floor(n/2)) (A001405).
  • A056174 (program): Number of non-monotone maps from 1,…,n to 1,…,n.
  • A056175 (program): Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).
  • A056182 (program): First differences of A003063.
  • A056188 (program): a(1) = 1; for n>1, sum of binomial(n,k) as k runs over RRS(n), the reduced residue system of n.
  • A056189 (program): a(n) = 2^n - A056188(n).
  • A056191 (program): Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(K*K) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2*L^2 and g divides F = gf.
  • A056192 (program): a(n) = n divided by its characteristic cube divisor A056191.
  • A056194 (program): Characteristic cube divisor of n!: a(n) = A056191(n!).
  • A056195 (program): a(n) = n! divided by its characteristic cube divisor A056194.
  • A056196 (program): Numbers n such that A055229(n) = 2.
  • A056197 (program): Diagonal of A056151.
  • A056199 (program): a(n) = n * a(n-1) - Sum_{k=1..n-2} a(k) with a(1) = 0 and a(2) = 1.
  • A056200 (program): a(n) = 2^n - A056045(n).
  • A056203 (program): Triangle of numbers related to congruum problem: T(n,k)=n^2+2nk-k^2 with n>k>0, starting at T(2,1)=7.
  • A056220 (program): a(n) = 2*n^2 - 1.
  • A056221 (program): Image of primes (A000040) under “little Hankel” transform that sends [c_0, c_1, …] to [d_0, d_1, …] where d_n = c_n^2 - c_{n+1}*c_{n-1}.
  • A056236 (program): a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.
  • A056237 (program): a(n) = 2*n^2 + 9*n - 5.
  • A056239 (program): If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} k*c_k.
  • A056241 (program): Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).
  • A056242 (program): Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,…,n} (1 <= k <= n).
  • A056243 (program): Third diagonal of triangle A056242.
  • A056267 (program): Number of primitive (aperiodic) words of length n which contain exactly two different symbols.
  • A056272 (program): Word structures of length n using a 5-ary alphabet.
  • A056273 (program): Word structures of length n using a 6-ary alphabet.
  • A056274 (program): Number of primitive (aperiodic) word structures of length n using a 3-ary alphabet.
  • A056278 (program): Number of primitive (aperiodic) word structures of length n which contain exactly two different symbols.
  • A056283 (program): Number of n-bead necklaces with exactly three different colored beads.
  • A056284 (program): Number of n-bead necklaces with exactly four different colored beads.
  • A056285 (program): Number of n-bead necklaces with exactly five different colored beads.
  • A056286 (program): Number of n-bead necklaces with exactly six different colored beads.
  • A056295 (program): Number of n-bead necklace structures using exactly two different colored beads.
  • A056303 (program): Number of primitive (period n) n-bead necklace structures using exactly two different colored beads.
  • A056308 (program): Number of reversible strings with n beads using a maximum of six different colors.
  • A056309 (program): Number of reversible strings with n beads using exactly two different colors.
  • A056310 (program): Number of reversible strings with n beads using exactly three different colors.
  • A056311 (program): Number of reversible strings with n beads using exactly four different colors.
  • A056312 (program): Number of reversible strings with n beads using exactly five different colors.
  • A056313 (program): Number of reversible strings with n beads using exactly six different colors.
  • A056323 (program): Number of reversible string structures with n beads using a maximum of four different colors.
  • A056326 (program): Number of reversible string structures with n beads using exactly two different colors.
  • A056342 (program): Number of bracelets of length n using exactly two different colored beads.
  • A056357 (program): Number of bracelet structures using exactly two different colored beads.
  • A056449 (program): a(n) = 3^floor((n+1)/2).
  • A056450 (program): a(n) = (3*2^n - (-2)^n)/2.
  • A056451 (program): Number of palindromes using a maximum of five different symbols.
  • A056452 (program): a(n) = 6^floor((n+1)/2).
  • A056453 (program): Number of palindromes of length n using exactly two different symbols.
  • A056454 (program): Number of palindromes of length n using exactly three different symbols.
  • A056455 (program): Palindromes using exactly four different symbols.
  • A056456 (program): Number of palindromes of length n using exactly five different symbols.
  • A056457 (program): Palindromes using exactly six different symbols.
  • A056468 (program): a(n) = Sum_{k=1..n} k^6*binomial(n,k).
  • A056469 (program): Number of elements in the continued fraction for Sum_{k=0..n} 1/2^2^k.
  • A056470 (program): Number of palindromic structures using a maximum of five different symbols.
  • A056471 (program): Number of palindromic structures using a maximum of six different symbols.
  • A056473 (program): Number of palindromic structures using exactly four different symbols.
  • A056474 (program): Number of palindromic structures using exactly five different symbols.
  • A056486 (program): a(n) = (9*2^n + (-2)^n)/4 for n>0.
  • A056487 (program): a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd.
  • A056488 (program): Number of periodic palindromes using a maximum of six different symbols.
  • A056489 (program): Number of periodic palindromes using exactly three different symbols.
  • A056490 (program): Number of periodic palindromes using exactly four different symbols.
  • A056491 (program): Number of periodic palindromes using exactly five different symbols.
  • A056503 (program): Number of periodic palindromic structures of length n using a maximum of two different symbols.
  • A056508 (program): Number of periodic palindromic structures of length n using exactly two different symbols.
  • A056520 (program): a(n) = (n + 2)*(2*n^2 - n + 3)/6.
  • A056524 (program): Palindromes with even number of digits.
  • A056525 (program): Palindromes with odd number of digits.
  • A056526 (program): First differences of Flavius Josephus’s sieve.
  • A056527 (program): Numbers where iterated sum of digits of square settles down to a cyclic pattern (in fact 13, 16, 13, 16, …).
  • A056528 (program): Sum of digits of square of sum of digits of square.
  • A056530 (program): Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.
  • A056531 (program): Sequence remaining after a fourth round of Flavius Josephus sieve; remove every fifth term of A056530.
  • A056533 (program): Even sieve: start with natural numbers, remove every 2nd term, remove every 4th term from what remains, remove every 6th term from what remains, etc.
  • A056536 (program): Mapping from half-antidiagonal reading of the triangle (as used in A028297) to the column-by-column reading of the triangular tables.
  • A056537 (program): Mapping from the column-by-column reading to the half-antidiagonal reading of the triangular tables. Inverse of A056536.
  • A056538 (program): Irregular triangle read by rows: row n lists the divisors of n in decreasing order.
  • A056539 (program): Self-inverse permutation: reverse the bits in binary expansion of n and also complement them (0->1, 1->0) if the run count (A005811) is even.
  • A056541 (program): a(n) = 2n*a(n-1) + 1 with a(0)=0.
  • A056542 (program): a(n) = n*a(n-1) + 1, a(1) = 0.
  • A056543 (program): a(n) = n*a(n-1) - 1 with a(1)=1.
  • A056545 (program): a(n) = 4*n*a(n-1) + 1 with a(0)=1.
  • A056546 (program): a(n) = 5*n*a(n-1) + 1 with a(0)=1.
  • A056547 (program): a(n) = 6*n*a(n-1) + 1 with a(0)=1.
  • A056548 (program): a(n) = Sum_{k>=1} round(n/k) where round(1/2) = 0.
  • A056549 (program): a(n) = Sum_{k>=1} round(n/k) where round(1/2)=1.
  • A056551 (program): Smallest cube divisible by n divided by largest cube which divides n.
  • A056552 (program): Powerfree kernel of cubefree part of n.
  • A056553 (program): Smallest 4th-power divisible by n divided by largest 4th-power which divides n.
  • A056554 (program): Powerfree kernel of 4th-powerfree part of n.
  • A056555 (program): Smallest number k (k>0) such that n*k is a perfect 4th power.
  • A056556 (program): First tetrahedral coordinate; repeat m (m+1)*(m+2)/2 times.
  • A056557 (program): Second tetrahedral coordinate.
  • A056558 (program): Third tetrahedral coordinate, i.e., tetrahedron with T(t,n,k)=k; succession of growing finite triangles with increasing values towards bottom right.
  • A056559 (program): Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.
  • A056560 (program): Tetrahedron with T(t,n,k)=n-k; succession of growing finite triangles with increasing values towards bottom left.
  • A056562 (program): Number of primes which are the difference between two triangular numbers, where the smaller is the n-th triangular number.
  • A056563 (program): Number of primes which are the difference between two triangular numbers, where the larger is the n-th triangular number.
  • A056565 (program): Fibonomial coefficients.
  • A056566 (program): Fibonomial coefficients.
  • A056567 (program): Fibonomial coefficients.
  • A056568 (program): Fibonomial coefficients.
  • A056570 (program): Third power of Fibonacci numbers (A000045).
  • A056571 (program): Fourth power of Fibonacci numbers A000045.
  • A056572 (program): Fifth power of Fibonacci numbers A000045.
  • A056573 (program): Sixth power of Fibonacci numbers A000045.
  • A056574 (program): Seventh power of Fibonacci numbers A000045.
  • A056576 (program): Highest k with 2^k <= 3^n.
  • A056577 (program): Difference between 3^n and highest power of 2 less than or equal to 3^n.
  • A056578 (program): a(n) = 1 + 2n + 3n^2 + 4n^3.
  • A056579 (program): 1+2n+3n^2+4n^3+5n^4.
  • A056585 (program): Eighth power of Fibonacci numbers A000045.
  • A056586 (program): Ninth power of Fibonacci numbers A000045.
  • A056587 (program): Tenth power of Fibonacci numbers A000045.
  • A056589 (program): Third column sequence of unsigned triangle A056588.
  • A056594 (program): Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).
  • A056595 (program): Number of nonsquare divisors of n.
  • A056596 (program): Number of nonsquare divisors of n!.
  • A056603 (program): Squarefree kernels of distinct values of lcm(1,…,m) (A051451).
  • A056604 (program): a(0)=1; thereafter a(n) = lcm(1, 2, 3, 4, …, prime(n)).
  • A056606 (program): Squarefree kernel of lcm(binomial(n,0), …, binomial(n,n)).
  • A056607 (program): a(n) is the n-th primorial divided by squarefree kernel of corresponding central binomial coefficient.
  • A056608 (program): Least prime factor of the n-th composite number.
  • A056609 (program): a(n) = rad(n!)/rad(A001142(n)) where rad(n) is the squarefree kernel of n, A007947(n).
  • A056610 (program): Quotient: squarefree kernel of lcm(1,..,n) (or of n!) divided by kernel of central binomial coefficient.
  • A056611 (program): Quotient: squarefree kernel of A002944(n) divided by that of A001405.
  • A056612 (program): a(n) = gcd(n!, n!*(1 + 1/2 + 1/3 + … + 1/n)).
  • A056615 (program): Binomial(2*n - 1, n - 1) - 1 (mod n^2).
  • A056616 (program): Numerator of binomial(2n,n)/(2n+1).
  • A056617 (program): Denominator of binomial(2*n,n) / (2*n+1).
  • A056618 (program): Concatenate factorials.
  • A056622 (program): Square root of largest unitary square divisor of n.
  • A056623 (program): Largest unitary square divisor of n: if n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1.
  • A056624 (program): Number of unitary square divisors of n.
  • A056627 (program): Square root of largest unitary square divisor of n!.
  • A056628 (program): Largest unitary square divisor of n!.
  • A056629 (program): Number of unitary square divisors of n!.
  • A056640 (program): At stage 1, start with a unit square. At each successive stage add 4*(n-1) new squares around outside with edge-to-edge contacts. Sequence gives number of squares (regardless of size) at n-th stage.
  • A056641 (program): Least positive integer k for which (b+1)^k is not palindromic in base b, b = 2, 3, 4, …
  • A056653 (program): Composite numbers together with 1 but excluding 4.
  • A056665 (program): Number of equivalence classes of n-valued Post functions of 1 variable under action of complementing group C(1,n).
  • A056671 (program): 1 + the number of unitary and squarefree divisors of n = number of divisors of reduced squarefree part of n.
  • A056672 (program): Number of unitary and squarefree divisors of n! Also, number of divisors of the special squarefree part of n!, A055773(n).
  • A056673 (program): Number of unitary and squarefree divisors of binomial(n, floor(n/2)). Also the number of divisors of the powerfree part of A001405(n), A056060(n).
  • A056674 (program): Number of squarefree divisors which are not unitary. Also number of unitary divisors which are not squarefree.
  • A056675 (program): Number of non-unitary but squarefree divisors of n!. Also number of unitary but not-squarefree divisors of n!.
  • A056691 (program): Number of divisors k of n with gcd(k+1, n) = 1.
  • A056692 (program): Number of divisors k of n with gcd(k-1, n) = 1.
  • A056699 (program): First differences are 2,1,-2,3 (repeated).
  • A056709 (program): Naught-y primes, primes with noughts (or zeros).
  • A056731 (program): First differences of A030124.
  • A056737 (program): Minimum nonnegative integer m such that n = k*(k+m) for some positive integer k.
  • A056738 (program): Positions where 2’s occur in A056731.
  • A056758 (program): Numbers n for which d(n)^3 < n.
  • A056771 (program): a(n) = a(-n) = 34*a(n-1) - a(n-2), and a(0)=1, a(1)=17.
  • A056788 (program): a(n) = n^n + (n-1)^(n-1).
  • A056789 (program): a(n) = Sum_{k=1..n} lcm(k,n)/gcd(k,n).
  • A056791 (program): Weight of binary expansion of n + length of binary expansion of n.
  • A056792 (program): Minimal number of steps to get from 0 to n by (a) adding 1 or (b) multiplying by 2.
  • A056793 (program): Number of divisors of lcm(1..n).
  • A056795 (program): Number of divisors of k as k runs through sequence of distinct values of LCM(1,..,n).
  • A056798 (program): Prime powers with even nonnegative exponents.
  • A056810 (program): Numbers whose fourth power is a palindrome.
  • A056811 (program): Number of primes not exceeding square root of n: primepi(sqrt(n)).
  • A056812 (program): Number of unitary prime factors of lcm[1..n], i.e., primes in LCM with exponent 1.
  • A056813 (program): Largest non-unitary prime factor of LCM(1,…,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).
  • A056819 (program): a(n) = Product_{k|n} (n+1-k).
  • A056822 (program): Nearest integer to n^2/16.
  • A056823 (program): Number of compositions minus number of partitions: A011782(n) - A000041(n).
  • A056827 (program): a(n) = floor(n^2/6).
  • A056829 (program): Nearest integer to n^2/6.
  • A056830 (program): Alternate digits 1 and 0.
  • A056832 (program): All a(n) = 1 or 2; a(1) = 1; get next 2^k terms by repeating first 2^k terms and changing last element so sum of first 2^(k+1) terms is odd.
  • A056833 (program): Nearest integer to n^2/7.
  • A056834 (program): a(n) = floor(n^2/7).
  • A056838 (program): a(n) = floor(n^2/9).
  • A056847 (program): Nearest integer to n - sqrt(n).
  • A056849 (program): Final digit of n^n.
  • A056850 (program): Minimal absolute difference of 3^n and 2^k.
  • A056854 (program): a(n) = Lucas(4*n).
  • A056855 (program): a(n) = (Product k) * (Sum 1/k), where both the product and the sum are over those positive integers k, where k <= n and gcd(k,n) = 1.
  • A056857 (program): Triangle read by rows: T(n,c) = number of successive equalities in set partitions of n.
  • A056860 (program): Triangle T(n,k) = number of element-subset partitions of {1..n} with n-k+1 equalities (n >= 1, 1 <= k <= n).
  • A056864 (program): Nearest integer to n^2/10.
  • A056865 (program): a(n) = floor(n^2/10).
  • A056866 (program): Orders of non-solvable groups, i.e., numbers that are not solvable numbers.
  • A056870 (program): Difference between partial sums of partition numbers (A026905) and partial sums of numbers of partitions into distinct parts (A026906).
  • A056871 (program): Sum of partial sums of partition numbers (A026905) and partial sums of numbers of partitions into distinct parts (A026906).
  • A056889 (program): Numerators of continued fraction for left factorial.
  • A056890 (program): Denominators of continued fraction for left factorial.
  • A056892 (program): a(n) = square excess of the n-th prime.
  • A056896 (program): Smallest prime which can be written as k^2 + n for k >= 0.
  • A056897 (program): Smallest square where a(n)+n is prime.
  • A056898 (program): a(n) = smallest number m such that m^2+n is prime.
  • A056899 (program): Primes of the form k^2 + 2.
  • A056900 (program): Numbers n where 36n^2+36n+11 is prime.
  • A056904 (program): Floor[p/24] where p is a prime which is 4 more than a square.
  • A056905 (program): Primes of the form k^2 + 5.
  • A056906 (program): Numbers k such that 36*k^2 + 5 is prime.
  • A056908 (program): Numbers k such that 36*k^2 + 36*k + 13 is prime.
  • A056909 (program): Primes of the form k^2+6.
  • A056911 (program): Odd squarefree numbers.
  • A056912 (program): Odd squarefree numbers for which the number of prime divisors is odd.
  • A056913 (program): Odd squarefree numbers for which the number of prime divisors is even.
  • A056914 (program): a(n) = Lucas(4*n+1).
  • A056916 (program): Product of the orders of the elements in a cyclic group with n elements.
  • A056918 (program): a(n) = 9*a(n-1)-a(n-2); a(0)=2, a(1)=9.
  • A056919 (program): Numerators of continued fraction for left factorial.
  • A056920 (program): Denominators of continued fraction for left factorial.
  • A056921 (program): a(0) = 0, a(1) = 1, a(2*n) = n*a(2*n-1) + a(2*n-2), a(2*n+1) = a(2*n) + a(2*n-1).
  • A056922 (program): Denominators of continued fraction for alternating factorial.
  • A056923 (program): Write the integers in groups: 0; 1,2; 3,4,5; 6,7,8,9; … and form the product of the members of each group.
  • A056924 (program): Number of divisors of n that are smaller than sqrt(n).
  • A056925 (program): Largest integer power of n which divides product of divisors of n.
  • A056926 (program): a(n) = sqrt(n) if n is a square, otherwise 1.
  • A056927 (program): Difference between n^2 and largest prime less than n^2.
  • A056928 (program): Average of smallest prime greater than n^2 and largest prime less than n^2.
  • A056929 (program): Difference between n^2 and average of smallest prime greater than n^2 and largest prime less than n^2.
  • A056932 (program): Antichains (or order ideals) in the poset 2*2*2*n or size of the distributive lattice J(2*2*2*n).
  • A056942 (program): Area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.
  • A056943 (program): Unused area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.
  • A056944 (program): Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area.
  • A056951 (program): Triangle whose rows show the result of flipping the first, first two, … and finally first n coins when starting with the stack (1,2,3,4,…,n) [starting with all heads up, where signs show whether particular coins end up heads or tails].
  • A056952 (program): Numerators of continued fraction for alternating factorial.
  • A056953 (program): Denominators of continued fraction for alternating factorial.
  • A056955 (program): Euclid set of class 2 and modulus 3.
  • A056956 (program): Numbers n such that 6n+1 and 6n+5 are both primes.
  • A056959 (program): In repeated iterations of function m -> m/2 if m even, m -> 3m+1 if m odd, a(n) is maximum value achieved if starting from n.
  • A056960 (program): Base 11 reversal of n (written in base 10).
  • A056961 (program): Base 12 reversal of n (written in base 10).
  • A056962 (program): Base 16 reversal of n (written in base 10).
  • A056963 (program): Base 20 reversal of n (written in base 10).
  • A056964 (program): a(n) = n + reversal of digits of n.
  • A056965 (program): a(n) = n - (reversal of digits of n).
  • A056967 (program): Write what is described (putting a leading zero on numbers which have an odd number of digits).
  • A056968 (program): 10^(n-1) modulo n.
  • A056969 (program): a(n) = 10^n modulo n.
  • A056973 (program): Number of blocks of {0,0} in the binary expansion of n.
  • A056974 (program): Number of blocks of {0, 0, 0} in the binary expansion of n.
  • A056975 (program): Number of blocks of {0, 0, 1} in binary expansion of n.
  • A056976 (program): Number of blocks of {0, 1, 0} in the binary expansion of n.
  • A056978 (program): Number of blocks of {1, 0, 0} in binary expansion of n.
  • A056979 (program): Number of blocks of {1, 0, 1} in binary expansion of n.
  • A056980 (program): Number of blocks of {1, 1, 0} in binary expansion of n.
  • A056981 (program): a(n) = A002596(n)^2.
  • A056982 (program): a(n) = 4^A005187(n). The denominators of the Landau constants.
  • A056986 (program): Number of permutations on {1,…,n} containing any given pattern alpha in the symmetric group S_3.
  • A056991 (program): Numbers with digital root 1, 4, 7 or 9.
  • A056992 (program): Digital roots of square numbers A000290.
  • A056998 (program): Erroneous version of A057348.
  • A057000 (program): a(n) = phi(n+1) - phi(n).
  • A057001 (program): (phi(n+1)-phi(n))/2.
  • A057003 (program): Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; … and multiply the members of each group.
  • A057009 (program): Number of conjugacy classes of subgroups of index 3 in free group of rank n.
  • A057010 (program): Number of conjugacy classes of subgroups of index 4 in free group of rank n.
  • A057016 (program): Consider recurrence b(0) = (2n+1)/2, b(n) = b(n-1)*floor(b(n-1)); sequence gives first integer reached.
  • A057020 (program): Numerator of (sum of divisors of n / number of divisors of n).
  • A057021 (program): Denominator of (sum of divisors of n / number of divisors of n).
  • A057022 (program): a(n) = floor((sum of divisors of n) / (number of divisors of n)), or floor(sigma_1(n)/sigma_0(n)).
  • A057023 (program): Largest odd factor of (n-th prime-1); k when n-th prime is written as k*2^m+1 [with k odd].
  • A057024 (program): Largest odd factor of (n-th prime+1); k when n-th prime is written as k*2^m-1 [with k odd].
  • A057025 (program): Smallest prime of form (2n+1)*2^m+1 for some m.
  • A057027 (program): Triangle T read by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form an increasing sequence and the others a decreasing sequence.
  • A057028 (program): Triangle T read by rows: row n consists of the numbers C(n,2)+1 to C(n+1,2); numbers in odd-numbered places form a decreasing sequence and the others an increasing sequence.
  • A057029 (program): Central column of arrays in A057027 and A057028.
  • A057030 (program): Let P(n) of a sequence s(1),s(2),s(3),… be obtained by leaving s(1),…,s(n-1) fixed and reversing every n consecutive terms thereafter; apply P(2) to 1,2,3,… to get PS(2), then apply P(3) to PS(2) to get PS(3), then apply P(4) to PS(3), etc. The limit of PS(n) is A057030.
  • A057031 (program): Sequence of differences of A057030.
  • A057032 (program): Let P(n) of a sequence s(1), s(2), s(3), … be obtained by leaving s(1), …, s(n-1) fixed and forward-cyclically permuting every n consecutive terms thereafter; apply P(2) to 1, 2, 3, … to get PS(2), then apply P(3) to PS(2) to get PS(3), then apply P(4) to PS(3), etc. The limit of PS(n) as n -> oo is this sequence.
  • A057033 (program): Let P(n) of a sequence s(1),s(2),s(3),… be obtained by leaving s(1),…s(n-1) fixed and reverse-cyclically permuting every n consecutive terms thereafter; apply P(2) to 1,2,3,… to get PS(2), then apply P(3) to PS(2) to get PS(3), then apply P(4) to PS(3), etc. The limit of PS(n) is A057033.
  • A057036 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057036(n)=i(2n-1).
  • A057037 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057037(n)=j(2n-1).
  • A057038 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057038(n)=i(2n).
  • A057039 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057039(n)=j(2n).
  • A057040 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057040(n)=i(F(n)), where F(n) is the n-th Fibonacci number.
  • A057041 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057041(n)=j(F(n)), where F(n) is the n-th Fibonacci number.
  • A057042 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; the n-th Fibonacci number is in antidiagonal a(n).
  • A057043 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057043(n)=i(L(n)), where L(n) is the n-th Lucas number.
  • A057044 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057044(n)=j(L(n)), where L(n) is the n-th Lucas number.
  • A057045 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; the n-th Lucas number is in antidiagonal a(n).
  • A057046 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057046(n)=i(2^n).
  • A057047 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057047(n)=j(2^n).
  • A057048 (program): a(n) = A017911(n+1) = round(sqrt(2)^(n+1)).
  • A057049 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057049(n) = i(n^2).
  • A057050 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057050(n)=j(n^2).
  • A057052 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057052(n) = i(n^3).
  • A057053 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057053(n)=j(n^3).
  • A057054 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; n^3 is in antidiagonal a(n).
  • A057055 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057055(n)=i(C(n,3)).
  • A057056 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057056(n)=j(C(n,3)).
  • A057057 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; C(n,3) is in antidiagonal a(n).
  • A057058 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and a(n)=i(A057027(n))
  • A057059 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; … Define i(m) and j(m) by R(i(m),j(m)) = m. Then a(n) = j(A057027(n)).
  • A057060 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057058(n)=i(n-th prime)).
  • A057061 (program): Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; …; each k is an R(i(k),j(k)) and A057058(n)=j(n-th prime)).
  • A057062 (program): Let R(i,j) be the infinite square array with antidiagonals 1; 2,3; 4,5,6; …; the n-th prime is in antidiagonal a(n).
  • A057063 (program): Let P(n) of a sequence s(1),s(2),s(3),… be obtained by leaving s(1),…,s(n) fixed and reverse-cyclically permuting every n consecutive terms thereafter; apply P(2) to 1,2,3,… to get PS(2), then apply P(3) to PS(2) to get PS(3), then apply P(4) to PS(3), etc. The limit of PS(n) is A057063.
  • A057064 (program): Let P(n) of a sequence s(1),s(2),s(3),… be obtained by leaving s(1),…,s(n) fixed and forward-cyclically permuting every n consecutive terms thereafter; apply P(2) to 1,2,3,… to get PS(2), then apply P(3) to PS(2) to get PS(3), then apply P(4) to PS(3), etc. The limit of PS(n) is A057064.
  • A057065 (program): a(n) = floor(n^n/2).
  • A057066 (program): Floor[4^4/n].
  • A057067 (program): a(n) = floor(5^5/n).
  • A057068 (program): floor[6^6/n].
  • A057069 (program): floor[7^7/n].
  • A057070 (program): floor[8^8/n].
  • A057071 (program): floor[9^9/n].
  • A057072 (program): floor[10^10/n].
  • A057073 (program): floor[11^11/n].
  • A057074 (program): floor[12^12/n].
  • A057075 (program): Table read by antidiagonals of T(n,k)=floor[n^n/k] with n,k >= 1.
  • A057076 (program): A Chebyshev or generalized Fibonacci sequence.
  • A057077 (program): Periodic sequence 1,1,-1,-1; expansion of (1+x)/(1+x^2).
  • A057078 (program): Periodic sequence 1,0,-1,…; expansion of (1+x)/(1+x+x^2).
  • A057079 (program): Periodic sequence: repeat [1,2,1,-1,-2,-1]; expansion of (1+x)/(1-x+x^2).
  • A057080 (program): Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2.
  • A057081 (program): Even indexed Chebyshev U-polynomials evaluated at sqrt(11)/2.
  • A057083 (program): Scaled Chebyshev U-polynomials evaluated at sqrt(3)/2; expansion of 1/(1 - 3*x + 3*x^2).
  • A057084 (program): Scaled Chebyshev U-polynomials evaluated at sqrt(2).
  • A057085 (program): a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.
  • A057086 (program): Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.
  • A057087 (program): Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.
  • A057088 (program): Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.
  • A057089 (program): Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
  • A057090 (program): Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence.
  • A057091 (program): Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence.
  • A057092 (program): Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci sequence.
  • A057093 (program): Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.
  • A057094 (program): Coefficient triangle for certain polynomials (rising powers).
  • A057103 (program): Triangle of congrua: T(n,k) = 4*n*k(n^2-k^2) with n>k>0 and starting at T(2,1) = 24. A055096(n)^2 + a(n) is a square, as is A055096(n)^2 - a(n).
  • A057104 (program): The non-octal numbers: numbers containing an 8 or 9 (they cannot be mistaken for octal numbers).
  • A057105 (program): Triangle of numbers (when unsigned) related to congruum problem: T(n,k)=k^2+2nk-n^2 with n>k>0 and starting at T(2,1)=1.
  • A057108 (program): Difference between the smallest number S(n) with S(n)! a multiple of n and the largest prime factor of n [taking a(1)=0].
  • A057109 (program): Numbers n that are not factors of P(n)!, where P(n) is the largest prime factor of n.
  • A057126 (program): Numbers n such that 2 is a square mod n.
  • A057129 (program): -4 is a square mod n.
  • A057130 (program): Product of first n primes of form 6k-1.
  • A057131 (program): One less than six times product of first n primes of form 6k-1.
  • A057137 (program): Concatenate next digit at right hand end (where the next digit after 9 is again 0).
  • A057138 (program): Add (n mod 10)*10^(n-1) to the previous term, with a(0) = 0.
  • A057139 (program): Odd number of digits palindrome based on sequential digits.
  • A057145 (program): Square array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, read by antidiagonals upwards.
  • A057146 (program): The sequence 2, floor(a), floor(a^2), floor(a^3), …, with a = 1+sqrt(5).
  • A057147 (program): a(n) = n times sum of digits of n.
  • A057148 (program): Palindromes only using 0 and 1 (i.e., base-2 palindromes).
  • A057168 (program): Next larger integer with same binary weight (number of 1 bits) as n.
  • A057174 (program): a(n+3)=a(n)+a(n+1)-a(n+2), starting with 1,2,3.
  • A057198 (program): a(n) = (5*3^(n-1)+1)/2.
  • A057211 (program): Alternating runs of ones and zeros, where the n-th run has length n.
  • A057212 (program): n-th run has length n.
  • A057213 (program): Second term of continued fraction for exp(n).
  • A057215 (program): [1->01, 2->10, 3->01]-transform of 3-symbol Thue-Morse A026600.
  • A057217 (program): a(n) = smallest positive integer k such that 1+n*k! is a prime.
  • A057218 (program): a(n) = least prime of the form n*k! + 1.
  • A057227 (program): Smallest member of smallest set S(n) of positive integers containing n which satisfies “k is in S, iff 2k-1 is in S, iff 4k is in S”.
  • A057237 (program): Maximum k <= n such that 1, 2, …, k are all relatively prime to n.
  • A057268 (program): a(n-1)+k=a(n) => a(n)*k=a(n+1).
  • A057300 (program): Binary counter with odd/even bit positions swapped; base-4 counter with 1’s replaced by 2’s and vice versa.
  • A057334 (program): In A000120, replace each entry k with the k-th prime and replace 0 with 1.
  • A057335 (program): a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.
  • A057347 (program): Leap years in the Islamic calendar starting year 1 AH (Anno Hegirae) = 622 CE (Common Era or AD). There are 11 leap years in a 30 year cycle.
  • A057348 (program): Days in months in the Islamic calendar starting from Muharram, 1 AH. The twelfth month has 30 days in a leap year.
  • A057349 (program): Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra-month.
  • A057353 (program): a(n) = floor(3n/4).
  • A057354 (program): a(n) = floor(2*n/5).
  • A057355 (program): a(n) = floor(3*n/5).
  • A057356 (program): a(n) = floor(2*n/7).
  • A057357 (program): a(n) = floor(3*n/7).
  • A057358 (program): a(n) = floor(4*n/7).
  • A057359 (program): a(n) = floor(5*n/7).
  • A057360 (program): a(n) = floor(3*n/8).
  • A057361 (program): a(n) = floor(5*n/8).
  • A057362 (program): a(n) = floor(5*n/13).
  • A057363 (program): a(n) = floor(8*n/13).
  • A057364 (program): a(n) = floor(8*n/21).
  • A057365 (program): a(n) = floor(13*n/21).
  • A057366 (program): a(n) = floor(7*n/19).
  • A057367 (program): a(n) = floor(11*n/30).
  • A057427 (program): a(n) = 1 if n > 0, a(n) = 0 if n = 0; series expansion of x/(1-x).
  • A057428 (program): Sign(-n): a(n) = 1 if -n > 0, = -1 if -n < 0, = 0 if n = 0.
  • A057432 (program): Obtained by reading first the numerator then the denominator of fractions in left-hand half of Stern-Brocot tree (A007305/A007306).
  • A057434 (program): a(n) = Sum_{k=1..n} phi(k)^2.
  • A057436 (program): Contains digits 1 through 6 only.
  • A057438 (program): a(1) = 1; a(n+1) = product_{k = 1 to n} [a(k)] *sum_{j = 1 to n} [1/a(j)].
  • A057449 (program): Product of differences between consecutive positive divisors of n.
  • A057458 (program): Number of k, 1 <= k <= n, where {k (n+1-k) + 1} is prime.
  • A057467 (program): GCD of n-th and (n+1)-st term in the sequence of first differences between primes, A001223.
  • A057470 (program): Let p(i) =i-th prime, let twin(n) = (P,Q) be n-th pair of twin primes; sequence gives p(P).
  • A057473 (program): Let p(i) =i-th prime, let twin(n) = (P,Q) be n-th pair of twin primes; sequence gives p(Q).
  • A057475 (program): Number of k, 1 <= k <= n, such that gcd(n,k) = gcd(n+1,k) = 1.
  • A057500 (program): Number of connected labeled graphs with n edges and n nodes.
  • A057514 (program): Number of peaks in mountain ranges encoded by A014486, number of leaves in the corresponding rooted plane trees (the root node is never counted as a leaf).
  • A057520 (program): a(n) = A014486(n)/2. In binary expansion there is one more 1 than 0 and reading from the left (the most significant bit) to right, the number of 0’s never exceed the number of 1’s.
  • A057521 (program): Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=b*c^2*d^3 then a(n)=c^2*d^3 when b is minimized.
  • A057524 (program): Number of 3 x n binary matrices without unit columns up to row and column permutations.
  • A057525 (program): Number of applications of f to reduce n to 1, where f(k) is the integer among k/2,(k+1)/4, (k+3)/4.
  • A057526 (program): Number of applications of f to reduce n to 1, where f(k) is the integer among k/2,(k-1)/4, (k+1)/4.
  • A057531 (program): Numbers whose sum of digits and number of divisors are equal.
  • A057532 (program): n is odd and sum of digits of n equals the numbers of divisors of n.
  • A057536 (program): Minimal number of coins needed to pay n Euro-cents using the Euro currency.
  • A057537 (program): Number of ways of making change for n Euro-cents using the Euro currency.
  • A057538 (program): Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.
  • A057543 (program): Maximum cycle length (orbit size) in the rotation permutation of 2n non-crossing handshakes.
  • A057544 (program): Maximum cycle length (orbit size) in the rotation permutation of n+2 side polygon triangularizations.
  • A057547 (program): A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.
  • A057552 (program): a(n) = Sum_{k=0..n} C(2k+2,k).
  • A057553 (program): Rank of (1,1,…,1) (n 1’s) when {0,1,2,…}^n is lexicographically ordered.
  • A057554 (program): Lexicographic ordering of MxM, where M={0,1,2,…}.
  • A057555 (program): Lexicographic ordering of N x N, where N = {1,2,3…}.
  • A057566 (program): Number of collinear triples in a 3 X n rectangular grid.
  • A057569 (program): Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2.
  • A057570 (program): Numbers of the form n*(7n+-1)/2.
  • A057585 (program): Area under Motzkin excursions.
  • A057587 (program): Nonnegative numbers of form n*(n^2+-1)/2.
  • A057588 (program): Kummer numbers: -1 + product of first n consecutive primes.
  • A057590 (program): Numbers of the form n*(n^3+-1)/2.
  • A057591 (program): Maximal size of binary code of length n that corrects 2 deletions.
  • A057592 (program): a(n) = Fibonacci(n+1)^2 + 4*Fibonacci(n).
  • A057597 (program): a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=0, a(1)=0, a(2)=1.
  • A057599 (program): a(n) = (n^2)!/(n!)^(n+1); number of ways of dividing n^2 labeled items into n unlabeled boxes of n items each.
  • A057602 (program): a(1)=2, a(n+1) is the smallest integer > a(n) such that the smallest prime factor of a(n+1) is the largest prime factor of a(n).
  • A057604 (program): Primes of the form 4*k^2 + 163.
  • A057615 (program): ATS: Add Then Sort (i.e., double previous term and then sort digits).
  • A057625 (program): a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.
  • A057627 (program): Number of nonsquarefree numbers not exceeding n.
  • A057651 (program): a(n) = (3 * 5^n - 1)/2.
  • A057653 (program): Odd numbers of form x^2 + y^2.
  • A057655 (program): The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.
  • A057656 (program): Number of points (x,y) in square lattice with (x-1/2)^2+y^2 <= n.
  • A057658 (program): a(n) = n*(n+1)^2*(n+2)^3*(n+3)^2*(n+4).
  • A057660 (program): a(n) = Sum_{k=1..n} n/gcd(n,k).
  • A057661 (program): a(n) = Sum_{k=1..n} lcm(n,k)/n.
  • A057666 (program): n*(n+1)^2*(n+2)*(n+3)^2*(n+4).
  • A057667 (program): Primes q of form q = 10p + 3, where p is an odd prime.
  • A057670 (program): a(n) = Sum_{k|n} lcm(k, n/k).
  • A057671 (program): a(n) equals floor(Vc(n) - Vs(n)), where Vc(n) is the volume of the cube with side length n and Vs(n) is the volume of the sphere of diameter n.
  • A057672 (program): a(n) equals floor(As(n) - Ac(n)), where As(n) is the area of the square with side length n and Ac(n) is the area of the circle of diameter n.
  • A057675 (program): 1 - (5/6)*n + (5/2)*n^2 + (10/3)*n^3 + n^4.
  • A057677 (program): a(n) is the numerator of b(n) where b(n)=1/b(n-1)+1/b(n-2) with b(1)=1 and b(2)=2.
  • A057681 (program): a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j).
  • A057682 (program): a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1).
  • A057693 (program): Number of permutations on n letters that have only cycles of length 3 or less.
  • A057703 (program): a(n) = n*(94 + 5*n + 25*n^2 - 5*n^3 + n^4)/120.
  • A057711 (program): a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.
  • A057716 (program): The nonpowers of 2.
  • A057717 (program): The non-powers of 10.
  • A057718 (program): A036917/8 (omitting leading term of A036917).
  • A057721 (program): a(n) = n^4 + 3*n^2 + 1.
  • A057722 (program): a(n) = n^4 - 3*n^2 + 1.
  • A057723 (program): Sum of positive divisors of n that are divisible by every prime that divides n.
  • A057728 (program): A triangular table of decreasing powers of two (with first column all ones).
  • A057744 (program): Expansion of (1-2*x^3)/(1-2*x-x^3+2*x^4).
  • A057769 (program): a(n) = 4*n^4 + 8*n^3 - 4*n - 1 = (2*n^2 - 1)*(2*n^2 + 4*n + 1).
  • A057773 (program): a(n) = Sum_{i=1..n} nu_2 ( prime(i) - 1), where prime(i) is the i-th prime and nu_2(m) = exponent of highest power of 2 dividing m.
  • A057777 (program): a(n) is the smallest number such that a(n)+1 is a prime and the largest power of 2 which divides it is 2^n.
  • A057780 (program): Multiples of 3 that are one less than a perfect square.
  • A057781 (program): a(n) = n^4+4 = (n^2-2*n+2)*(n^2+2*n+2) = ((n-1)^2+1)*((n+1)^2+1).
  • A057788 (program): Expansion of (1+x)/(1-x)^12.
  • A057789 (program): a(n) = Sum_{k = 1..n, gcd(k,n)=1} k*(n-k).
  • A057791 (program): Sum[k^(n-k)], where sum is over positive integers, k, where k <= n and gcd(k,n) = 1.
  • A057792 (program): Sum[k^k], where sum is over positive integers, k, where k <= n and gcd(k,n) = 1.
  • A057795 (program): Sum k!, where sum is over positive integers k <= n with gcd(k,n) = 1.
  • A057811 (program): pi(n) is even.
  • A057812 (program): pi(n) is odd.
  • A057813 (program): a(n) = (2*n+1)*(4*n^2+4*n+3)/3.
  • A057814 (program): Number of partitions of an n-set into blocks of size > 4.
  • A057815 (program): a(n) = gcd(n,binomial(n,floor(n/2))).
  • A057817 (program): Moebius invariant of cographic hyperplane arrangement for complete graph K_n. Also value of Tutte dichromatic polynomial T_G(0,1) for G=K_n. Also alternating sum F_{n,1} - F_{n,2} + F_{n,3} - …, where F_{n,k} is the number of labeled forests on n nodes with k connected components.
  • A057819 (program): a(0)=4, a(1)=9, a(n) = 4a(n-1) - a(n-2).
  • A057820 (program): First differences of sequence of consecutive prime powers (A000961).
  • A057827 (program): a(0) = 1; a(n) = LCM(n, sum{k=0 to n-1}[a(k)]).
  • A057828 (program): Number of perfect squares, k^2, where k^2 <= n and gcd(k,n) = 1.
  • A057837 (program): Number of partitions of a set of n elements where the partitions are of size > 3.
  • A057843 (program): a(n) = floor(n*tau^2) - 3, where tau = (1+sqrt(5))/2.
  • A057854 (program): Non-Lucas numbers: the complement of A000032.
  • A057855 (program): Greatest k such that (k-th prime) <= (n times n-th prime).
  • A057857 (program): Number of residue classes modulo n-th primorial number which contain a prime.
  • A057858 (program): Number of residue classes modulo n-th primorial number which contain only composite numbers.
  • A057859 (program): Number of residue classes modulo n which contain a prime.
  • A057860 (program): Number of residue classes modulo n which contain only composite numbers.
  • A057861 (program): floor[2^n/Fibonacci(n)].
  • A057862 (program): a(n) = 2^n mod Fibonacci(n).
  • A057863 (program): a(n) = Product_{k=1..n} (2k-1)!!.
  • A057868 (program): Denominator of “modified Bernoulli number” b(2n) = Bernoulli(2*n)/(2*n*n!).
  • A057884 (program): A square array based on tetrahedral numbers (A000292) with each term being the sum of 2 consecutive terms in the previous row.
  • A057886 (program): Number of integer 4-tuples that give the lengths of the sides of a nondegenerate quadrilateral with perimeter n.
  • A057889 (program): Bit-reverse of n, including as many leading as trailing zeros.
  • A057890 (program): In base 2, either a palindrome or becomes a palindrome if trailing 0’s are omitted.
  • A057891 (program): In base 2, neither a palindrome nor becomes a palindrome if trailing 0’s are omitted.
  • A057901 (program): a(n) = 3^prime(n).
  • A057902 (program): a(n) = 5^prime(n).
  • A057903 (program): Positive integers that are not the sum of exactly two positive cubes.
  • A057904 (program): Positive integers that are not the sum of exactly three positive cubes.
  • A057918 (program): Number of pairs of numbers (a,b) each less than n where (a,b,n) is in geometric progression.
  • A057921 (program): d(n+1) divides d(n), where d(n) is number of positive divisors of n.
  • A057922 (program): d(n) divides d(n+1), where d(n) is number of positive divisors of n.
  • A057932 (program): a(n) = floor(10^(n+1)/81).
  • A057933 (program): Floor[(80/81)*10^n].
  • A057944 (program): Largest triangular number less than or equal to n; write m-th triangular number m+1 times.
  • A057945 (program): Number of triangular numbers needed to represent n with greedy algorithm.
  • A057947 (program): n has ambiguous representations in “bad hexadecimal”: numbers with the digit 1 followed by a digit less than 6.
  • A057948 (program): S-primes: let S = {1,5,9, … 4i+1, …}; then an S-prime is in S but is not divisible by any members of S except itself and 1.
  • A057960 (program): Number of base 5 (n+1)-digit numbers starting with a zero and with adjacent digits differing by one or less.
  • A057961 (program): Number of points in square lattice covered by a disc centered at (0,0) as its radius increases.
  • A057962 (program): Number of points (x,y) in square lattice with (x-1/2)^2+(y-1/2)^2 <= n.
  • A057963 (program): Triangle T(n,k) of number of minimal 2-covers of a labeled n-set that cover k points of that set uniquely (k=2,..,n).
  • A057977 (program): GCD of consecutive central binomial coefficients: a(n) = gcd(A001405(n+1), A001405(n)).
  • A057979 (program): a(n) = 1 for even n and (n-1)/2 for odd n.
  • A058005 (program): a(n) = gcd(2*n, binomial(2*n, n)).
  • A058006 (program): Alternating factorials: 0! - 1! + 2! - … + (-1)^n n!
  • A058008 (program): Numbers k such that (2*k - 1)!/(k!)^2 is an integer.
  • A058026 (program): Number of positive integers, k, where k <= n and gcd(k,n) = gcd(k+1,n) = 1.
  • A058031 (program): a(n) = n^4 - 2*n^3 + 3*n^2 - 2*n + 1, the Alexander polynomial for reef and granny knots.
  • A058033 (program): Number of powers of 2 between (but not including) two consecutive primorials.
  • A058034 (program): Number of numbers whose cube root rounds to n.
  • A058038 (program): a(n) = Fibonacci(2*n)*Fibonacci(2*n+2).
  • A058039 (program): a(n) = a(n-1) + 2*a(floor(n/2)) if n > 0, otherwise 1.
  • A058042 (program): Trajectory of binary number 10110 under the operation ‘Reverse and Add!’ carried out in base 2.
  • A058043 (program): a(n) = nextprime(n^2) - prevprime(n^2).
  • A058053 (program): Number of 3-rowed binary matrices with n ones and no zero columns, up to row and column permutation.
  • A058060 (program): Number of distinct prime factors of d(n), the number of divisors of n.
  • A058061 (program): Number of prime factors (counted with multiplicity) of d(n), the number of divisors of n.
  • A058062 (program): Number of distinct prime factors of sigma(n), the sum of the divisors of n.
  • A058063 (program): Number of prime factors (when counted with multiplicity) of sigma(n), the sum of divisors of n.
  • A058065 (program): Complement of A057843.
  • A058066 (program): a(n) = floor(n*t), t = 1 + sqrt(5)/2.
  • A058067 (program): Number of polynomial functions from Z to Z/nZ.
  • A058071 (program): A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.
  • A058074 (program): Integers m such that gcd(d(m),d(m+1)) = 1, where d(m) is number of positive divisors of m.
  • A058075 (program): Numbers k such that gcd(sigma(k), sigma(k+1)) = 1, where sigma(k) is sum of positive divisors of k.
  • A058077 (program): Binomial coefficients formed from consecutive primes: a(n) = binomial( prime(n+1), prime(n) ).
  • A058078 (program): Greatest common divisor of two binomial coefficients formed from consecutive primes: a(n) = gcd(C(prime(n+2), prime(n+1)), C(prime(n+1), prime(n))).
  • A058080 (program): Numbers whose product of divisors exceeds their square.
  • A058085 (program): Coefficients of ménage hit polynomials.
  • A058086 (program): Coefficients of ménage hit polynomials.
  • A058089 (program): Coefficients of ménage hit polynomials.
  • A058090 (program): Coefficients of ménage hit polynomials.
  • A058094 (program): Number of 321-hexagon-avoiding permutations in S_n, i.e., permutations of 1..n with no submatrix equivalent to 321, 56781234, 46781235, 56718234 or 46718235.
  • A058095 (program): McKay-Thompson series of class 9c for the Monster group.
  • A058126 (program): a(n) = n^n - n^2 with 0^0=1.
  • A058127 (program): Triangle read by rows: T(j,k) is the number of acyclic functions from {1,…,j} to {1,…,k}. For n >= 1, a(n) = (k-j)*k^(j-1), where k is such that C(k,2) < n <= C(k+1,2) and j = (n-1) mod C(k,2). Alternatively, table T(k,j) read by antidiagonals with k >= 1, 0 <= j <= k: T(k,j) = number of acyclic-function digraphs on k vertices with j vertices of outdegree 1 and (k-j) vertices of outdegree 0; T(k,j) = (k-j)*k^(j-1).
  • A058128 (program): a(1) = 1, a(n) = (n^n-n)/(n-1)^2 for n >= 2.
  • A058161 (program): Number of labeled cyclic groups with a fixed identity.
  • A058181 (program): Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.
  • A058182 (program): a(n) = a(n-1)^2 + a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.
  • A058183 (program): Number of digits in concatenation of first n positive integers.
  • A058184 (program): “Real rabbits”: a(n) Real(c(n) where complex c(n)=a(n)+ib(n) and c(0)=i, c(1)=-i, c(n)=c(n-1)+ic(n-2).
  • A058185 (program): Numbers (written in decimal) which appear the same when written in base 5 and base 10/2.
  • A058186 (program): Numbers (written in base 5) which appear the same when written in base 5 and base 10/2.
  • A058187 (program): Expansion of (1+x)/(1-x^2)^4: duplicated tetrahedral numbers.
  • A058189 (program): Number of increasing geometric progressions ending in n (in the positive integers), including those of length 1 or 2.
  • A058190 (program): Number of increasing geometric progressions ending in n (in the positive integers), excluding those of length 1 or 2.
  • A058195 (program): Areas of a sequence of right-angled figures described below.
  • A058200 (program): Coefficients of the highest power of r in a sequence of parametric solutions for the Diophantine equation x^3+y^3+z^3=1.
  • A058202 (program): Triangle in which n-th row gives the numbers which when subtracted from 2^n produce primes.
  • A058207 (program): Three steps forward, two steps back.
  • A058212 (program): a(n) = 1 + floor(n*(n-3)/6).
  • A058224 (program): Largest d such that the linear programming bound for quantum codes of length n is feasible for some real K>1.
  • A058250 (program): GCD of n-th primorial number and its totient.
  • A058251 (program): LCM of n-th primorial number and its Euler totient.
  • A058261 (program): a(n) = n times the Collatz number of n (as given in A006577).
  • A058263 (program): a(n) = gcd(prime(n) - 1, prime(n+1) - 1).
  • A058265 (program): Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.
  • A058278 (program): Expansion of (1 - x^2)/(1 - x - x^3).
  • A058279 (program): a(0)=a(1)=1, a(n)=a(n-2)+(n+1)*a(n-1).
  • A058281 (program): Continued fraction for square root of e.
  • A058294 (program): Successive rows of a triangle, the columns of which are generalized Fibonacci sequences S(j).
  • A058296 (program): Average of consecutive primes.
  • A058298 (program): Triangle n!/(n-k), 1 <= k < n, read by rows.
  • A058307 (program): a(n) = (n+1)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.
  • A058308 (program): a(n) = (n+2)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.
  • A058309 (program): a(n) = (n+3)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.
  • A058310 (program): (1/2)*(n^2+n+2)*(n^2+3*n+1).
  • A058312 (program): Denominator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.
  • A058313 (program): Numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.
  • A058315 (program): Apply inverse of “INVERT” transform to primes with prime exponents.
  • A058318 (program): Number of energy levels in atoms of n-th element of periodic table.
  • A058319 (program): Coefficients (multiplied by 48) in Alternative Extended Simpson’s rule for numerical integration.
  • A058321 (program): Number of x such that phi(x) = 2^n.
  • A058331 (program): a(n) = 2*n^2 + 1.
  • A058333 (program): Number of 3 X 3 matrices with elements from [0,…,(n-1)] satisfying the condition that the middle element of each row or column is the difference of the two end elements (in absolute value).
  • A058344 (program): Difference between the sum of the odd aliquot divisors of n and the sum of the even aliquot divisors of n.
  • A058355 (program): Coefficients in the series (1 + x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + … )/(1 - x - x^4 - x^6 - x^8 - x^9 - x^10 - x^12 - x^14 - … ).
  • A058356 (program): Coefficients in the series (1 + 2x^2 + 3x^3 + 5x^5 + 7x^7 + 11x^11 + 13x^13 + … )/(1 - x - 4x^4 - 6x^6 - 8x^8 - 9x^9 - 10x^10 - 12x^12 - 14x^14 - … ).
  • A058358 (program): Coefficients in the series (1 + x + 4x^4 + 6x^6 + 8x^8 + 9x^9 + 10x^10 + 12x^12 + 14x^14 + … )/(1 - 2x^2 - 3x^3 - 5x^5 - 7x^7 - 11x^11 - 13x^13 - … ).
  • A058363 (program): Numbers whose reduced system of residues forms an arithmetic progression. It consists of primes, twice primes, and powers of 2.
  • A058364 (program): Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.
  • A058365 (program): Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.
  • A058366 (program): Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 7 sites wide.
  • A058367 (program): Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 6 sites wide.
  • A058368 (program): Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 5 sites wide.
  • A058372 (program): a(n) = -(n + 1)*(2*n^2 + n - 12)/6.
  • A058373 (program): a(n) = (1/6)*(2*n - 3)*(n + 2)*(n + 1).
  • A058384 (program): Largest power of 2 which is a divisor of p(n)-1, where p(n) = n-th prime.
  • A058393 (program): A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.
  • A058394 (program): A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.
  • A058395 (program): A square array based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.
  • A058396 (program): Expansion of ((1-x)/(1-2*x))^3.
  • A058481 (program): a(n) = 3^n - 2.
  • A058482 (program): Number of 3 X n binary matrices with no zero rows or columns.
  • A058484 (program): McKay-Thompson series of class 12F for Monster.
  • A058485 (program): McKay-Thompson series of class 12G for Monster.
  • A058487 (program): McKay-Thompson series of class 12I for the Monster group.
  • A058500 (program): Primes of the form p*2^k + 1, where p is an odd prime and k > 0.
  • A058501 (program): Primes p such that largest odd factor of p-1 is not a prime (i.e., is composite or 1).
  • A058511 (program): McKay-Thompson series of class 15D for the Monster group.
  • A058515 (program): GCD of totients of consecutive integers.
  • A058517 (program): Positive even numbers not of the form prime + 3^x.
  • A058529 (program): Numbers whose prime factors are all congruent to +1 or -1 modulo 8.
  • A058538 (program): McKay-Thompson series of class 18c for Monster.
  • A058539 (program): McKay-Thompson series of class 18d for the Monster group.
  • A058541 (program): Trajectory of 1 under map that sends x to 3x - sigma(x).
  • A058545 (program): Trajectory of 23 under map that sends x to 3x - sigma(x), where sigma(x) is the sum of the divisors of x.
  • A058550 (program): Eisenstein series E_14(q) (alternate convention E_7(q)).
  • A058576 (program): McKay-Thompson series of class 24F for Monster.
  • A058577 (program): a(n) = floor(e^sqrt(n)).
  • A058581 (program): (4*n^2+2*n-3)*(2*n-1)*n/3.
  • A058582 (program): Expansion of (1+3*x+4*x^2)/(1-4*x^2+4*x^4).
  • A058607 (program): a(n) = (1 + 1/2 + 1/3 + … + 1/n)*(2n-1)!/(n-1)!.
  • A058620 (program): Lesser of two consecutive primes whose difference divided by two is a prime: ( prime(next prime after n) - prime(n) )/2 is prime.
  • A058621 (program): a(n) = 1/2*binomial(2*n,n) - (1+(-1)^n)/4*(binomial(n,floor(n/2)))^2.
  • A058622 (program): a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2).
  • A058633 (program): Partial sums of the Collatz sequence.
  • A058642 (program): Number of unlabeled graphs with n edges, no nodes of degree 1 or 2, no multiple edges and no cut nodes, under “series-equivalence”.
  • A058645 (program): a(n) = 2^(n-3)*n^2*(n+3).
  • A058649 (program): a(n) = 2^(n-4)*n*(n+1)*(n^2+5*n-2).
  • A058654 (program): The sum of a prime and a nonzero square.
  • A058656 (program): a(n) = gcd(n+1, phi(n)).
  • A058663 (program): a(n) = gcd(n-1, n-phi(n)).
  • A058665 (program): a(n) = gcd(n+1, n-phi(n)).
  • A058667 (program): 2^(n-2)*n*(n+2)!/3.
  • A058681 (program): Number of matroids of rank 2 on n labeled points.
  • A058682 (program): a(n) = p(0) + p(1) + … + p(n) - n - 1, where p = partition numbers, A000041.
  • A058692 (program): a(n) = B(n) - 1, where B(n) = Bell numbers, A000110.
  • A058694 (program): Partial products p(0)*p(1)*…*p(n) of partition numbers A000041.
  • A058695 (program): Number of ways to partition 2n+1 into positive integers.
  • A058696 (program): Number of ways to partition 2n into positive integers.
  • A058738 (program): a(n) = floor(n*exp(n)).
  • A058748 (program): a(n) = round(n*exp(n)).
  • A058749 (program): a(n) = ceiling(n*exp(n)).
  • A058764 (program): Smallest number x such that cototient(x) = 2^n.
  • A058794 (program): Row 3 of A007754.
  • A058795 (program): Row 4 of A007754.
  • A058796 (program): Row 5 of A007754.
  • A058797 (program): a(n) = n*a(n-1) - a(n-2), with a(-1) = 0, a(0) = 1.
  • A058798 (program): a(n) = n*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.
  • A058799 (program): Column 2 of A007754.
  • A058806 (program): a(n) = n! * H_n(n) where H_0(n) = 1/n, H_m(n) = Sum_{k=1..n} H_{m-1}(k).
  • A058809 (program): The sequence lambda(3,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly three starting and/or finishing points.
  • A058810 (program): The sequence lambda(n,n), where lambda is defined in A055203.
  • A058838 (program): a(n) = 1 + sum of the anti-divisors of n.
  • A058840 (program): From Renyi’s “beta expansion of 1 in base 3/2”: sequence gives y(0), y(1), …
  • A058842 (program): From Renyi’s “beta expansion of 1 in base 3/2”: sequence gives a(1), a(2), … where x(n) = a(n)/2^n, with 0 < a(n) < 2^n, a(1) = 1, a(n) = 3*a(n-1) modulo 2^n.
  • A058872 (program): Number of 2-colored labeled graphs with n nodes.
  • A058877 (program): Number of labeled acyclic digraphs with n nodes containing exactly n-1 points of in-degree zero.
  • A058884 (program): Partial sums of the partition function (A000041), with the last term subtracted. Also the sum of the row of the character table for S_n corresponding to the partition n-1,1 for n>1. Also the sum over all partitions lambda of n of one less than the number of 1’s in lambda.
  • A058886 (program): Sum of the row of the character table of S_n corresponding to the partition 2,1^{n-2}.
  • A058888 (program): Number of terms in the set invphi(2*p(n)), where p(n) is the n-th prime.
  • A058895 (program): a(n) = n^4 - n.
  • A058896 (program): a(n) = 4^n - 4.
  • A058919 (program): a(n) = n^4/2 - n^3 + 3n^2/2 - n + 1.
  • A058920 (program): a(n) = 2n^4 + 2n^3 + 3n^2 + 2n + 1.
  • A058922 (program): a(n) = n*2^n - 2^n.
  • A058923 (program): a(n) = binomial(n,0) - binomial(n,2) + binomial(n,4).
  • A058927 (program): Numerators of series related to triangular cacti.
  • A058928 (program): Denominators of series related to triangular cacti.
  • A058932 (program): Number of unlabeled claw-free cubic graphs with 2n nodes and connectivity 1.
  • A058937 (program): Maximal exponent of x in all terms of Somos polynomial of order n.
  • A058957 (program): Numbers having at least two representations as b^2 - c^2 with b > c >= 0.
  • A058962 (program): a(n) = 2^(2*n)*(2*n+1).
  • A058966 (program): a(3) = 1, otherwise a(n) = n*2^(n-3) - 2^(n-2) - 2.
  • A058968 (program): a(n) = 2^n + 2^(n - 1) - n - 8.
  • A058974 (program): a(n) = 0 if n = 1 or a prime, otherwise a(n) = s + a(s) iterated until no change occurs, where s (A008472) is sum of distinct primes dividing n.
  • A058978 (program): Minimal number of (non-consecutive) Fibonacci numbers needed to get n by addition and subtraction.
  • A058984 (program): Number of partitions of n in which number of parts is not 2.
  • A058986 (program): Sorting by prefix reversal (or “flipping pancakes”). You can only reverse segments that include the initial term of the current permutation; a(n) is the number of reversals that are needed to transform an arbitrary permutation of n letters to the identity permutation.
  • A058987 (program): a(n) = Catalan(n) - Motzkin(n-1).
  • A058992 (program): Gossip Problem: there are n people and each of them knows some item of gossip not known to the others. They communicate by telephone and whenever one person calls another, they tell each other all that they know at that time. How many calls are required before each gossip knows everything?
  • A059007 (program): Numbers m such that m^2 reversed is a prime.
  • A059009 (program): Numbers having an odd number of zeros in their binary expansion.
  • A059010 (program): Natural numbers having an even number of nonleading zeros in their binary expansion.
  • A059011 (program): Odd number of 0’s and 1’s in binary expansion.
  • A059012 (program): Numbers that have an even number of 0’s and 1’s in their binary expansion.
  • A059013 (program): Odd number of 0’s and even number of 1’s in binary expansion.
  • A059014 (program): Numbers that have an even number of 0’s and an odd number of 1’s in binary expansion.
  • A059015 (program): Total number of 0’s in binary expansions of 0, …, n.
  • A059016 (program): Number of 0’s in binary expansion of Fibonacci(n).
  • A059018 (program): Write 10*n in base 4; a(n) = sum of digits mod 4.
  • A059019 (program): Number of Dyck paths of semilength n with no peak at height 3.
  • A059020 (program): Number of 2 X n checkerboards (with at least one red square) in which the set of red squares is edge connected.
  • A059026 (program): Table B(n,m) read by rows: B(n,m) = LCM(n,m)/n + LCM(n,m)/m - 1 for all 1<=m<=n.
  • A059027 (program): Number of Dyck paths of semilength n with no peak at height 4.
  • A059028 (program): Row sums of A059026: a(n) = sum( lcm(n,m)/n + lcm(n,m)/m - 1, m = 1..n ).
  • A059029 (program): a(n) = n if n is even, 2*n + 1 if n is odd.
  • A059030 (program): Fourth main diagonal of A059026: a(n) = B(n+3,n) = lcm(n+3,n)/(n+3) + lcm(n+3,n)/n - 1 for all n >= 1.
  • A059031 (program): Fifth main diagonal of A059026: a(n) = B(n+4,n) = lcm(n+4,n)/(n+4) + lcm(n+4,n)/n - 1 for all n >= 1.
  • A059036 (program): In a triangle of numbers (such as that in A059032, A059033, A059034) how many entries lie above position (n,k)? Answer: T(n,k) = (n+1)*(k+1)-1 (n >= 0, k >= 0).
  • A059045 (program): Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + … + n*k^(n-1).
  • A059046 (program): Numbers n such that sigma(n)-n divides n-1.
  • A059053 (program): Number of chiral pairs of necklaces with n beads and two colors (color complements being equivalent); i.e., turning the necklace over neither leaves it unchanged nor simply swaps the colors.
  • A059064 (program): Card-matching numbers (Dinner-Diner matching numbers).
  • A059078 (program): Number of orientable necklaces with 2n beads and two colors which when turned over produce their own color complement.
  • A059094 (program): Numbers whose sum of digits is a cube.
  • A059100 (program): a(n) = n^2 + 2.
  • A059109 (program): Numbers m such that m*phi(m)-1 is prime, where phi is the Euler function (A000010).
  • A059110 (program): Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L’(n,i)*binomial(i,m), m=0..n.
  • A059114 (program): Triangle T(n,m)= Sum_{i=0..n} L’(n,i)*Product_{j=1..m} (i-j+1), read by rows.
  • A059115 (program): Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).
  • A059116 (program): The sequence lambda(4,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly four starting and/or finishing points.
  • A059125 (program): A self-generated dragon-like folding sequence.
  • A059126 (program): A hierarchical sequence (W2{2} according to the description in the attached file - see link).
  • A059129 (program): A hierarchical sequence (W2{2}* - see A059126).
  • A059130 (program): A hierarchical sequence (W2{3}* - see A059126).
  • A059132 (program): A hierarchical sequence (W2{2}c - see A059126).
  • A059133 (program): A hierarchical sequence (S(W2{2}c) - see A059126).
  • A059134 (program): A hierarchical sequence (W2{3}c - see A059126).
  • A059135 (program): A hierarchical sequence (S(W2{3}c) - see A059126).
  • A059136 (program): A hierarchical sequence (W3{2,2}c - see A059126).
  • A059137 (program): A hierarchical sequence (W3{2,2}cc - see A059126).
  • A059138 (program): A hierarchical sequence (S(W3{2,2}cc) - see A059126).
  • A059139 (program): A hierarchical sequence (W2{2}*c - see A059126).
  • A059140 (program): A hierarchical sequence (S(W2{2}*c) - see A059126).
  • A059141 (program): A hierarchical sequence (W2{3}*c - see A059126).
  • A059142 (program): A hierarchical sequence (S(W2{3}*) - see A059126).
  • A059143 (program): A hierarchical sequence (W3{2,2}*c - see A059126).
  • A059144 (program): A hierarchical sequence (W3{2,2}*cc - see A059126).
  • A059145 (program): A hierarchical sequence (S(W3{2,2}*cc) - see A059126).
  • A059146 (program): A hierarchical sequence (W’2{2} - see A059126).
  • A059147 (program): A hierarchical sequence (W’2{3} - see A059126).
  • A059149 (program): A hierarchical sequence (W’2{2}* - see A059126).
  • A059150 (program): A hierarchical sequence (W’2{3}* - see A059126).
  • A059152 (program): A hierarchical sequence (W’2{2}c - see A059126).
  • A059153 (program): a(n) = 2^(n+2)*(2^(n+1)-1).
  • A059154 (program): A hierarchical sequence (W’2{3}c - see A059126).
  • A059155 (program): A hierarchical sequence (S(W’2{3}c) - see A059126).
  • A059156 (program): A hierarchical sequence (W’3{2,2}c - see A059126).
  • A059157 (program): A hierarchical sequence (W’3{2,2}cc - see A059126).
  • A059158 (program): A hierarchical sequence (S(W’3{2,2}cc) - see A059126).
  • A059159 (program): A hierarchical sequence (W’2{2}*c) - see A059126).
  • A059161 (program): A hierarchical sequence (W’2{3}*c - see A059126).
  • A059162 (program): A hierarchical sequence (S(W’2{3}*c) - see A059126).
  • A059163 (program): A hierarchical sequence (W’3{2,2}*c - see A059126).
  • A059164 (program): A hierarchical sequence (W’3{2,2}*cc - see A059126).
  • A059165 (program): a(n) = (n+1)*2^(n+4).
  • A059169 (program): Number of partitions of n into 3 parts which form the sides of a nondegenerate isosceles triangle.
  • A059171 (program): Size of largest conjugacy class in S_n, the symmetric group on n symbols.
  • A059173 (program): Maximal number of regions into which 4-space can be divided by n hyper-spheres.
  • A059174 (program): Maximal number of regions into which 5-space can be divided by n hyper-spheres.
  • A059193 (program): Engel expansion of 1/e = 0.367879… .
  • A059204 (program): Number of non-unimodal permutations of n items (i.e., those which do not simply go up for the first part and then down for the rest, but at some point go down then up).
  • A059214 (program): Square array T(k,n) = C(n-1,k) + Sum_{i=0..k} C(n,i) read by antidiagonals (k >= 1, n >= 1).
  • A059222 (program): Minimal number of disjoint edge-paths into which the graph of the n-ary cube can be partitioned.
  • A059224 (program): a(n) = 2^(n-3)*(n + 3)*(2*n - 3).
  • A059231 (program): Number of different lattice paths running from (0,0) to (n,0) using steps from S = {(k,k) or (k,-k): k positive integer} that never go below the x-axis.
  • A059238 (program): Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.
  • A059246 (program): Numerator of Sum_{j=1..n} d(j)/n, where d = number of divisors function (A000005).
  • A059247 (program): Denominator of Sum_{j=1..n} d(j)/n, where d = number of divisors function (A000005).
  • A059248 (program): Numerator of 1/F(1) + 1/F(2) + 1/F(3) + … + 1/F(n), where F(n) is the n-th Fibonacci number (A000045).
  • A059249 (program): Tersum n + (n-1); write n and n-1 in base 3 and add mod 3 with no carries.
  • A059250 (program): Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0.
  • A059255 (program): Both sum of n+1 consecutive squares and sum of the immediately following n consecutive squares.
  • A059258 (program): Primes p such that x^53 = 2 has no solution mod p.
  • A059259 (program): Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-x-x*y-y^2) = 1/((1+y)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), …
  • A059260 (program): Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-y-x*y-x^2) = 1/((1+x)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), …
  • A059267 (program): Numbers n with 2 divisors d1 and d2 having difference 2: d2 - d1 = 2; equivalently, numbers that are 0 (mod 4) or have a divisor d of the form d = m^2 - 1.
  • A059268 (program): Concatenate subsequences [2^0, 2^1, …, 2^n] for n = 0, 1, 2, …
  • A059269 (program): Numbers m for which the number of divisors, tau(m), is divisible by 3.
  • A059270 (program): a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.
  • A059278 (program): G.f. is G(x*(1-x)/(1-2*x)) where G(x) is g.f. for Catalan numbers A000108.
  • A059279 (program): G.f. is ((1-x)/(1-2*x)) * G(x*(1-x)/(1-2*x)) where G(x) is g.f. for Catalan numbers A000108.
  • A059280 (program): Expansion of e.g.f. exp(x*(1-x)/(1-2*x)).
  • A059281 (program): E.g.f.: ((1-x)/(1-2*x)) * exp(x*(1-x)/(1-2*x)).
  • A059284 (program): Right edge of triangle in A059283.
  • A059288 (program): a(n) = binomial(2*n,n) mod n.
  • A059289 (program): a(n) = 1 + (binomial(2n,n) mod n).
  • A059290 (program): a(n) = round(1/144*n^2*(n + 3)).
  • A059291 (program): a(n) = round((n-1)^2*(n+5)/144).
  • A059292 (program): a(n) = n + 2 - (number of divisors of n).
  • A059293 (program): a(n) = round(n*(5*n - 14)/12) + 1.
  • A059297 (program): Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.
  • A059298 (program): Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.
  • A059299 (program): Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 3.
  • A059300 (program): Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.
  • A059302 (program): A diagonal of A008296.
  • A059304 (program): a(n) = 2^n * (2*n)! / (n!)^2.
  • A059306 (program): Number of 2 X 2 singular integer matrices with elements from {0,…,n}.
  • A059316 (program): Least integer m such that between m and 2m (including endpoints) there are exactly n primes.
  • A059321 (program): Smallest number m such that m^2+1 is divisible by A002144(n)^2 (= squares of primes congruent to 1 mod 4).
  • A059322 (program): First differences of sequence of consecutive safe primes.
  • A059324 (program): Numbers n such that 6n + 5 is composite.
  • A059325 (program): Numbers n such that 6n + 5 is prime.
  • A059328 (program): Table T(n,k) = T(n - 1,k) + T(n,k - 1) + T(n - 1,k)*T(n,k - 1) starting with T(0,0)=1, read by antidiagonals.
  • A059329 (program): Number of 3 X 3 matrices, with elements from {0,…,n}, having the property that the middle element of each of the eight 3-element horizontal, vertical and diagonal lines equals the average of the two end elements.
  • A059332 (program): Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.
  • A059338 (program): a(n) = Sum_{k=1..n} k^5 * binomial(n,k).
  • A059342 (program): Triangle giving denominators of coefficients of Euler polynomials, highest powers first.
  • A059343 (program): Triangle of nonzero coefficients of Hermite polynomials H_n(x) in increasing powers of x.
  • A059344 (program): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.
  • A059346 (program): Difference array of Catalan numbers A000108 read by antidiagonals.
  • A059347 (program): Difference array of Motzkin numbers A001006 read by antidiagonals.
  • A059348 (program): Third diagonal of array in A059347.
  • A059358 (program): Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^4.
  • A059365 (program): Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1) - binomial(2*r-s-1,r), r>=0, 0 <= s <= r.
  • A059366 (program): Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.
  • A059371 (program): a(n) = (n-1)! + ((n+1)/2)*a(n-1), a(1)=0.
  • A059374 (program): Triangle T(n,k)=Sum_{i=0..n} L’(n,n-i)*binomial(i,k), k=0..n-1.
  • A059375 (program): Number of seating arrangements for the ménage problem.
  • A059376 (program): Jordan function J_3(n).
  • A059377 (program): Jordan function J_4(n).
  • A059378 (program): Jordan function J_5(n).
  • A059381 (program): Product J_2(i), i=1..n.
  • A059382 (program): Product J_3(i), i=1..n.
  • A059383 (program): Product J_4(i), i=1..n.
  • A059384 (program): a(n) = Product_{i=1..n} J_5(i).
  • A059387 (program): Jordan function J_n(6) (see A059379).
  • A059389 (program): Sums of two nonzero Fibonacci numbers.
  • A059390 (program): Numbers that are not the sum of two nonzero Fibonacci numbers.
  • A059396 (program): Number of primes less than square root of n-th prime; i.e., number of trial divisions by smaller primes to show that n-th prime is indeed prime.
  • A059398 (program): Row sums of triangle in A059397.
  • A059399 (program): Triangular hopscotch.
  • A059403 (program): Quarter-squared applied twice.
  • A059404 (program): Numbers with different exponents in their prime factorizations.
  • A059409 (program): a(n) = 4^n * (2^n - 1).
  • A059410 (program): J_n(9) (see A059379).
  • A059412 (program): Number of distinct minimal unary DFA’s with exactly n states.
  • A059413 (program): Number of distinct languages accepted by unary DFA’s with n states.
  • A059417 (program): Start with 1; square; add 2; subtract 1; repeat.
  • A059420 (program): A diagonal of A059419.
  • A059421 (program): A diagonal of A059419.
  • A059422 (program): Difference between number of even equivalence classes and odd classes of terms in a symmetric determinant of order n.
  • A059425 (program): Primes of form n^2 + 19n + 17.
  • A059426 (program): First differences of A026273.
  • A059428 (program): Number of points of rotation in a prime block spiral.
  • A059435 (program): Number of lattice paths in plane starting at (0,0) and ending at (n,n) with steps from {(i,j): i+j > 0, i, j >= 0} that never go below the line y = x.
  • A059439 (program): A diagonal of A059438.
  • A059446 (program): Triangle T(n,k) = coefficient of x^n*y^k/(n!*k!) in 1/(1-x-y-x*y), read by rows in order 00, 10, 01, 20, 11, 02, …
  • A059448 (program): The parity of the number of zero digits when n is written in binary.
  • A059451 (program): Number of ways n can be written as the sum of two numbers whose binary expansions have even numbers of zeros; also number of ways n can be written as the sum of two numbers whose binary expansions have odd numbers of zeros.
  • A059452 (program): Safe primes (A005385) which are not Sophie Germain primes.
  • A059453 (program): Sophie Germain primes (A005384) which are not safe primes (A005385).
  • A059455 (program): Safe primes which are also Sophie Germain primes.
  • A059456 (program): Unsafe primes: primes not in A005385.
  • A059457 (program): Numerator of Sum_{k=0..n} (-1)^k/(3*k+1).
  • A059460 (program): Iteration of unitary-sigma function: a(1) = 2, a(n) = usigma(a(n-1)).
  • A059473 (program): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, …
  • A059474 (program): Triangle read by rows: T(n,k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w + 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, …
  • A059479 (program): Number of 3 X 3 matrices with elements from {0,…,n-1} such that the middle element of each of the eight lines of three (rows, columns and diagonals) is the square (mod n) of the difference of the end elements.
  • A059480 (program): a(0) = a(1) = 1; a(n) = a(n-1) + (n+1)*a(n-2).
  • A059481 (program): Triangle T(n,k) = binomial(n+k-1,k), 0 <= k <= n, read by rows.
  • A059482 (program): a(0)=1, a(n) = a(n-1) + 8*10^(n-1).
  • A059485 (program): Highest prime factor is greater than 3.
  • A059502 (program): a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.
  • A059505 (program): Transform of A059502 applied to sequence 2,3,4,…
  • A059506 (program): Transform of A059502 applied to sequence 3,4,5,…
  • A059507 (program): Transform of A059502 applied to sequence 4,5,6,…
  • A059508 (program): Transform of A059502 applied to sequence 5,6,7,…
  • A059509 (program): Main diagonal of the array A059503.
  • A059512 (program): For n>=2, the number of (s(0), s(1), …, s(n-1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,….,n-1, s(0) = 2, s(n-1) = 2.
  • A059516 (program): Number of different relations between n intervals (possibly of zero length) on a line.
  • A059517 (program): The sequence A059515(3,n). Number of ways of placing n identifiable nonnegative intervals with a total of exactly three starting and/or finishing points.
  • A059522 (program): a(1) = 1, then a(n) = n*(n^(n-1)-1)*(n-2)!/(n-1).
  • A059531 (program): Beatty sequence for 1 + 1/Pi.
  • A059532 (program): Beatty sequence for 1 + Pi.
  • A059533 (program): Beatty sequence for 1 + Catalan’s constant.
  • A059534 (program): Beatty sequence for 1 + 1/Catalan’s constant.
  • A059535 (program): Beatty sequence for Pi^2/6, or zeta(2).
  • A059536 (program): Beatty sequence for zeta(2)/(zeta(2)-1).
  • A059537 (program): Beatty sequence for zeta(3).
  • A059539 (program): Beatty sequence for 3^(1/3).
  • A059540 (program): Beatty sequence for 3^(1/3)/(3^(1/3)-1).
  • A059541 (program): Beatty sequence for 1 + log(2).
  • A059542 (program): Beatty sequence for 1 + 1/log(2).
  • A059544 (program): Beatty sequence for log(3)/(log(3)-1).
  • A059545 (program): Beatty sequence for log(10).
  • A059546 (program): Beatty sequence for log(10)/(log(10)-1).
  • A059547 (program): Beatty sequence for 1 + 1/log(3).
  • A059549 (program): Beatty sequence for 1 + 1/log(10).
  • A059550 (program): Beatty sequence for 1 + log(10).
  • A059551 (program): Beatty sequence for Gamma(1/3).
  • A059552 (program): Beatty sequence for Gamma(1/3)/(Gamma(1/3)-1).
  • A059553 (program): Beatty sequence for Gamma(2/3).
  • A059554 (program): Beatty sequence for Gamma(2/3)/(Gamma(2/3)-1).
  • A059555 (program): Beatty sequence for 1 + gamma A001620.
  • A059556 (program): Beatty sequence for 1 + 1/gamma.
  • A059557 (program): Beatty sequence for 1 + gamma^2, (gamma is the Euler-Mascheroni constant A001620).
  • A059558 (program): Beatty sequence for 1 + 1/gamma^2.
  • A059559 (program): Beatty sequence for 1 + log(1/gamma), (gamma is the Euler-Mascheroni constant A001620).
  • A059560 (program): Beatty sequence for 1 - 1/log(gamma).
  • A059561 (program): Beatty sequence for log(Pi).
  • A059562 (program): Beatty sequence for log(Pi)/(log(Pi)-1).
  • A059563 (program): Beatty sequence for e + 1/e.
  • A059564 (program): Beatty sequence for (e^2 + 1)/(e^2 - e + 1).
  • A059565 (program): Beatty sequence for e^gamma (gamma is the Euler-Mascheroni constant A001620).
  • A059566 (program): Beatty sequence for e^gamma/(e^gamma-1).
  • A059567 (program): Beatty sequence for 1 - log(log(2)).
  • A059568 (program): Beatty sequence for 1 - 1/log(log(2)).
  • A059570 (program): Number of fixed points in all 231-avoiding involutions in S_n.
  • A059576 (program): Summatory Pascal triangle T(n,k) (0 <= k <= n) read by rows. Top entry is 1. Each entry is the sum of the parallelogram above it.
  • A059582 (program): First differences give digits of Pi = 3.1415926…
  • A059585 (program): Number of labeled 3-node T_0-hypergraphs with n hyperedges (empty hyperedges and multiple hyperedges included).
  • A059590 (program): Numbers obtained by reinterpreting base-2 representation of n in the factorial base: a(n) = Sum_{k>=0} A030308(n,k)*A000142(k+1).
  • A059591 (program): Squarefree part of n^2+1.
  • A059592 (program): Square-full part of n^2+1.
  • A059593 (program): Number of degree-n permutations of order exactly 5.
  • A059594 (program): Convolution triangle based on A008619 (positive integers repeated).
  • A059595 (program): Seventh column (m=6) of convolution triangle A059594(n,m).
  • A059596 (program): Eighth column (m=7) of convolution triangle A059594(n,m).
  • A059597 (program): Ninth column (m=8) of convolution triangle A059594(n,m).
  • A059598 (program): Tenth column (m=9) of convolution triangle A059594(n,m).
  • A059599 (program): Expansion of (3+x)/(1-x)^6.
  • A059600 (program): Expansion of (1+6*x+x^2)/(1-x)^8.
  • A059601 (program): Expansion of (1+10*x+5*x^2)/(1-x)^10.
  • A059602 (program): Expansion of (5+10*x+x^2)/(1-x)^10.
  • A059603 (program): Expansion of (1+15*x+15*x^2+x^3)/(1-x)^12.
  • A059605 (program): a(n) = (1/3!)*(n^3 + 24*n^2 + 107*n + 90), compare A059604.
  • A059606 (program): Expansion of (1/2)*(exp(2*x)-1)*exp(exp(x)-1).
  • A059616 (program): Numerator of (n*(n-1)/(8*(2*n+1)).
  • A059617 (program): Denominator of (n*(n-1)/(8*(2*n+1)).
  • A059620 (program): Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, start with A0 = the 0th key.
  • A059624 (program): Expansion of (3+10*x+3*x^2)/(1-x)^12.
  • A059625 (program): Eleventh column (m=10) of convolution triangle A059594.
  • A059626 (program): Generalized nim sum n + n + n in base 10; carryless multiplication 3 X n base 10.
  • A059627 (program): Generalized nim sum n + n + n + n in base 10; carryless multiplication 4 X n base 10.
  • A059628 (program): Carryless multiplication 5 X n base 10.
  • A059629 (program): Carryless multiplication 6 X n base 10.
  • A059630 (program): Carryless multiplication 7 X n base 10.
  • A059631 (program): Carryless multiplication 8 X n base 10.
  • A059633 (program): G.f.: x^3/(1 - 2*x + x^3 - x^4). Recurrence: a(n) = 2*a(n-1) - a(n-3) + a(n-4).
  • A059648 (program): a(n) = [[(k^2)*n]-(k*[k*n])], where k = sqrt(2) and [] is the floor function.
  • A059649 (program): Positions of ones in A059648.
  • A059650 (program): First differences of A059649.
  • A059655 (program): Positions of minus ones (-1’s) in A059652.
  • A059656 (program): First differences of A059655.
  • A059672 (program): Sum of binary numbers with n 1’s and one (possibly leading) 0.
  • A059673 (program): Sum of binary numbers with n 1’s and one (non-leading) 0.
  • A059674 (program): Square array a(m,n) = binomial(max(m,n), min(m,n)) (m>=0, n>=0) read by antidiagonals.
  • A059678 (program): Triangle T(n,k) giving number of fixed 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).
  • A059708 (program): Numbers n such that all digits have same parity.
  • A059712 (program): Number of stacked directed animals on the square lattice.
  • A059714 (program): Number of stacked directed animals on the triangular lattice.
  • A059716 (program): Number of column convex polyominoes with n hexagonal cells.
  • A059721 (program): Mean of first six positive powers of n, i.e., (n + n^2 + n^3 + n^4 + n^5 + n^6)/6.
  • A059722 (program): a(n) = n*(2*n^2 - 2*n + 1).
  • A059727 (program): a(n) = Fibonacci(n)*(Fibonacci(n) + 1).
  • A059730 (program): Third diagonal of A059922.
  • A059734 (program): Carryless 11^n base 10; a(n) is carryless sum of 10*a(n-1) and a(n-1).
  • A059738 (program): Binomial transform of A054341 and inverse binomial transform of A049027.
  • A059753 (program): Minimal degree of a height one multiple of (x-1)^n.
  • A059760 (program): a(n) is the number of edges (one-dimensional faces) in the convex polytope of real n X n doubly stochastic matrices.
  • A059765 (program): Possible sizes of the torsion group of an elliptic curve over the rationals Q. This is a finite sequence.
  • A059769 (program): Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Fibonacci numbers.
  • A059777 (program): Number of self-conjugate three-quadrant Ferrers graphs that partition n.
  • A059778 (program): Expansion of 1 / product((1+q^(2*n+3))/(1-q^(2*n+2)), n=0..inf).
  • A059781 (program): Triangle T(n,k) giving exponent of power of 2 dividing entry (n,k) of trinomial triangle A027907.
  • A059782 (program): Triangle T(n,k) giving exponent of power of 3 dividing entry (n,k) of trinomial triangle A027907.
  • A059786 (program): Smallest prime after 2*(n-th prime).
  • A059787 (program): Distance between 2*(n-th prime) and next prime.
  • A059788 (program): a(n) = largest prime < 2*prime(n).
  • A059789 (program): Distance of 2*Prime[n] from previous prime.
  • A059793 (program): Stationary value of quotient in the continued fraction expansion of sqrt(prime) when the quotient-cycle-length = 1.
  • A059794 (program): a(n) = n* - 2^(n-1), where n* (A003418) = least common multiple of the numbers [1,…,n].
  • A059797 (program): Second in a series of arrays counting standard tableaux by partition type.
  • A059811 (program): Let g_n be the ball packing n-width for the manifold torus X interval; sequence gives numerator of (g_n/Pi)^2.
  • A059813 (program): Let s_n be the simplex packing n-width for the manifold torus X interval; sequence gives numerator of s_n/Pi.
  • A059817 (program): Let s_n be the simplex packing n-width for the manifold torus X square; sequence gives numerator of s_n/Pi.
  • A059826 (program): a(n) = (n^2 - n + 1)*(n^2 + n + 1).
  • A059827 (program): Cubes of triangular numbers: (n*(n+1)/2)^3.
  • A059830 (program): a(n) = n^6 + n^4 + n^2 + 1.
  • A059831 (program): Determinant of Wilkinson’s eigenvalue test matrix of order 2n+1.
  • A059832 (program): A ternary tribonacci triangle: form the triangle as follows: start with 3 single values: 1, 2, 3. Each succeeding row is a concatenation of the previous 3 rows.
  • A059833 (program): “Madonna’s Sequence”: add 1 (mod 10) to each digit of Pi.
  • A059834 (program): Sum of squares of entries of Wilkinson’s eigenvalue test matrix of order 2n+1.
  • A059835 (program): Form triangle as follows: start with three single digits: 0, 1, 2. Each succeeding row is a concatenation of the previous three rows.
  • A059836 (program): Triangle T(s,t), s>=1, 1<=t<=s (see formula line).
  • A059837 (program): Diagonal T(s,s) of triangle A059836.
  • A059838 (program): Number of permutations in the symmetric group S_n that have even order.
  • A059839 (program): a(n) = n^8 + n^6 + n^4 + n^2 + 1.
  • A059840 (program): a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.
  • A059841 (program): Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.
  • A059843 (program): a(n) is the smallest prime p such that p-n is a nonzero square.
  • A059844 (program): a(n) = smallest nonzero square x^2 such that n+x^2 is prime.
  • A059845 (program): a(n) = n*(3*n + 11)/2.
  • A059848 (program): As a square table by antidiagonals, the n-digit number which in base k starts 1010101…
  • A059851 (program): a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + … (this is a finite sum).
  • A059855 (program): Period of continued fraction for sqrt(n^2+4), n >= 1.
  • A059857 (program): Alternating clock-face numbers.
  • A059859 (program): Sum of squares of first n quarter-squares (A002620).
  • A059860 (program): a(n) = binomial(n+1, 2)^5.
  • A059861 (program): a(n) = Product_{i=2..n} (prime(i) - 2).
  • A059862 (program): a(n) = Product_{i=3..n} (prime(i) - 3).
  • A059863 (program): a(n) = Product_{i=3..n} (prime(i)-4).
  • A059864 (program): a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.
  • A059865 (program): Product_{i=4..n} (prime(i) - 6).
  • A059867 (program): Number of irreducible representations of the symmetric group S_n that have odd degree.
  • A059893 (program): Reverse the order of all but the most significant bit in binary expansion of n: if n = 1ab..yz then a(n) = 1zy..ba.
  • A059894 (program): Complement and reverse the order of all but the most significant bit in binary expansion of n. n = 1ab..yz -> 1ZY..BA = a(n), where A = 1-a, B = 1-b, … .
  • A059905 (program): Index of first half of decomposition of integers into pairs based on A000695.
  • A059906 (program): Index of second half of decomposition of integers into pairs based on A000695.
  • A059920 (program): If m/n = q + r/n (r < n, n,m >=1), then array a(m,n) = qr (meaning q followed by r). Read by antidiagonals.
  • A059924 (program): Write the numbers from 1 to n^2 in a spiraling square; a(n) is the total of the sums of the two diagonals.
  • A059929 (program): a(n) = Fibonacci(n)*Fibonacci(n+2).
  • A059937 (program): Sum of binary numbers with n 1’s and two (possibly leading) 0’s.
  • A059938 (program): Sum of binary numbers with n 1’s and two (non-leading) 0’s.
  • A059939 (program): a(n) = floor(log_2(n+1) - 1).
  • A059944 (program): Denominators of Maclaurin series coefficients for 2*cos(x/sqrt(3) + arctan(-sqrt(3))) = cos(x/sqrt(3)) + sqrt(3)*sin(x/sqrt(3)).
  • A059945 (program): Number of 4-block bicoverings of an n-set.
  • A059952 (program): Ordering of a deck of 52 cards after an in-shuffle.
  • A059953 (program): Ordering of a deck of 52 cards after an out-shuffle.
  • A059956 (program): Decimal expansion of 6/Pi^2.
  • A059957 (program): Sum of distinct prime factors of n and n+1, or number of prime factors of n(n+1) or of lcm(n,n+1).
  • A059966 (program): a(n) = (1/n) * Sum_{ d divides n } mu(n/d) * (2^d - 1).
  • A059967 (program): Number of 9-ary trees.
  • A059968 (program): 10-ary trees.
  • A059973 (program): Expansion of x*(1 + x - 2*x^2) / ( 1 - 4*x^2 - x^4).
  • A059974 (program): a(n)=a(p)+a(q) where p and q are the two primes less than n and closest to n; with a(1)=1, a(2)=1.
  • A059975 (program): a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors.
  • A059977 (program): a(n) = binomial(n+2, 2)^4.
  • A059978 (program): a(n) = binomial(n+2,n)^6.
  • A059979 (program): Number of 7-dimensional cage assemblies.
  • A059980 (program): Number of 8-dimensional cage assemblies.
  • A059981 (program): Order of compositeness for the n-th composite number.
  • A059985 (program): Łukasiewicz words as integers written in factorial base.
  • A059986 (program): Number of rods required to make a 3-D cube of side length n.
  • A059988 (program): a(n) = (10^n - 1)^2.
  • A059989 (program): Numbers n such that 3*n+1 and 4*n+1 are both squares.
  • A059990 (program): Number of points of period n under the dual of the map x->2x on Z[1/6].
  • A059991 (program): a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).
  • A059993 (program): Pinwheel numbers: a(n) = 2*n^2 + 6*n + 1.
  • A059995 (program): Drop the final digit of n.
  • A059997 (program): a(n) = (n/2)*(n + 1)*(3*n + 11).
  • A059999 (program): a(n) = (1/6)*n^5 - (19/8)*n^4 + (51/4)*n^3 - (253/8)*n^2 + (445/12)*n - 14.
  • A060001 (program): a(n) = Fibonacci(n)!.
  • A060006 (program): Decimal expansion of real root of x^3 - x - 1 (the plastic constant).
  • A060007 (program): Decimal expansion of v_4, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
  • A060008 (program): a(n) = 9*binomial(n,4) = 3n*(n-1)*(n-2)*(n-3)/8.
  • A060011 (program): Schizophrenic sequence: these are the repeating digits in the decimal expansion of sqrt(f(2n+1)), where f(m) = A014824(m).
  • A060013 (program): New record highs reached in A060000.
  • A060018 (program): a(n) = floor(2*sqrt(n-2)).
  • A060019 (program): a(n) = floor(2*sqrt(prime(n)-2)) where prime(n) = n-th prime.
  • A060020 (program): Maximal size of a nonspanning subset of any Abelian group of order n.
  • A060022 (program): Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)…(1-x^N)) for N = 3.
  • A060036 (program): Triangular array T read by rows: T(n,k) = k^2 mod n, for k = 1,2,…,n-1, n = 2,3,…
  • A060037 (program): Triangular array T read by rows: T(n,k)=k^2 mod n, for k=1,2,…,[n/2], n=2,3,…
  • A060040 (program): Square array T(n,k) (n >= 2, k >= 1) giving smallest positive integer m such that any set of m points in general position in R^n contains k points in convex position, read by antidiagonals.
  • A060054 (program): Numerators of numbers appearing in the Euler-Maclaurin summation formula.
  • A060055 (program): Denominators of nonzero numbers appearing in the Euler-Maclaurin summation formula. (See A060054 for the definition of these numbers.)
  • A060056 (program): Nonzero numbers in expansion of ((tan(x))^4)/4! in (x^n)/n!.
  • A060060 (program): Third column of triangle A060058.
  • A060065 (program): Smallest mode of the sequence { C(n-k,k), k=0..n/2 }.
  • A060067 (program): Largest power of n which divides n!.
  • A060068 (program): Divide n! by largest power of n which will leave the result an integer.
  • A060072 (program): a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.
  • A060073 (program): a(n) = (n^(n-1)-1)/(n-1)^2.
  • A060075 (program): Third column of triangle A060074.
  • A060076 (program): Fourth column of triangle A060074.
  • A060080 (program): Scaled sums of squares.
  • A060086 (program): Convolution triangle A059594 with extra first column.
  • A060091 (program): Number of 4-block ordered bicoverings of an unlabeled n-set.
  • A060097 (program): Denominator of coefficients of Euler polynomials (rising powers).
  • A060098 (program): Triangle of partial sums of column sequences of triangle A060086.
  • A060099 (program): G.f.: 1/((1-x^2)^3*(1-x)^4).
  • A060100 (program): Fifth column (m=4) of triangle A060098.
  • A060101 (program): Sixth column (m=5) of triangle A060098.
  • A060102 (program): Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).
  • A060103 (program): Fourth column (m=3) of triangle A060102.
  • A060104 (program): Fifth column (m=4) of triangle A060102.
  • A060105 (program): Sixth column (m=5) of triangle A060102.
  • A060106 (program): Numbers that are congruent to {1, 4, 6, 9, 11} mod 12. The ebony keys on a piano, starting with A0 = the 0th key.
  • A060107 (program): Numbers that are congruent to {0, 2, 3, 5, 7, 8, 10} mod 12. The ivory keys on a piano, start with A0 = the 0th key.
  • A060108 (program): Sequence of sums based on primes = 7 mod 8.
  • A060112 (program): Sums of nonconsecutive factorial numbers.
  • A060130 (program): Number of nonzero digits in factorial base representation (A007623) of n; minimum number of transpositions needed to compose each permutation in the lists A060117 & A060118.
  • A060135 (program): Sequence of adjacent transpositions (a[n] a[n]+1), which, when starting from the identity permutation and applied successively, produce a Hamiltonian circuit through all permutations of S_4, in such a way that S_{n-1} is always traversed before the rest of S_n. Furthermore, each subsequence from the first to the (n!-1)-th term is palindromic.
  • A060136 (program): Square array read by antidiagonals with T(n,k)=T(n,k-1)^2+n*T(n,k-1)+1 and T(n,0)=0.
  • A060137 (program): Square array read by antidiagonals with T(n,k)=T(n,k-1)^2-n*T(n,k-1)+1 and T(n,0)=0.
  • A060138 (program): Ordered set S defined by these rules: 0 and 2 are in S and if x is a nonzero number in S, then 2x-1 and 4x are in S.
  • A060140 (program): Ordered set S defined by these rules: 0 and 1 are in S and if x is a nonzero number in S, then 3x and 9x+1 are in S.
  • A060141 (program): Ordered set S defined by these rules: 0 and 1 are in S and if x is a nonzero number in S, then 3x and 9x+2 are in S.
  • A060142 (program): Ordered set S defined by these rules: 0 is in S and if x is in S then 2x+1 and 4x are in S.
  • A060143 (program): a(n) = floor(n/tau), where tau = (1 + sqrt(5))/2.
  • A060144 (program): a(n) = floor(n/(1+tau)), or equivalently floor(n/(tau)^2), where tau is the golden ratio (A001622).
  • A060145 (program): a(n) = floor(n/tau) - floor(n/(1 + tau)).
  • A060149 (program): Number of homogeneous generators of degree n for graded algebra associated with meanders.
  • A060150 (program): a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.
  • A060151 (program): Number of base n digits required to write n!.
  • A060154 (program): Table T(n,k) by antidiagonals of n^k mod k [n,k >= 1].
  • A060155 (program): Table T(n,k) by antidiagonals of floor[n^k/k] [n,k >= 1].
  • A060156 (program): a(n) = floor(10^n/n).
  • A060157 (program): Number of permutations of [n] with 3 sequences.
  • A060158 (program): Number of permutations of [n] with 4 sequences.
  • A060160 (program): a(n) = 2^n - 1 + Fibonacci(n-1)*2^(n+1).
  • A060161 (program): a(n) = 2^n - 1 + 2*Fibonacci(n-1).
  • A060163 (program): a(n) = (n^3 + 5*n + 18)/6.
  • A060175 (program): Table T(n,k) by antidiagonals of exponent of largest power of k-th prime which divides n.
  • A060176 (program): Table T(n,k) by antidiagonals of value of largest power of k-th prime which divides n.
  • A060182 (program): a(0) = 1, a(1) = 5, a(2) = 13; a(n) = 2*a(n-1) + 2, n > 2.
  • A060183 (program): a(0)=1, a(n) = 100*a(n-1) + 36*n - 128.
  • A060187 (program): Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).
  • A060188 (program): A column and diagonal of A060187.
  • A060189 (program): A column and diagonal of A060187 (k=3).
  • A060190 (program): A column and diagonal of A060187 (k=4).
  • A060191 (program): Union_i p(4i), p(4i+1), where p(k) = k-th prime.
  • A060192 (program): Union_i p(4i+2), p(4i+3), where p(k) = k-th prime.
  • A060193 (program): Partial products of A060191.
  • A060194 (program): Partial products of A060192.
  • A060195 (program): a(n) = 8^(n-1)*(2^n-1).
  • A060196 (program): Decimal expansion of 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + …
  • A060197 (program): Start at n, repeatedly apply pi(x) until reach 0; a(n) = number of steps to reach 0.
  • A060202 (program): Let G = complete graph on 4 vertices, create the sequence G, L(G), L(L(G)), L(L(L(G))), … where each graph in this sequence is the line graph of the previous graph; a(n) is number of vertices of the n-th graph in this sequence.
  • A060203 (program): Least cube root of unity mod p, greater than 1, where p is the n-th prime congruent to 1 mod 3.
  • A060208 (program): a(n) = 2*pi(n) - pi(2*n), where pi(i) = A000720(i).
  • A060210 (program): Largest prime factor of 1+smaller term of twin primes.
  • A060212 (program): Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.
  • A060213 (program): Lesser of twin primes whose average is 6 times a prime.
  • A060226 (program): a(n) = n^n - n*(n-1)^(n-1).
  • A060229 (program): Smaller member of a twin prime pair whose mean is a multiple of A002110(3)=30.
  • A060230 (program): Smaller of twin primes whose middle term is a multiple of A002110(4)=210.
  • A060231 (program): Smaller of twin primes whose middle term is a multiple of A002110(5)=2310.
  • A060234 (program): a(n) = (prime(n) mod (prime(n+1)-prime(n))).
  • A060236 (program): If n mod 3 = 0 then a(n) = a(n/3), otherwise a(n) = n mod 3.
  • A060237 (program): a(n) = n!^2 * Sum_{m=1..n}( Sum_{k=1..m} 1/(k*m) ).
  • A060238 (program): det(M) where M is an n X n matrix with M[i,j] = lcm(i,j).
  • A060242 (program): a(n) = (2^n - 1)*(4^n - 1).
  • A060249 (program): Size of the automorphism group of the symmetric group S_n.
  • A060254 (program): Primes which are the sum of two consecutive composite numbers.
  • A060264 (program): First prime after 2n.
  • A060265 (program): Largest prime less than 2n.
  • A060266 (program): Difference between 2n and the following prime.
  • A060267 (program): Difference between 2 closest primes surrounding 2n.
  • A060271 (program): Difference between smallest prime following and largest prime preceding 2*(n-th prime).
  • A060275 (program): At least two ordered triples of positive numbers have sum n and equal products.
  • A060278 (program): Sum of composite divisors of n less than n.
  • A060279 (program): Number of labeled rooted trees with all 2n nodes of odd degree.
  • A060285 (program): Number of partitions of n objects of 2 colors with parts size >1.
  • A060286 (program): 2^(p-1)*(2^p-1) where p is a prime.
  • A060293 (program): Expected coupon collection numbers rounded up; i.e., if aiming to collect a set of n coupons, the expected number of random coupons required to receive the full set.
  • A060294 (program): Decimal expansion of Buffon’s constant 2/Pi.
  • A060296 (program): Number of regular convex polytopes in n-dimensional space, or -1 if the number is infinite.
  • A060300 (program): a(n) = (2n(n+1))^2.
  • A060305 (program): Pisano periods for primes: period of Fibonacci numbers mod prime(n).
  • A060308 (program): Largest prime <= 2n.
  • A060311 (program): Expansion of e.g.f. exp((exp(x)-1)^2/2).
  • A060312 (program): Number of distinct ways to tile a 2 X n rectangle with dominoes (solutions are identified if they are rotations or reflections of each other).
  • A060313 (program): Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) on n labeled nodes.
  • A060318 (program): Powers of 3 in the odd Catalan numbers Catalan(2^n - 1).
  • A060325 (program): a(n) = n-th prime prime(n) subtracted from sum of all composites between prime(n) and prime(n-1).
  • A060328 (program): Primes which are the sum of three consecutive composite numbers.
  • A060336 (program): Number of n X n {-1,0,1} matrices modulo rows permutation (by symmetry this is the same as the number of {-1,0,1} matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other.
  • A060337 (program): Number of labeled acyclic digraphs with n nodes containing exactly n-2 points of in-degree zero.
  • A060340 (program): Possible sizes of the torsion group of an elliptic curve over some quadratic extension of the rationals Q. This is the full sequence.
  • A060343 (program): Smallest prime which is the sum of n composite numbers.
  • A060344 (program): For n >= 2, let N_n denote the set of all unipotent upper-triangular real n X n matrices A such that for every k=1,2,…,n-1 the minor of A with rows 1,2,…,k and columns n-k+1,…,n is nonzero. a(n) is the number of connected components of N_n.
  • A060348 (program): a(n) = n^n * (n^2 - 1)/24.
  • A060352 (program): a(n) = n*3^n - 1.
  • A060354 (program): The n-th n-gonal number: a(n) = n*(n^2 - 3*n + 4)/2.
  • A060356 (program): Expansion of e.g.f.: -LambertW(-x/(1+x)).
  • A060365 (program): Multiples of one thousand which are described by single words in American English.
  • A060366 (program): Powers of one thousand which are described by single words in dated British English usage, extended by using “-ard” beyond 10^9.
  • A060367 (program): Average order of an element in a cyclic group of order n rounded down.
  • A060368 (program): Number of irreducible representations of the symmetric group S_n that have even degree.
  • A060371 (program): a(n) = (prime(n) - 1)! + 1.
  • A060372 (program): p(n), positive part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.
  • A060373 (program): q(n), negative part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.
  • A060374 (program): a(n)=p+q, where n=p-q and p, q, p+q are in A005836 (integers written without 2 in base 3).
  • A060375 (program): a(n) = (n+2)^(n+3) - n^(n+1).
  • A060378 (program): Even-odd sieve.
  • A060381 (program): a(n) = prime(n)*prime(n+1)*…*prime(2*n-1), where prime(i) is the i-th prime.
  • A060384 (program): Number of decimal digits in n-th Fibonacci number.
  • A060389 (program): a(1)=p_1, a(2)=p_1 + p_1*p_2, a(3)=p_1 + p_1*p_2 + p_1*p_2*p_3, … where p_i is the i-th prime.
  • A060390 (program): Fibonacci sieve: using Fibonacci numbers, strike out every 2nd, 3rd, 5th, 8th, 13th, 21st, 34th… of those remaining.
  • A060401 (program): a(n) = minimal m such that m>n, n divides m, n-1 divides m-1, n-2 divides m-2 and so on down to 1 divides m-n+1.
  • A060404 (program): G.f.: Sum_{k >= 1} (phi(k)/k)*log(1-f(x^k)), where f(x) = (1 - sqrt(1 - 4*x)) / (2*x) - 1 is the g.f. for the Catalan numbers (A000108) C_1, C_2, C_3, …
  • A060405 (program): Sum of Lucas (A000032) and Pell (A000129) numbers.
  • A060416 (program): a(n) = n*4^n - 1.
  • A060418 (program): Largest decimal digit in n-th prime.
  • A060420 (program): Least decimal digit in n-th prime.
  • A060422 (program): Number of acute triangles made from vertices of a regular n-gon.
  • A060423 (program): Number of obtuse triangles made from vertices of a regular n-gon.
  • A060428 (program): Numbers of form x^2 + xy + y^2 (with repetitions if more than one representation is possible).
  • A060429 (program): a(n) = 4*prime(n)^2+1.
  • A060431 (program): Number of cubefree numbers <= n.
  • A060432 (program): Partial sums of A002024.
  • A060435 (program): Number of functions f: {1,2,…,n} -> {1,2,…,n} with even cycles only.
  • A060446 (program): Number of ways to color vertices of a pentagon using <= n colors, allowing rotations and reflections.
  • A060453 (program): Dot product of the squares and the quarter-squares: a(n) = sum(i=1..n, i^2 * floor(i^2/4)).
  • A060455 (program): 7th-order Fibonacci numbers with a(0)=…=a(6)=1.
  • A060458 (program): Maximal value seen in the final n decimal digits of 2^j for all values of j.
  • A060459 (program): a(n) = (n*(n+1))^3.
  • A060460 (program): Consider the final n decimal digits of 2^j for all values of j. They are periodic. Sequence gives position (or phase) of the maximal value seen in these n digits.
  • A060461 (program): Numbers k such that 6*k-1 and 6*k+1 are twin composites.
  • A060462 (program): Integers k such that k! is divisible by k*(k+1)/2.
  • A060464 (program): Numbers that are not congruent to 4 or 5 mod 9.
  • A060469 (program): Smallest positive a(n) such that number of solutions to a(n) = a(j)+a(k) j<k<n is one or less.
  • A060470 (program): Smallest positive a(n) such that number of solutions to a(n)=a(j)+a(k) j<k<n is two or less.
  • A060471 (program): Smallest positive a(n) such that number of solutions to a(n)=a(j)+a(k) j<k<n is three or less.
  • A060473 (program): a(n) = numerator of phi(n)/(n+1), where phi(n) is Euler’s phi, A000010.
  • A060474 (program): a(n) = denominator of phi(n)/(n+1), where phi(n) is Euler’s phi, A000010.
  • A060475 (program): Triangular array formed from successive differences of factorial numbers, then with factorials removed.
  • A060476 (program): Let n = 2^e_2 * 3^e_3 * 5^e_5 * … be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, …} is nonprime.
  • A060477 (program): Number of orbits of length n in map whose periodic points are A000051.
  • A060482 (program): New record highs reached in A060030.
  • A060483 (program): Number of 5-block tricoverings of an n-set.
  • A060488 (program): Number of 4-block ordered tricoverings of an unlabeled n-set.
  • A060493 (program): A diagonal of A036969.
  • A060494 (program): a(n) = floor(n^4/64).
  • A060505 (program): a(n) = floor(2^n/(n^2)).
  • A060507 (program): Denominators of the asymptotic expansion of the Airy function Ai(x).
  • A060508 (program): Exponent of largest power of n < 2^n.
  • A060509 (program): Largest power of n not exceeding 2^n.
  • A060510 (program): Alternating with hexagonal stutters: if n is hexagonal (2k^2 - k, i.e., A000384) then a(n)=a(n-1), otherwise a(n) = 1 - a(n-1).
  • A060511 (program): Hexagonal excess: smallest amount by which n exceeds a hexagonal number (2k^2-k, A000384).
  • A060521 (program): Number of 3 X n grids of black and white cells, no 3 of same color vertically or horizontally contiguous.
  • A060531 (program): 9th binomial transform of (1,0,1,0,1,…), A059841.
  • A060532 (program): Number of ways to color vertices of a heptagon using <= n colors, allowing rotations and reflections.
  • A060538 (program): Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.
  • A060539 (program): Table by antidiagonals of number of ways of choosing k items from n*k.
  • A060540 (program): Square array read by antidiagonals downwards: T(n,k) = (n*k)!/(k!^n*n!), (n>=1, k>=1), the number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.
  • A060541 (program): a(n) = binomial(4*n, 4).
  • A060542 (program): a(n) = (1/6)*multinomial(3*n;n,n,n).
  • A060543 (program): Triangle, read by antidiagonals, where T(n,k) = C(n+n*k+k, n*k+k).
  • A060544 (program): Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.
  • A060545 (program): a(n) = C(n^2,n)/n.
  • A060546 (program): a(n) = 2^ceiling(n/2).
  • A060547 (program): a(n) = 2^(floor(n/3) + ((n mod 3) mod 2)).
  • A060548 (program): a(n) is the number of D3-symmetric patterns that may be formed with a top-down equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal’s triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
  • A060549 (program): a(n) is the number of distinct patterns (modulo geometric D3-operations) with strict median-reflective (palindrome) symmetry (i.e., having no other symmetry) which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal’s triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
  • A060550 (program): a(n) is the number of distinct patterns (modulo geometric D_3-operations) with no other than strict 120-degree rotational symmetry which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal’s triangle modulo 2, where n is the number of cells in each edge of the arrangement.
  • A060553 (program): a(n) is the number of distinct (modulo geometric D3-operations) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal’s triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
  • A060556 (program): Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).
  • A060557 (program): Row sums of triangle A060556.
  • A060558 (program): Fifth column (m=4) of triangle A060556.
  • A060559 (program): One half of sixth column (m=5) of triangle A060556.
  • A060560 (program): Number of ways to color vertices of an octagon using <= n colors, allowing rotations and reflections.
  • A060561 (program): Number of ways to color vertices of a 9-gon using <= n colors, allowing rotations and reflections.
  • A060563 (program): First n digits after decimal point in the expansion of sqrt(n), or 0 if n is a square. Leading zeros omitted.
  • A060566 (program): a(n) = n^2 - 79*n + 1601.
  • A060569 (program): Consider Pythagorean triples which satisfy X^2+(X+7)^2=Z^2; sequence gives increasing values of Z.
  • A060571 (program): Tower of Hanoi: the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1 is on move n to move disk A001511 from peg A060571 (here) to peg A060572.
  • A060572 (program): Tower of Hanoi: the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1 is on move n to move disk A001511 from peg A060571 to peg A060572 (here).
  • A060573 (program): Tower of Hanoi: using the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1, a(n) is the smallest disk on peg 0 after n moves.
  • A060574 (program): Tower of Hanoi: using the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1, a(n) is the smallest disk on peg 1 after n moves (or 0 if there are no disks on peg 1).
  • A060575 (program): Tower of Hanoi: using the optimal way to move an even number of disks from peg 0 to peg 2 or an odd number from peg 0 to peg 1, a(n) is the smallest disk on peg 2 after n moves (or 0 if there are no disks on peg 2).
  • A060576 (program): Number of homeomorphically irreducible general graphs on 1 labeled node and with n edges.
  • A060577 (program): Number of homeomorphically irreducible general graphs on 2 labeled nodes and with n edges.
  • A060582 (program): If the final digit of n in base 3 is the same as a([n/3]) then this digit, otherwise a(n)= mod 3-sum of these two digits, with a(0)=0.
  • A060584 (program): Compare ultimate and penultimate digits of n base 3, i.e., 0 if n mod 3 = floor(n/3) mod 3, 1 otherwise; also 0 if (n mod 9) is a multiple of 4, 1 otherwise.
  • A060585 (program): Write n in base 3, then (working from left to right) if the k-th digit of n is not equal to the digit to its left then the k-th digit of a(n) is 1, otherwise it is 0, and finally read the result as a base-2 number.
  • A060587 (program): A ternary code: inverse of A060583.
  • A060588 (program): If the final two digits of n written in base 3 are the same then this digit, otherwise mod 3-sum of these two digits.
  • A060589 (program): a(n) = 2*(2^n-1)*3^(n-1).
  • A060590 (program): Numerator of the expected time to finish a random Tower of Hanoi problem with n disks using optimal moves.
  • A060593 (program): a(n) is the number of ways that a cycle of length 2n+1 in the symmetric group S_(2n+1) can be decomposed as the product of two cycles of length 2n+1.
  • A060594 (program): Number of solutions to x^2 == 1 (mod n), that is, square roots of unity modulo n.
  • A060602 (program): Number of tilings of the d-dimensional zonotope constructed from d+3 vectors.
  • A060603 (program): Number of ways of expressing an n-cycle in the symmetric group S_n as a product of n+1 transpositions.
  • A060604 (program): a(n) = binomial(prime(n), n) where prime(n) = n-th prime.
  • A060605 (program): a(n) = sum of lengths of the iteration sequences of Euler totient function from 1 to n.
  • A060606 (program): The n-th term is the sum of lengths of iteration chains to get fixed points(=1) for the Euler totient function from 1 to n.
  • A060607 (program): Number of iterations of phi(x) at prime(n) needed to reach 1.
  • A060610 (program): Repeatedly apply Euler phi to the n-th prime; a(n) is the number of terms in the resulting iteration chain which are not powers of 2 (number of initial iterations until reaching the first power of 2).
  • A060615 (program): Number of conjugacy classes in the group GL_2(K) when K is a finite field with q = p^m for a prime p and m >= 1.
  • A060620 (program): Average of the first n primes rounded down.
  • A060621 (program): Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here the codimension D-d is equal to 3 and d varies.
  • A060626 (program): Number of right triangles of a given area required to form successively larger squares.
  • A060629 (program): 1/2+Sum_{n >= 1) a(n)*x^(2*n)/(4^n*(2*n)!) = 1/Pi*EllipticK(x).
  • A060632 (program): a(n) = 2^wt(floor(n/2)) (i.e., 2^A000120(floor(n/2)), or A001316(floor(n/2))).
  • A060633 (program): Surround numbers of an n X 1 rectangle.
  • A060634 (program): Union of Fibonacci numbers and prime numbers.
  • A060635 (program): a(n) is the number of 2 X 1 domino tilings of the set S in the plane R^2 consisting of the union of the following two rectangles: rectangle1: |x| <= n, |y| <= 1, rectangle2: |x| <= 1, |y| <= n.
  • A060640 (program): If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + … + (e_i+1)*p_i^e_i).
  • A060641 (program): Surround numbers of a length 2n zig-zag.
  • A060643 (program): Number of conjugacy classes in the symmetric group S_n that have even number of elements.
  • A060644 (program): a(n) = floor((n+1)^(n+1)/n^n).
  • A060645 (program): a(0) = 0, a(1) = 4, then a(n) = 18*a(n-1) - a(n-2).
  • A060646 (program): Bonse sequence: a(n) = minimal j such that n-j+1 < prime(j).
  • A060647 (program): Number of alpha-beta evaluations in a tree of depth n and branching factor b=3.
  • A060648 (program): Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).
  • A060655 (program): Pack n integer-sided rectangles into the smallest possible square so that no sides of the rectangle are the same. Sequence gives the side of the smallest square.
  • A060656 (program): a(n) = 2*a(n-1)*a(n-2)/a(n-3), with a(0)=a(1)=1.
  • A060658 (program): Even numbers n such that sigma(x) = n has no solution.
  • A060659 (program): a(n)= smallest number of squares on a checkerboard that has exactly n domino tilings.
  • A060677 (program): Number of linear n-celled polyominoes, those with the property that a line can be drawn that intersects the interior of every cell.
  • A060679 (program): Orders of non-cyclic groups.
  • A060681 (program): Largest difference between consecutive divisors of n (ordered by size).
  • A060685 (program): Largest difference between consecutive divisors (ordered by size) of 2n+1.
  • A060687 (program): Numbers n such that there exist exactly 2 Abelian groups of order n, i.e., A000688(n) = 2.
  • A060690 (program): a(n) = binomial(2^n + n - 1, n).
  • A060691 (program): Expansion of AGM(1,1-8x) (where AGM denotes the arithmetic-geometric mean).
  • A060692 (program): Number of parts if 3^n is partitioned into parts of size 2^n as far as possible and into parts of size 1^n.
  • A060693 (program): Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.
  • A060696 (program): Number of permutations in S_n avoiding the strings 123, 321 and 231.
  • A060697 (program): The sum of the first n composite numbers plus 1 is a prime.
  • A060698 (program): The sum of the first n composite numbers minus 1 is a prime.
  • A060704 (program): Singular n X n matrices over GF(2).
  • A060706 (program): For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(4n) consisting of permutations whose cycle decomposition is a product of n disjoint 4-cycles.
  • A060708 (program): The Reuleaux Triangle constant.
  • A060710 (program): Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.
  • A060715 (program): Number of primes between n and 2n exclusive.
  • A060719 (program): a(0) = 1; a(n+1) = a(n) + Sum_{i=0..n} binomial(n,i)*(a(i)+1).
  • A060722 (program): a(n) = 3^(n^2).
  • A060723 (program): a(n) is the denominator of r(n) where r(n) is the sequence of rational numbers defined by the recursion: r(0) = 0, r(1) = 1 and for n>1 r(n) = r(n-1) + r(n-2)/2. From this definition it is clear that a(n) is always a power of 2 (see A060755).
  • A060724 (program): Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).
  • A060725 (program): E.g.f.: exp(-(x^5/5))/(1-x).
  • A060726 (program): For n >= 1, a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.
  • A060727 (program): For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.
  • A060734 (program): Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.
  • A060735 (program): a(1)=1, a(2)=2; thereafter, a(n) is the smallest number m not yet in the sequence such that every prime that divides a(n-1) also divides m.
  • A060736 (program): Array of square numbers read by antidiagonals in up direction.
  • A060739 (program): a(n) = (-1)^(n(n-1)/2) * Product_{k=0,…,n-1} (n+k-1)!/((k!)^2 * (n-1-k)!).
  • A060746 (program): Absolute value of numerator of non-Euler-constant term of Laurent expansion of Gamma function at s=-n.
  • A060747 (program): a(n) = 2*n - 1.
  • A060753 (program): Denominator of 1*2*4*6*…*(prime(n-1)-1) / (2*3*5*7*…*prime(n-1)).
  • A060755 (program): a(n) = log_2(A060723(n)).
  • A060757 (program): a(n) = 4^(n^2).
  • A060758 (program): a(n) = 5^(n^2).
  • A060761 (program): a(n) = 9^(n^2).
  • A060762 (program): Number of conjugacy classes (the same as the number of irreducible representations) in the dihedral group with 2n elements.
  • A060774 (program): a(n) = number of lattice paths from (0,0,0) to (n,n,n) along the cracks on the surface of a Rubik-ized n X n X n cube so that no step increases distance from goal.
  • A060775 (program): The greatest divisor d|n such that d < n/d, with a(1) = 1.
  • A060778 (program): a(n) = gcd(A000005(n+1), A000005(n)).
  • A060779 (program): a(n) = lcm(A000005(n+1), A000005(n)).
  • A060780 (program): a(n) = gcd(sigma(n+1), sigma(n)) = gcd(A000203(n+1), A000203(n)).
  • A060781 (program): Numbers n such that lcm(sigma(n+1), sigma(n)) = lcm(A000203(n+1), A000203(n)).
  • A060783 (program): Number of conics which pass through 3 points and are bitangent to a general curve of order n.
  • A060784 (program): Number of double tangents of order n.
  • A060785 (program): a(n) = 3*(n-2)*(5*n -11).
  • A060786 (program): a(n) = 9*(n-2)*(5*n-13)*(5*n^2 - 19*n + 16)/2.
  • A060787 (program): a(n) = 18*(n-2)*(2*n-5).
  • A060788 (program): a(n) = 9*(n-2)^2 * (n^2 - 2*n - 1).
  • A060789 (program): a(n) = n / (gcd(n,2) * gcd(n,3)).
  • A060790 (program): Inscribe two circles of curvature 2 inside a circle of curvature -1. Sequence gives curvatures of the smallest circles that can be sequentially inscribed in such a diagram.
  • A060791 (program): a(n) = n / gcd(n,5).
  • A060797 (program): Integer part of square root of n-th primorial, A002110(n).
  • A060798 (program): Numbers k such that difference between the upper and lower central divisors of k is 1.
  • A060800 (program): a(n) = p^2 + p + 1 where p runs through the primes.
  • A060801 (program): Invert transform of odd numbers: a(n) = Sum_{k=1..n} (2*k+1)*a(n-k), a(0)=1.
  • A060805 (program): Numerators of special continued fraction for 2*zeta(3).
  • A060806 (program): Denominators of special continued fraction for 2*zeta(3).
  • A060816 (program): a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.
  • A060817 (program): Size of the automorphism group of the alternating group A_n.
  • A060818 (program): a(n) = 2^(floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + …).
  • A060819 (program): a(n) = n / gcd(n,4).
  • A060820 (program): (2*n-1)^2 + (2*n)^2.
  • A060821 (program): Triangle T(n,k) read by rows giving coefficients of Hermite polynomial of order n (n >= 0, 0 <= k <= n).
  • A060822 (program): a(n) = prime(n) + n^3 + n^2 + 4n - 1.
  • A060823 (program): 4-wave sequence beginning with 2’s with middles dropped.
  • A060827 (program): 3-wave sequence beginning with 2’s.
  • A060828 (program): Size of the Sylow 3-subgroup of the symmetric group S_n.
  • A060831 (program): a(n) = Sum_{k=1..n} (number of odd divisors of k) (cf. A001227).
  • A060832 (program): a(n) = Sum_{k>0} floor(n/k!).
  • A060833 (program): Separate the natural numbers into disjoint sets A, B with 1 in A, such that the sum of any 2 distinct elements of the same set never equals 2^k + 2. Sequence gives elements of set A.
  • A060834 (program): a(n) = 6*n^2 + 6*n + 31.
  • A060836 (program): Number of permutations of n letters where exactly 5 change position.
  • A060839 (program): Number of solutions to x^3 == 1 (mod n).
  • A060842 (program): (C(2p,p)-2)/p^2 where p runs through the primes.
  • A060844 (program): Primes of the form 6*k^2 + 6*k + 31.
  • A060845 (program): Largest prime < a nontrivial power of a prime.
  • A060846 (program): Smallest prime > a nontrivial power of a prime.
  • A060847 (program): Difference between a nontrivial prime power (A025475) and the previous prime.
  • A060848 (program): Difference between a nontrivial prime power (A025475) and the next prime.
  • A060851 (program): a(n) = (2n-1) * 3^(2n-1).
  • A060862 (program): a(n) = 0 if n is deficient, 1 if n is abundant, 2 if n is perfect.
  • A060865 (program): a(n) is the exact power of 2 that divides the n-th Fibonacci number (A000045).
  • A060866 (program): Sum of (d+d’) over all unordered pairs (d,d’) with d*d’ = n.
  • A060867 (program): a(n) = (2^n - 1)^2.
  • A060868 (program): Number of n X n matrices over GF(3) with rank 1.
  • A060869 (program): Number of n X n matrices over GF(4) with rank 1.
  • A060870 (program): Number of n X n matrices over GF(5) with rank 1.
  • A060871 (program): Number of n X n matrices over GF(7) with rank 1.
  • A060872 (program): Sum of d*d’ over all unordered pairs (d,d’) with d*d’ = n.
  • A060880 (program): Compositorial numbers (A036691) - 1.
  • A060881 (program): n-th primorial (A002110) + prime(n + 1).
  • A060882 (program): a(n) = n-th primorial (A002110) minus next prime.
  • A060883 (program): a(n) = n^6 + n^3 + 1.
  • A060884 (program): a(n) = n^4 - n^3 + n^2 - n + 1.
  • A060885 (program): a(n) = Sum_{j=0..10} n^j.
  • A060886 (program): a(n) = n^4 - n^2 + 1.
  • A060887 (program): a(n) = Sum_{j=0..12} n^j.
  • A060888 (program): a(n) = n^6 - n^5 + n^4 - n^3 + n^2 - n + 1.
  • A060889 (program): n^8-n^7+n^5-n^4+n^3-n+1.
  • A060890 (program): n^8 + 1.
  • A060891 (program): a(n) = n^6 - n^3 + 1.
  • A060892 (program): n^8-n^6+n^4-n^2+1.
  • A060893 (program): n^8 - n^4 + 1.
  • A060894 (program): n^8+n^7-n^5-n^4-n^3+n+1.
  • A060895 (program): n^16 + 1.
  • A060896 (program): n^12 - n^6 + 1.
  • A060899 (program): Number of walks of length n on square lattice, starting at origin, staying on points with x+y >= 0.
  • A060901 (program): Exact power of 3 that divides the n-th Fibonacci number (sequence A000045).
  • A060902 (program): Number of ordered factorizations of the identity permutation in the symmetric group S_n into 2n-2 transpositions such that the factors generate S_n.
  • A060904 (program): Largest power of 5 that divides n.
  • A060917 (program): Expansion of e.g.f.: exp((-1)^k/k*LambertW(-x)^k)/(k-1)!, k=3.
  • A060919 (program): Number of corners in a 4-sided fractal.
  • A060920 (program): Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).
  • A060921 (program): Bisection of Fibonacci triangle A037027: odd-indexed members of column sequences of A037027 (not counting leading zeros).
  • A060922 (program): Convolution triangle for Lucas numbers A000032(n+1), n >= 0.
  • A060923 (program): Bisection of Lucas triangle A060922: even indexed members of column sequences of A060922 (not counting leading zeros).
  • A060924 (program): Bisection of Lucas triangle A060922: odd-indexed members of column sequences of A060922 (not counting leading zeros).
  • A060925 (program): a(n) = 2*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 4.
  • A060926 (program): Row sums of triangle A060923 (even part of bisection of Lucas triangle).
  • A060927 (program): Row sums of triangle A060924 (odd part of bisection of Lucas triangle).
  • A060928 (program): Expansion of 1/(1 - 5*x - 4*x^3).
  • A060929 (program): Second convolution of Lucas numbers A000032(n+1), n >= 0.
  • A060930 (program): Third convolution of Lucas numbers A000032(n+1), n >= 0.
  • A060931 (program): Fourth convolution of Lucas numbers A000032(n+1), n >= 0.
  • A060932 (program): Fifth convolution of Lucas numbers A000032(n+1), n >= 0.
  • A060933 (program): Sixth convolution of Lucas numbers A000032(n+1), n >= 0.
  • A060934 (program): Second column of Lucas bisection triangle (even part).
  • A060937 (program): Number of iterations of d(n) (A000005) needed to reach 2 starting at n (n is counted).
  • A060939 (program): a(n) = (Sum of the first n primes) + n.
  • A060941 (program): Duchon’s numbers: the number of paths of length 5*n from the origin to the line y = 2*x/3 with unit East and North steps that stay below the line or touch it.
  • A060943 (program): a(n) = n!^n * Sum_{k=1..n} 1/k^n.
  • A060944 (program): a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.
  • A060945 (program): Number of compositions (ordered partitions) of n into 1’s, 2’s and 4’s.
  • A060946 (program): Trace of Vandermonde matrix of numbers 1,2,…,n, i.e., the matrix A with A[i,j] = i^(j-1), 1 <= i <= n, 1 <= j <= n.
  • A060954 (program): Largest prime factor of 10*n + 1.
  • A060956 (program): Leading digit of 3^n.
  • A060959 (program): Table by antidiagonals of generalized Fibonacci numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=0 and T(n,1)=1.
  • A060961 (program): Number of compositions (ordered partitions) of n into 1’s, 3’s and 5’s.
  • A060964 (program): Table by antidiagonals where T(n,k) = n*T(n,k-1) - T(n,k-2) with T(n,0) = 2 and T(n,1) = n.
  • A060968 (program): Number of solutions to x^2 + y^2 == 1 (mod n).
  • A060973 (program): a(2*n+1) = a(n+1)+a(n), a(2*n) = 2*a(n), with a(1)=0 and a(2)=1.
  • A060979 (program): |First digit - second digit + third digit - fourth digit …| = 11.
  • A060983 (program): Number of primitive sublattices of index n in generic 3-dimensional lattice.
  • A060992 (program): a(n) = Sum_{gcd(i,j) | 0 < i <= j < n and i+j = n}.
  • A060995 (program): Number of routes of length 2n on the sides of an octagon from a point to opposite point.
  • A060996 (program): Stirling2 transform of [2,3,3,3,3,3,3,3,…].
  • A060997 (program): Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, …
  • A060998 (program): Squares of 1 and primes, written backwards.
  • A060999 (program): Nearest integer to (n+1)^3/9.
  • A061000 (program): x.v where x = first n terms of A060999, v = [1,8,27,…,n^3].
  • A061001 (program): x.x, x = first n terms of A060999.
  • A061003 (program): Nearest integer to n^5/25.
  • A061004 (program): Nearest integer to n^6/36.
  • A061005 (program): (Nearest integer to n^6/36) / 2.
  • A061006 (program): a(n) = (n-1)! mod n.
  • A061007 (program): a(n) = -(n-1)! mod n.
  • A061008 (program): a(n) = Sum_{j=1..n} (-(n-1)! mod n).
  • A061009 (program): a(n) = -2 + Sum_{j=1..n} (-(n-1)!) mod n.
  • A061015 (program): Numerator of Sum_{i=1..n} 1/p(i)^2, p(i) = i-th prime.
  • A061017 (program): List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.
  • A061018 (program): Triangle: a(n,m) = number of permutations of (1,2,…,n) with one or more fixed points in the m first positions.
  • A061019 (program): Negate primes in factorization of n.
  • A061020 (program): Negate primes in factorizations of divisors of n, then sum.
  • A061021 (program): a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.
  • A061024 (program): a(n) = (prime(n)!)^2.
  • A061035 (program): Triangle T(m,n) = numerator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,…,1.
  • A061037 (program): Numerator of 1/4 - 1/n^2.
  • A061038 (program): Denominator of 1/4 - 1/n^2.
  • A061039 (program): Numerator of 1/9 - 1/n^2.
  • A061040 (program): Denominator of 1/9 - 1/n^2.
  • A061041 (program): Numerator of 1/16 - 1/n^2.
  • A061042 (program): Denominator of 1/16 - 1/n^2.
  • A061043 (program): Numerator of 1/25 - 1/n^2.
  • A061044 (program): Denominator of 1/25 - 1/n^2.
  • A061045 (program): Numerator of 1/36 - 1/n^2.
  • A061046 (program): Denominator of 1/36 - 1/n^2.
  • A061047 (program): Numerator of 1/49 - 1/n^2.
  • A061048 (program): Denominator of 1/49 - 1/n^2.
  • A061049 (program): Numerator of 1/64 - 1/n^2.
  • A061050 (program): Denominator of 1/64 - 1/n^2.
  • A061054 (program): Floor(n+n^(3/4)).
  • A061062 (program): Sum of squared factorials: (0!)^2 + (1!)^2 + (2!)^2 + (3!)^2 +…+ (n!)^2.
  • A061065 (program): For n <= 6, entry of maximal modulus in the inverse of the n-th Hilbert matrix. For n >= 3, this is the (n-1,n-1)-th entry.
  • A061066 (program): a(n) = (prime(n)^2 - 1)/8.
  • A061076 (program): a(n) is the sum of the products of the digits of all the numbers from 1 to n.
  • A061077 (program): a(n) is the sum of the products of the digits of the first n odd numbers.
  • A061078 (program): Sum of the products of the digits of the first n positive even numbers.
  • A061079 (program): Denominators in the series for sin integral Si(x).
  • A061082 (program): a(n) = A053061(n)/n.
  • A061083 (program): Fibonacci-type sequence based on division: a(0) = 1, a(1) = 2 and a(n) = a(n-2)/a(n-1) but ignore decimal point.
  • A061084 (program): Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
  • A061085 (program): a(n) = A019550(n) / 3.
  • A061086 (program): a(n) is the concatenation of n with n^3.
  • A061087 (program): a(n) = A061086(n) / n.
  • A061088 (program): a(n) = A053062(n) / n.
  • A061091 (program): Number of k with 1 <= k <= n relatively prime to phi(k).
  • A061094 (program): The alternating group A_n contains an element x which is not conjugate to its inverse (equivalently not all the entries in the character table of A_n are real numbers).
  • A061099 (program): Squares with digital root 1.
  • A061100 (program): Squares with digital root 4.
  • A061101 (program): Squares with digital root 7.
  • A061104 (program): Smallest number whose digit sum is n^2.
  • A061107 (program): a(0) = 0, a(1) = 1, a(n) is the concatenation of a(n-2) and a(n-1) for n > 1.
  • A061142 (program): Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.
  • A061146 (program): Decimal expansion of 11*Pi/10.
  • A061147 (program): Product of all distinct numbers formed by permuting digits of n.
  • A061148 (program): Smallest positive integer for which the number of divisors is a product of 2 distinct primes: Min{x; d[x]=pq}.
  • A061150 (program): a(n) = Sum_{d|n} d*prime(d).
  • A061159 (program): Numerators in expansion of Euler transform of b(n) = 1/2.
  • A061160 (program): Numerators in expansion of Euler transform of b(n) = 1/3.
  • A061161 (program): Numerators in expansion of Euler transform of b(n) = 1/4.
  • A061162 (program): a(n) = (6n)!n!/((3n)!(2n)!^2).
  • A061163 (program): a(n) = (10n)!*n!/((5n)!*(4n)!*(2n)!).
  • A061165 (program): Polynomial extrapolation of 2, 3, 5, 7, 11.
  • A061167 (program): a(n) = n^5 - n.
  • A061168 (program): Partial sums of floor(log_2(k)) (= A000523(k)).
  • A061169 (program): Third column of Lucas bisection triangle (even part).
  • A061170 (program): Fourth column of Lucas bisection triangle (even part).
  • A061171 (program): One half of second column of Lucas bisection triangle (odd part).
  • A061172 (program): Third column of Lucas bisection triangle (odd part).
  • A061173 (program): One-fourth of fourth column of Lucas bisection triangle (odd part).
  • A061174 (program): Fifth column of Lucas bisection triangle (odd part).
  • A061175 (program): One half of sixth column of Lucas bisection triangle (odd part).
  • A061176 (program): Coefficients of polynomials ( (1 -x +sqrt(x))^n + (1 -x -sqrt(x))^n )/2.
  • A061177 (program): Coefficients of polynomials ( (1 -x +sqrt(x))^(n+1) - (1 -x -sqrt(x))^(n+1) )/(2*sqrt(x)).
  • A061178 (program): Third column (m=2) of triangle A060920 (bisection of Fibonacci triangle, even part).
  • A061179 (program): Fourth column (m=3) of triangle A060920 (bisection of Fibonacci triangle, even part).
  • A061180 (program): Fifth column (m=4) of triangle A060920 (bisection of Fibonacci triangle, even part).
  • A061181 (program): Sixth column (m=5) of triangle A060920 (bisection of Fibonacci triangle, even part).
  • A061182 (program): Third column (m=2) of triangle A060921 (bisection of Fibonacci triangle, odd part).
  • A061183 (program): One-fourth of the fourth (m=3) column of triangle A060921 (bisection of Fibonacci triangle, odd part).
  • A061184 (program): Fifth (m=4) column of triangle A060921 (bisection of Fibonacci triangle, odd part).
  • A061185 (program): One half of sixth (m=5) column of triangle A060921 (bisection of Fibonacci triangle, odd part).
  • A061190 (program): a(n) = n^n - n.
  • A061200 (program): tau_5(n) = number of ordered 5-factorizations of n.
  • A061201 (program): Partial sums of A007425: (tau<=)_3(n).
  • A061202 (program): (tau<=)_4(n).
  • A061203 (program): (tau<=)_5(n).
  • A061204 (program): (tau<=)_6(n).
  • A061205 (program): a(n) = n times R(n) where R(n) (A004086) is the digit reversal of n.
  • A061206 (program): a(n) = total number of occurrences of the consecutive pattern 1324 in all permutations of [n+3].
  • A061213 (program): a(n) = product of first n triangular numbers (A000217) + 1.
  • A061214 (program): Product of composite numbers between the n-th and (n+1)st primes.
  • A061217 (program): Number of zeros in the concatenation n(n-1)(n-2)(n-3)…321.
  • A061219 (program): a(n) is the largest number which can be formed with no zeros, using least number of digits and having digit sum = n.
  • A061222 (program): a(n) = n^2 + (n + 1)^3 + (n + 2)^4.
  • A061223 (program): a(n) = n^3 + (n + 1)^4 + (n + 2)^5.
  • A061224 (program): a(n) = n^2 + (n + 1)^3 + (n + 2)^4 + (n + 3)^5.
  • A061225 (program): Numbers of the form k^3 + (k + 1)^4 + (k + 2)^5 + (k + 3)^6.
  • A061226 (program): a(n) = n^2 + (n^2 with digits reversed).
  • A061227 (program): a(n) = p + R{p} where R{p} is the digit reversal of n-th prime p.
  • A061228 (program): a(1) = 2, a(n) = smallest number greater than n which is not coprime to n.
  • A061229 (program): Floor of geometric mean of n and the reversal of n.
  • A061234 (program): Smallest number with prime(n)^2 divisors where prime(n) is the n-th prime.
  • A061237 (program): Prime numbers == 1 (mod 9).
  • A061238 (program): Prime numbers == 2 (mod 9).
  • A061239 (program): Prime numbers == 4 (mod 9).
  • A061240 (program): Prime numbers == 5 (mod 9).
  • A061241 (program): Prime numbers == 7 (mod 9).
  • A061242 (program): Primes of the form 9*k - 1.
  • A061243 (program): a(n) = n+r where r is the smallest number such that n divides (n+1)(n+2)(n+3)…(n+r).
  • A061249 (program): Smallest number with digit sum = Fibonacci(n).
  • A061250 (program): (n-2)*(n-1)^n.
  • A061252 (program): a(n) = 16^n - 15^n.
  • A061253 (program): Let G_n be the elementary Abelian group G_n = (C_3)^n; a(n) is the number of times the number 1 appears in the character table of G_n.
  • A061255 (program): Euler transform of Euler totient function phi(n), cf. A000010.
  • A061256 (program): Euler transform of sigma(n), cf. A000203.
  • A061259 (program): a(n)=Sum_{d|n} d*numbpart(d), where numbpart(d)=number of partitions of d, cf. A000041.
  • A061263 (program): a(n) = floor(n^3/9).
  • A061265 (program): Number of squares between n-th prime and (n+1)st prime.
  • A061266 (program): Number of squares between n^3 and (n+1)^3.
  • A061278 (program): a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) with a(1) = 1 and a(k) = 0 if k <= 0.
  • A061279 (program): a(n) = Sum_{k >= 0} 2^k * binomial(k+2,n-2*k).
  • A061282 (program): Minimal number of steps to get from 0 to n by (a) adding 1 or (b) multiplying by 3. A stopping problem: begin with n and at each stage if a multiple of 3 divide by 3, otherwise subtract 1.
  • A061283 (program): Smallest number with exactly 2n-1 divisors.
  • A061285 (program): a(n) = 2^((prime(n) - 1)/2).
  • A061286 (program): Smallest integer for which the number of divisors is the n-th prime.
  • A061287 (program): Integer part of square root of n-th Fibonacci number.
  • A061288 (program): Integer part of square root of n-th triangular number.
  • A061290 (program): Square array read by antidiagonals of T(n,k) = T(n-1,k) + T(n-1, floor(k/2)) with T(0,0)=1.
  • A061291 (program): a(1) = 1; a(n+1) = a(1) + a(2)*(a(2) + a(3)*(a(3) +…+a(n-1)*(a(n-1) + a(n))…)).
  • A061292 (program): a(n) = a(n-1)*a(n-2)*a(n-3) - a(n-4) for n>3 with a(0) = a(1) = a(2) = a(3) = 2.
  • A061295 (program): Minimal number of steps to get from 0 to n by (a) adding 1 or (b) subtracting 1 or (c) multiplying by 3.
  • A061298 (program): Table by antidiagonals of rows of sequences where each row is binomial transform of preceding row and row 1 is (1,2,1,2,1,2,1,2,…).
  • A061299 (program): Least number whose number of divisors is A007304(n) (the n-th number that is the product of 3 distinct primes).
  • A061302 (program): n*(n-1)^(n-2).
  • A061304 (program): Squarefree triangular numbers.
  • A061312 (program): Triangle T[n,m]: T[n,-1] = 0; T[0,0] = 0; T[n,0] = n*n!; T[n,m] = T[n,m-1] - T[n-1,m-1].
  • A061313 (program): Minimal number of steps to get from 1 to n by (a) subtracting 1 or (b) multiplying by 2.
  • A061314 (program): Table by antidiagonals where T(n,k)=T(n,k-1)+T(n,k-1)^2/k^2 and T(n,0)=n.
  • A061315 (program): Array read by antidiagonals: T(n,k)=T(n,k-1)*(T(n,k-1)+k-1)/k with T(n,1)=n.
  • A061316 (program): a(n) = n*(n+1)*(n^2 + n + 4)/4.
  • A061317 (program): Split positive integers into extending even groups and sum: 1+2, 3+4+5+6, 7+8+9+10+11+12, 13+14+15+16+17+18+19+20, …
  • A061318 (program): Column 3 of A061314.
  • A061319 (program): Column 4 of A061315.
  • A061320 (program): Column 4 of A061314.
  • A061321 (program): Column 5 of A061315.
  • A061322 (program): a(n) = a(n-1) * (1 + a(n-1)/n^2) with a(0) = 2.
  • A061338 (program): Increase in maximal number of comparisons for sorting n elements by list merging.
  • A061339 (program): Minimal number of steps to get from 0 to n by (a) adding 1 or (b) subtracting 1 or (c) multiplying by 2.
  • A061340 (program): a(n) = n*omega(n)^n where omega(n) is the number of distinct prime divisors of n.
  • A061344 (program): Numbers of form p^m + 1, p odd prime, m >= 1.
  • A061345 (program): Odd prime powers.
  • A061346 (program): Odd numbers that are neither primes nor prime powers.
  • A061347 (program): Period 3: repeat [1, 1, -2].
  • A061349 (program): Sum of antidiagonals of A060736.
  • A061350 (program): Maximal size of Aut(G) where G is a finite Abelian group of order n.
  • A061352 (program): First row of array shown below.
  • A061353 (program): First column of array shown in A061352.
  • A061354 (program): Numerator of Sum_{k=0..n} 1/k!.
  • A061355 (program): Denominator of Sum_{k=0..n} 1/k!.
  • A061356 (program): Triangle read by rows. T(n,k) are the labeled trees on n nodes with maximal node degree k (0 < k < n).
  • A061357 (program): Number of 0<k<n such that n-k and n+k are both primes.
  • A061358 (program): Number of ways of writing n = p+q with p, q primes and p >= q.
  • A061369 (program): a(n) = smallest square in the arithmetic progression {nk+1 : k >= 0}.
  • A061370 (program): a(n) = floor(ratio of product and sum of first n numbers).
  • A061371 (program): Composite numbers with all prime digits.
  • A061373 (program): “Natural” logarithm, defined inductively by a(1)=1, a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if n, m>1.
  • A061376 (program): a(n) = f(n) + f(f(n)) where f(n) = 0 if n <= 1 or a prime, otherwise f(n) = sum of distinct primes dividing n.
  • A061377 (program): a(1) = 1, a(n+1) = numerator of the continued fraction [1; 2, 4, 8, …, 2^n].
  • A061378 (program): Product of all numbers formed by permuting the digits of n.
  • A061380 (program): Triangular numbers with product of digits also a triangular number.
  • A061384 (program): Numbers n such that sum of digits = number of digits.
  • A061386 (program): Sum of digits = 3 times number of digits.
  • A061391 (program): a(n) = t(n,3) = Sum_{d|n} tau(d^3), where tau(n) = number of divisors of n, cf. A000005.
  • A061392 (program): a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.
  • A061393 (program): Number of appearances of n in sequence defined by b(k) = b(floor(k/3)) + b(ceiling(k/3)) with b(0)=0 and b(1)=1, i.e., in A061392.
  • A061395 (program): Let p be the largest prime factor of n; if p is the k-th prime then set a(n) = k; a(1) = 0 by convention.
  • A061397 (program): Characteristic function sequence of primes multiplied componentwise by N, the natural numbers.
  • A061398 (program): Number of squarefree integers between prime(n) and prime(n+1).
  • A061399 (program): Number of nonsquarefree integers between primes p(n) and p(n+1).
  • A061402 (program): a(n) = floor(n*sqrt(e)).
  • A061403 (program): Denominators in the series for Bessel function J4(x).
  • A061404 (program): Denominators in the series for Bessel function J5(x).
  • A061405 (program): Denominators in the series for Bessel function J6(x).
  • A061406 (program): Denominators in the series for Bessel function J7(x).
  • A061407 (program): Denominators in the series for Bessel function J8(x).
  • A061408 (program): For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both positive squares; list all such pairs (x,y) ordered by values of y; sequence gives y values.
  • A061418 (program): a(n) = floor(a(n-1)*3/2) with a(1) = 2.
  • A061419 (program): a(n) = ceiling(a(n-1)*3/2) with a(1) = 1.
  • A061420 (program): a(n) = a(ceiling((n-1)*2/3)) + 1 with a(0) = 0.
  • A061423 (program): Sum of digits = 6 times number of digits.
  • A061431 (program): a(n) = LCM of the n consecutive numbers n(n-1)/2 + 1, …, n(n+1)/2.
  • A061432 (program): a(n) = smallest n-digit square.
  • A061433 (program): Largest n-digit square.
  • A061434 (program): a(n) is the smallest n-digit cube.
  • A061435 (program): a(n) is the largest n-digit cube.
  • A061436 (program): Number of steps for trajectory of n to reach 1 under the map that sends x -> x/3 if x mod 3 = 0, x -> x+3-(x mod 3) if x is not 0 mod 3 (for a 2nd time when n starts at 1).
  • A061439 (program): Largest number whose cube has n digits.
  • A061440 (program): Denominators in the series for Bessel function J9(x).
  • A061446 (program): Primitive part of Fibonacci(n).
  • A061452 (program): n*bigomega(n)^n, where bigomega(n) is the number of prime divisors of n, counted with multiplicity.
  • A061453 (program): a(n) = numerator of the continued fraction [1; 2^2, 3^3, …, n^n].
  • A061454 (program): a(n) = denominator of the continued fraction [1; 2^2, 3^3, …, n^n].
  • A061462 (program): The exact power of 2 that divides the n-th Bell number (A000110). Has period 12.
  • A061463 (program): Numerator of 1 + 1/(2^2) + 1/(3^3) + … 1/(n^n).
  • A061464 (program): Denominator of 1 + 1/(2^2) + 1/(3^3) + … 1/(n^n).
  • A061466 (program): Product of primes prime(3*n+1), prime(3*n+2), prime(3*n+3).
  • A061468 (program): a(n) = d(n) + phi(n), where d(n) is the number of divisors (A000005) and phi(n) is Euler’s totient function (A000010).
  • A061470 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 1.
  • A061471 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 2.
  • A061472 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 3.
  • A061473 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 4.
  • A061474 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 5.
  • A061475 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 6.
  • A061476 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 7.
  • A061477 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 8.
  • A061478 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 9.
  • A061479 (program): Smallest number m such that first digit - second digit + third digit - fourth digit … (of m) = n.
  • A061480 (program): n-th digit in decimal expansion of 1/n.
  • A061482 (program): a(1) = 1, a(2) = 2, a(n) = sum of products of previous terms taking n-2 at a time.
  • A061483 (program): Numerator of 1 + 1/2 + 2/3 + 3/4 + … + (n-1)/n.
  • A061484 (program): Numerator of 1/3 + 3/5 + 5/7 + … + (2n - 1)/(2n + 1).
  • A061486 (program): Let the number of digits in n be k; a(n) = sum of the products of the digits of n taken r at a time where r ranges from 1 to k.
  • A061489 (program): Numbers that are Fibonacci numbers plus or minus 1.
  • A061495 (program): a(n) = lcm(3n+1, 3n+2, 3n+3).
  • A061496 (program): a(n) = gcd(abundant(n), abundant(n+1)) where abundant(n) is the n-th abundant number.
  • A061501 (program): a(1) = 1, a(n+1) = (a(n) + n) mod 10.
  • A061502 (program): a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.
  • A061503 (program): a(n) = Sum_{k=1..n} tau(k^2), where tau is the number of divisors function A000005.
  • A061504 (program): a(n+1) = le nombre des lettres dans a(n).
  • A061505 (program): Leading digit of n^n.
  • A061506 (program): a(n) = lcm(6n+2, 6n+4, 6n+6).
  • A061510 (program): Write n in decimal, omit 0’s, raise each digit k to k-th power and multiply.
  • A061518 (program): a(0) = 0; a(n) is obtained by incrementing each digit of a(n-1) by 5.
  • A061519 (program): a(0) = 1; a(n) is obtained by incrementing each digit of a(n-1) by 5.
  • A061524 (program): Surround numbers of an n X 2 rectangle when n is even.
  • A061525 (program): Surround numbers of an n X 2 rectangle when n is odd.
  • A061532 (program): Nearest integer to n^7/49.
  • A061534 (program): Expansion of (1-x^2)/(1-3*x-x^2+x^3).
  • A061536 (program): a(1) = 1 and a(n) = a(n-1) + (the number of primes <= n) for n > 1.
  • A061537 (program): Product of unitary divisors of n.
  • A061538 (program): Product of all divisors of n, divided by product of unitary divisors; or equivalently product of non-unitary divisors of n.
  • A061547 (program): Number of 132 and 213-avoiding derangements of {1,2,…,n}.
  • A061548 (program): Numerator of probability that there is no error when average of n numbers is computed, assuming errors of +1, -1 are possible and they each occur with p = 1/4.
  • A061549 (program): Denominator of probability that there is no error when average of n numbers is computed, assuming errors of +1, -1 are possible and they each occur with p=1/4.
  • A061550 (program): a(n) = (2n+1)*(2n+3)*(2n+5).
  • A061551 (program): Number of paths along a corridor width 8, starting from one side.
  • A061554 (program): Square table read by antidiagonals: a(n,k) = binomial(n+k, floor(k/2)).
  • A061555 (program): Integer part of sigma(n!)/n!.
  • A061557 (program): a(n) = (7*n+2)*C(n)/(n+2), where C(n) is the n-th Catalan number.
  • A061561 (program): Trajectory of 22 under the Reverse and Add! operation carried out in base 2.
  • A061563 (program): Start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.
  • A061568 (program): Number of primes <= sum of first n primes.
  • A061570 (program): a(1)=0, a(2)=1, a(n)=3*n-1 for n >= 3.
  • A061572 (program): a(n) = (n!)^2 * Sum_{k=1..n} 1/(k^2*(k-1)!).
  • A061573 (program): a(n) = (n!)^2*Sum_{k=1..n} 1/k!.
  • A061578 (program): First differences of A061577.
  • A061579 (program): Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.
  • A061600 (program): a(n) = n^3 - n + 1.
  • A061601 (program): 9’s complement of n: a(n) = 10^d - 1 - n where d is the number of digits in n. If a is a digit in n replace it with 9 - a.
  • A061602 (program): Sum of factorials of the digits of n.
  • A061638 (program): Primes p such that the greatest prime divisor of p-1 is 7.
  • A061639 (program): Number of planar planted trees with n non-root nodes and every 2-valent node isolated.
  • A061640 (program): a(n) = !n*n!.
  • A061646 (program): a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) with a(-1) = 1, a(0) = 1, a(1) = 1.
  • A061647 (program): Beginning at the well for the topograph of a positive definite quadratic form with values 1, 1, 1 at a superbase (i.e., 1, 1 and 1 are the vonorms of the superbase), these numbers indicate the labels of the edges of the topograph on a path of greatest ascent.
  • A061648 (program): Area of all nondecreasing Dyck paths of length 2n.
  • A061650 (program): a(n) = n*20^(n-1).
  • A061654 (program): a(n) = (3*16^n + 2)/5.
  • A061667 (program): a(n) = Fibonacci(2*n+1) - 2^(n-1).
  • A061669 (program): a(n) = n*(mu(n) + 1), where mu(n) is the Moebius function A008683.
  • A061673 (program): Even numbers k such that k+1 and k-1 are both composite.
  • A061679 (program): Concatenation of n^3 and 7.
  • A061680 (program): a(n) = gcd(d(n^2), d(n)).
  • A061681 (program): a(0)=1; a(n) = a(n-1) + lead(a(n-1)) for n > 0 where for an integer x lead(x) is the leading digit in base 10.
  • A061690 (program): Generalized Stirling numbers.
  • A061693 (program): Generalized Bell numbers.
  • A061695 (program): Generalized Bell numbers.
  • A061703 (program): G.f.: 2*x*(2-2*x-3*x^2+2*x^3)/((1-3*x-x^2+x^3)*(1-x)).
  • A061704 (program): Number of cubes dividing n.
  • A061705 (program): Number of matchings in the wheel graph with n spokes.
  • A061709 (program): Consider a (hollow) triangle with n cells on each edge, for a total of 3(n-1) cells if n>1, or 1 cell if n=1; a(n) is number of ways of labeling cells with 0’s and 1’s; triangle may be rotated and turned over.
  • A061711 (program): a(n) = n!*n^n.
  • A061714 (program): Number of types of (n-1)-swap moves for traveling salesman problem. Number of circular permutations on elements 0,1,…,2n-1 where every two elements 2i,2i+1 and no two elements 2i-1,2i are adjacent.
  • A061716 (program): Binary order of n-th prime.
  • A061717 (program): Binary order of n^n.
  • A061718 (program): a(n) = (n*(n+1)/2)^n.
  • A061720 (program): First differences of sequence of primorials.
  • A061722 (program): a(n) = 10 * n^2 + 7.
  • A061723 (program): Floor of arithmetic-geometric mean of n and 2*n - 1.
  • A061725 (program): p^2 + 2 where p is a prime.
  • A061726 (program): If n-th triangular number (A000217(n)) is odd, multiply it by 4; if even, multiply it by 5.
  • A061728 (program): Start with (a, b) = (2, 4). The next pair (a’, b’) is ((b + 1) mod 10, (a + 1) mod 10) where (a, b) is the previous pair.
  • A061742 (program): a(n) is the square of the product of first n primes.
  • A061743 (program): Numbers k such that k! is divisible by (k+1)^2.
  • A061751 (program): Numbers k such that k! is divisible by (k+1)^3.
  • A061752 (program): n! is divisible by (n+1)^4.
  • A061753 (program): n! is divisible by (n+1)^5.
  • A061754 (program): Numbers k such that k! is divisible by (k+1)^6.
  • A061755 (program): n! is divisible by (n+1)^7.
  • A061756 (program): n! is divisible by (n+1)^8.
  • A061757 (program): n! is divisible by (n+1)^9.
  • A061758 (program): n! is divisible by (n+1)^10.
  • A061759 (program): Numbers k such that k! is divisible by (k+1)^11.
  • A061761 (program): a(n) = 2^n + 2*n - 1.
  • A061762 (program): a(n) = (sum of digits of n) + (product of digits of n).
  • A061764 (program): n! is divisible by (n+1)^12.
  • A061765 (program): a(n) = usigma(sigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448) and sigma(n) is the sum of the divisors (A000203).
  • A061766 (program): a(1) = 4; a(n) = smallest composite number of the form k*a(n-1) + 1.
  • A061767 (program): a(1) = 4; a(n) = smallest composite number of the form k*n + 1.
  • A061771 (program): Primes p(k) such that p(k+1) - p(k) = 2^m for some m (smaller of a pair of consecutive primes which differ by a power of 2).
  • A061772 (program): a(n) is the number of n-digit multiples of n.
  • A061774 (program): a(n) = (n-1)!, as n runs through the prime powers >= 1.
  • A061775 (program): Number of nodes in rooted tree with Matula-Goebel number n.
  • A061776 (program): Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives number of triangles in n-th generation.
  • A061777 (program): Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives total population of triangles at n-th generation.
  • A061778 (program): a(n) = Product_{j=0..floor(n/2)} binomial(n,j).
  • A061779 (program): Primes p such that q-p = 22, where q is the next prime after p.
  • A061782 (program): a(n) = smallest positive number m such that m*n is a triangular number.
  • A061785 (program): a(n) = m such that 2^m < 5^n < 2^(m+1).
  • A061787 (program): a(n) = Sum_{k=1..n} (2k-1)^(2k-1).
  • A061788 (program): a(n) = Sum_{k=1..n} (2k)^(2k).
  • A061789 (program): a(n) = Sum_{k=1..n} prime(k)^prime(k).
  • A061792 (program): 49*(n*(n+1)/2)+6.
  • A061793 (program): a(n) = 25*n*(n + 1)/2 + 3.
  • A061799 (program): Smallest number with at least n divisors.
  • A061800 (program): a(n) = n + (-1)^(n mod 3).
  • A061801 (program): a(n) = (7*6^n - 2)/5.
  • A061802 (program): Sum of n-th row of triangle of primes: 2; 2 3 2; 2 3 5 3 2; 2 3 5 7 5 3 2; …; where n-th row contains 2n+1 terms.
  • A061803 (program): Sum of n-th row of triangle of 4th powers: 1; 1 16 1; 1 16 81 16 1; 1 16 81 256 81 16 1; …
  • A061804 (program): a(n) = 2*n*(2*n^2 + 1).
  • A061807 (program): Smallest positive multiple of n containing only even digits.
  • A061810 (program): Multiples of 3 with all odd digits.
  • A061811 (program): Multiples of 3 with all even digits.
  • A061814 (program): Multiples of 4 containing only even digits.
  • A061817 (program): Multiples of 9 containing only odd digits.
  • A061818 (program): Multiples of 2 containing only digits 0,1,2.
  • A061819 (program): Multiples of 3 containing only digits 0,1,2,3.
  • A061820 (program): Multiples of 4 containing only digits 0,…,4.
  • A061821 (program): Multiples of 5 containing only digits 0,…,5.
  • A061822 (program): Multiples of 6 containing only digits 0,…,6.
  • A061823 (program): Multiples of 7 containing only digits 0,…,7.
  • A061824 (program): Multiples of 8 containing only the digits 0, …, 8.
  • A061825 (program): Multiples of 7 containing only odd digits.
  • A061826 (program): Multiples of 7 containing only even digits.
  • A061829 (program): Multiples of 5 having only odd digits.
  • A061830 (program): Multiples of 5 having only even digits.
  • A061831 (program): Multiples of 9 having only even digits.
  • A061832 (program): Multiples of 11 having only even digits.
  • A061833 (program): Multiples of 11 having only odd digits.
  • A061834 (program): a(n) = binomial(n,2) * !n.
  • A061836 (program): a(n) = smallest k>0 such that k+n divides k!.
  • A061837 (program): Numbers k such that (k+2)^2 | k!.
  • A061838 (program): Numbers k such that (k+3)^3 | k!.
  • A061840 (program): Numbers k such that (k+4)^4 | k!.
  • A061841 (program): Numbers k such that (k+5)^5 | k!.
  • A061842 (program): Numbers k such that (k+6)^6 | k!.
  • A061861 (program): First two significant digits of 1/n written in decimal.
  • A061866 (program): a(n) is the number of solutions to x+y+z = 0 mod 3, where 1 <= x < y < z <= n.
  • A061867 (program): Squares whose product of digits is also a square (allowing zeros).
  • A061870 (program): Numbers such that |first digit - second digit + third digit - fourth digit …| = 1.
  • A061871 (program): |First digit - second digit + third digit - fourth digit …| = 2.
  • A061872 (program): |First digit - second digit + third digit - fourth digit …| = 3.
  • A061873 (program): Numbers n such that |first digit - second digit + third digit - fourth digit …| = 4.
  • A061874 (program): |First digit - second digit + third digit - fourth digit …| = 5.
  • A061875 (program): |First digit - second digit + third digit - fourth digit …| = 6.
  • A061876 (program): |First digit - second digit + third digit - fourth digit …| = 7.
  • A061877 (program): |First digit - second digit + third digit - fourth digit …| = 8.
  • A061878 (program): |First digit - second digit + third digit - fourth digit …| = 9.
  • A061879 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 10.
  • A061880 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 11.
  • A061881 (program): First (leftmost) digit - second digit + third digit - fourth digit …. = 12.
  • A061884 (program): a(n) = sum_{ d | n } phi(lcm(d,n/d)), where phi(n) = Euler totient A000010.
  • A061885 (program): n + largest triangular number less than or equal to n.
  • A061887 (program): n + largest square less than or equal to n; numbers in the range [2k^2,2k^2+2k] for some k.
  • A061891 (program): a(0) = 1; for n>0, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.
  • A061896 (program): Triangle of coefficients of Lucas polynomials.
  • A061900 (program): Triangular numbers that are not squarefree.
  • A061902 (program): Number of digits in n-th term of A061482.
  • A061904 (program): Numbers n such that the iterative cycle: n -> sum of digits of n^2 has only one distinct element.
  • A061909 (program): Skinny numbers: numbers n such that there are no carries when n is squared by “long multiplication”.
  • A061910 (program): Positive numbers n such that sum of digits of n^2 is a square.
  • A061924 (program): Number of combinations in card games with 4 suits and 4 players.
  • A061925 (program): a(n) = ceiling(n^2/2) + 1.
  • A061927 (program): a(n) = n(n+1)(2n+1)(n^2+n+3)/30.
  • A061928 (program): Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.
  • A061981 (program): a(n) = 3^n - 2*n - 1.
  • A061982 (program): a(n) = 3^n - (n+1)*(n+2)/2.
  • A061983 (program): 3^n - (3n^2 + n + 2)/2.
  • A061987 (program): Number of times n-th distinct value is repeated in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984; also number of times n-th distinct row is repeated in square array T(n,k) = T(n-1,k) + T(n-1,floor(k/2)) + T(n-1,floor(k/3)) with T(0,0) = 1, i.e., in A061980.
  • A061989 (program): Number of ways to place 3 nonattacking queens on a 3 X n board.
  • A061995 (program): Number of ways to place 2 nonattacking kings on an n X n board.
  • A061999 (program): a(n) = 2*a(n-1)^2 - 2*a(n-2)^2 with a(0) = 0, a(1) = 1.
  • A062000 (program): a(n) = a(n-1)^2 - a(n-2)^2 with a(0) = 0, a(1) = 2.
  • A062003 (program): Product of the k numbers formed by cyclically permuting digits of n (where k = number of digits of n).
  • A062004 (program): a(n) = 2*n*mu(n).
  • A062005 (program): Floor of arithmetic-geometric mean of n and 2n.
  • A062006 (program): a(n) = prime(n)^n + 1.
  • A062007 (program): a(n) = mu(n)*prime(n).
  • A062011 (program): a(n) = 2*tau(n) = 2*A000005(n).
  • A062018 (program): a(n) = n^n written backwards.
  • A062020 (program): a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i) - prime(j)).
  • A062023 (program): a(n) = (n^(n+1) + n^(n-1))/2.
  • A062024 (program): a(n) = ((n+1)^n + (n-1)^n)/2.
  • A062025 (program): a(n) = n*(13*n^2 - 7)/6.
  • A062026 (program): a(n) = n*(n+1)*(n^2 - 3*n + 6)/4.
  • A062028 (program): a(n) = n + sum of the digits of n.
  • A062029 (program): Group even numbers into (2), (4,6), (8,10,12), (14,16,18,20), …; a(n) = product of n-th group.
  • A062030 (program): Group even numbers into (2,4), (6,8,10,12), (14,16,18,20,22,24), …; a(n) = product of n-th group.
  • A062031 (program): Group odd numbers into (1), (3,5,7), (9,11,13,15,17), …; a(n) = product of n-th group.
  • A062032 (program): Group odd numbers into (1), (3,5), (7,9,11), (13,15,17,19), …; a(n) = product of n-th group.
  • A062033 (program): Binary expansion of n does not contain 1-bits at even positions. Integers whose base 4 representation consists of only 0’s and 2s.
  • A062042 (program): a(1) = 2, a(n) = least number greater than a(n-1) such that a(n-1) + a(n) is a prime.
  • A062044 (program): Primes arising in A062042.
  • A062046 (program): Sum of even numbers between consecutive primes.
  • A062047 (program): Sum of odd numbers between consecutive primes.
  • A062048 (program): a(n) = Sum_{k=1..n} floor(sqrt(prime(k))).
  • A062050 (program): n-th chunk consists of the numbers 1, …, 2^n.
  • A062051 (program): Number of partitions of n into powers of 3.
  • A062052 (program): Numbers with 2 odd integers in their Collatz (or 3x+1) trajectory.
  • A062054 (program): Numbers with 4 odd integers in their Collatz (or 3x+1) trajectory.
  • A062055 (program): Numbers with 5 odd integers in their Collatz (or 3x+1) trajectory.
  • A062056 (program): Numbers with 6 odd integers in their Collatz (or 3x+1) trajectory.
  • A062057 (program): Numbers with 7 odd integers in their Collatz (or 3x+1) trajectory.
  • A062058 (program): Numbers with 8 odd integers in their Collatz (or 3x+1) trajectory.
  • A062059 (program): Numbers with 9 odd integers in their Collatz (or 3x+1) trajectory.
  • A062060 (program): Numbers with 10 odd integers in their Collatz (or 3x+1) trajectory.
  • A062068 (program): a(n) = d(sigma(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisor function (A000203).
  • A062069 (program): a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).
  • A062071 (program): a(n) = [n/1] + [n/(2^2)] + [n/(3^3)] + [n/(4^4)] + … + [n/(k^k)] + …, up to infinity, where [ ] is the floor function.
  • A062074 (program): a(n) = n^3 * 3^n.
  • A062075 (program): a(n) = n^4 * 4^n.
  • A062076 (program): a(n) = (2n-1)^n * n^(2n-1).
  • A062077 (program): a(n) = (2n)^n * n^(2n).
  • A062079 (program): Group the odd numbers as (1), (3,5), (7,9,11), (13,15,17,19), (21,23,25,27,29), … then a(n) = LCM of the n-th group.
  • A062080 (program): Group the even numbers as 2, (4,6), (8,10,12), (14,16,18,20), (22,24,26,28,30), … then a(n) = LCM of the n-th group.
  • A062081 (program): Group the even numbers as (2,4), (6,8,10,12), (14,16,18,20,22,24), (26,28,30,32,34,36,38,40), … then a(n) = LCM of the n-th group.
  • A062085 (program): Squarefree numbers with all even digits.
  • A062086 (program): Squarefree numbers with all odd digits.
  • A062087 (program): Squarefree numbers with all prime digits.
  • A062090 (program): a(1) = 1, a(n) = smallest odd number that does not divide the product of all previous terms.
  • A062091 (program): a(1) = 2, a(n)= smallest even number which does not divide the product of all previous terms.
  • A062092 (program): a(n) = 2*a(n-1) - (-1)^n for n > 0, a(0)=2.
  • A062096 (program): a(1) = 2; for n > 1, a(n) is smallest number, greater than a(n-1), which is relatively prime to the sum of all previous terms.
  • A062097 (program): a(1) = 1; a(n) = sum of the sum and the product of all previous terms.
  • A062098 (program): a(n) = 7 * n!.
  • A062099 (program): Triangular numbers whose sum of digits is a triangular number.
  • A062106 (program): Number of ways a black pawn (from any starting square on the second back rank) can (theoretically) end on the n-th square of the leftmost file counted from the back rank.
  • A062107 (program): Diagonal of table A062104.
  • A062108 (program): a(n) = floor(n^(3/4)).
  • A062109 (program): Expansion of ((1-x)/(1-2*x))^4.
  • A062110 (program): A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.
  • A062111 (program): Upper-right triangle resulting from binomial transform calculation for nonnegative integers.
  • A062112 (program): a(0)=0; a(1)=1; a(n) = a(n-1) + (3 + (-1)^n)*a(n-2)/2.
  • A062113 (program): a(0)=1; a(1)=2; a(n) = a(n-1) + a(n-2)*(3 - (-1)^n)/2.
  • A062114 (program): a(n) = 2*Fibonacci(n) - (1 - (-1)^n)/2.
  • A062116 (program): a(n) = 2^n mod 17.
  • A062117 (program): Order of 3 mod n-th prime.
  • A062119 (program): a(n) = n! * (n-1).
  • A062123 (program): a(n) = 2 + 9*n*(1 + n)/2.
  • A062124 (program): Fourth column of A046741.
  • A062125 (program): Fifth column of A046741.
  • A062135 (program): Odd-numbered columns of Losanitsch triangle A034851 formatted as triangle with an additional first column.
  • A062136 (program): Twelfth column of Losanitsch’s triangle A034851 (formatted as lower triangular matrix).
  • A062137 (program): Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).
  • A062138 (program): Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).
  • A062139 (program): Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x).
  • A062140 (program): Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).
  • A062141 (program): Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
  • A062142 (program): Fourth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
  • A062143 (program): Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
  • A062144 (program): Sixth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
  • A062145 (program): Coefficient triangle of certain polynomials N(3; m,x).
  • A062146 (program): Row sums of signed triangle A062137 (generalized Laguerre, a=3).
  • A062147 (program): Row sums of unsigned triangle A062137 (generalized a=3 Laguerre).
  • A062148 (program): Second (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).
  • A062149 (program): Third column sequence of triangle A062138 (generalized a=5 Laguerre).
  • A062150 (program): Fourth (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).
  • A062151 (program): Fifth column sequence of triangle A062138 (generalized a=5 Laguerre).
  • A062152 (program): Sixth (unsigned) column of triangle A062138 (generalized a=5 Laguerre).
  • A062153 (program): a(n) = floor(log_3(n)).
  • A062157 (program): a(n) = 0^n-(-1)^n.
  • A062158 (program): a(n) = n^3 - n^2 + n - 1.
  • A062159 (program): a(n) = n^5 - n^4 + n^3 - n^2 + n - 1.
  • A062160 (program): Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by antidiagonals.
  • A062161 (program): Boustrophedon transform of n mod 2.
  • A062162 (program): Boustrophedon transform of (-1)^n.
  • A062169 (program): Triangle T(n, k) = k! mod n for n >= 1, 1 <= k <= n.
  • A062170 (program): Maximum value of factorials mod n.
  • A062171 (program): Number of non-unitary divisors of n (A048105) > number of distinct prime divisors of n (A001221).
  • A062172 (program): Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].
  • A062173 (program): a(n) = 2^(n-1) mod n.
  • A062174 (program): a(n) = 3^(n-1) mod n.
  • A062175 (program): a(n) = 4^(n-1) mod n.
  • A062176 (program): a(n) = 5^(n-1) mod n.
  • A062178 (program): a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1.
  • A062186 (program): a(n) = a(n-1) - a(floor(n/2)), with a(1)=1.
  • A062187 (program): a(n+1) = a(n) - a(floor(n/2)), with a(0)=0, a(1)=1.
  • A062188 (program): a(n+1) = a(n) + a(floor(n/2)), with a(0)=0, a(1)=1.
  • A062189 (program): a(n) = 2 * 3^(n-2)*n*(1+2*n).
  • A062190 (program): Coefficient triangle of certain polynomials N(5; m,x).
  • A062191 (program): Row sums of signed triangle A062138 (generalized a=5 Laguerre).
  • A062192 (program): Row sums of unsigned triangle A062138 (generalized a=5 Laguerre).
  • A062193 (program): Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).
  • A062194 (program): Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).
  • A062195 (program): Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).
  • A062196 (program): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  • A062197 (program): Row sums of signed triangle A062139 (generalized a=2 Laguerre).
  • A062198 (program): Sum of first n semiprimes.
  • A062199 (program): Second (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).
  • A062200 (program): Number of compositions of n such that two adjacent parts are not equal modulo 2.
  • A062205 (program): Number of alignments of n strings of length 4.
  • A062206 (program): a(n) = n^(2n).
  • A062207 (program): a(n) = 2*n^n-1.
  • A062208 (program): a(n) = Sum_{m>=0} binomial(m,3)^n*2^(-m-1).
  • A062234 (program): a(n) = 2*prime(n) - prime(n+1).
  • A062235 (program): a(n) = prime(n)^2 - prime(n+1).
  • A062236 (program): Sum of the levels of all nodes in all noncrossing trees with n edges.
  • A062242 (program): McKay-Thompson series of class 18D for the Monster group.
  • A062243 (program): McKay-Thompson series of class 24c for the Monster group.
  • A062244 (program): McKay-Thompson series of class 36B for the Monster group.
  • A062245 (program): Expansion of Hauptmodul for group G’_{27|3}.
  • A062246 (program): McKay-Thompson series of class 27c for the Monster group.
  • A062249 (program): a(n) = n + d(n), where d(n) = number of divisors of n, cf. A000005.
  • A062258 (program): Number of (0,1)-strings of length n not containing the substring 0100100.
  • A062260 (program): Third (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).
  • A062261 (program): Fourth (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).
  • A062262 (program): Fifth (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).
  • A062263 (program): Sixth (unsigned) column of triangle A062140 (generalized a=4 Laguerre).
  • A062264 (program): Coefficient triangle of certain polynomials N(4; m,x).
  • A062265 (program): Row sums of signed triangle A062140 (generalized a=4 Laguerre).
  • A062266 (program): Row sums of unsigned triangle A062140 (generalized a=4 Laguerre).
  • A062267 (program): Row sums of (signed) triangle A060821 (Hermite polynomials).
  • A062272 (program): Boustrophedon transform of (n+1) mod 2.
  • A062273 (program): a(n) is an n-digit number with digits in increasing order with 0 following 9 and this is maintained in the concatenation of any number of consecutive terms.
  • A062275 (program): Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.
  • A062276 (program): a(n) = floor(n^(n+1) / (n+1)^n).
  • A062278 (program): a(n) = floor(3^n / n^3).
  • A062281 (program): Smallest multiple of n-th prime with all even digits.
  • A062282 (program): Number of permutations of n elements with an even number of fixed points.
  • A062283 (program): Table by antidiagonals of floor[ n^k / k^n ].
  • A062284 (program): Primes p such that p + 50 is also prime.
  • A062287 (program): Palindromic numbers with even digits.
  • A062288 (program): Numbers k such that prime(k)+50 is also prime.
  • A062289 (program): Numbers n such that n-th row in Pascal triangle contains an even number, i.e., A048967(n) > 0.
  • A062296 (program): a(n) = number of entries in n-th row of Pascal’s triangle divisible by 3.
  • A062298 (program): Number of nonprimes <= n.
  • A062301 (program): Number of ways writing n-th prime as a sum of two primes.
  • A062302 (program): Number of ways writing n-th prime as a sum of a prime and a nonprime.
  • A062303 (program): Number of ways writing n-th prime as a sum of a nonprime and a composite.
  • A062312 (program): Nonprime numbers squared.
  • A062313 (program): Factorials of nonprime numbers.
  • A062316 (program): Neither the sum or difference of 2 squares.
  • A062317 (program): Numbers k such that 5*k-1 is a perfect square.
  • A062318 (program): Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.
  • A062319 (program): Number of divisors of n^n, or of A000312(n).
  • A062320 (program): Nonsquarefree numbers squared. A013929(n)^2.
  • A062322 (program): Factorials of nonsquarefree numbers, or A013929(n)!, (including 1).
  • A062323 (program): Triangle with a(n,n)=1, a(n,k)=(n-1)*a(n-1,k)+a(n-2,k) for n>k.
  • A062326 (program): Primes p such that p^2 - 2 is also prime.
  • A062329 (program): a(n) = (sum of digits of n) - (product of digits of n).
  • A062330 (program): a(n) = product of the sum and product of the digits of a(n-1) (0 is not to be considered a factor in the product).
  • A062331 (program): a(n) is the product of the sum and the product of the digits of n (0 is not to be considered a factor in the product).
  • A062332 (program): Primes starting and ending with 1.
  • A062335 (program): Primes starting and ending with 9.
  • A062337 (program): Primes whose sum of digits is 7.
  • A062340 (program): Primes whose sum of digits is a multiple of 5.
  • A062342 (program): Primes whose sum of digits is a multiple of 8.
  • A062344 (program): Triangle of binomial(2*n, k) with n >= k.
  • A062346 (program): Consider 2n tennis players; a(n) is the number of matches needed to let every possible pair play each other.
  • A062347 (program): a(n) = (product of first n primes) modulo prime(n+1).
  • A062348 (program): a(n) = n! / (number of distinct prime divisors of n).
  • A062349 (program): a(n) = n! / (number of prime divisors of n, counted with multiplicity).
  • A062350 (program): Primes containing digits 1, 2, 3 only.
  • A062354 (program): a(n) = sigma(n)*phi(n).
  • A062355 (program): a(n) = d(n) * phi(n), where d(n) is the number of divisors function.
  • A062356 (program): a(n) = floor(n/phi(n)).
  • A062357 (program): a(n) = n*p(n+1)-(n+1)*p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).
  • A062358 (program): a(n) = n! / number of divisors of n.
  • A062359 (program): a(n) = floor(n!/sigma(n)).
  • A062362 (program): a(n) = floor of Sum_{ d divides n } phi(d)/d.
  • A062363 (program): a(n) = Sum_{d|n} d!.
  • A062367 (program): Multiplicative with a(p^e) = (e+1)*(e+2)*(2*e+3)/6.
  • A062368 (program): Multiplicative with a(p^e) = (e+1)*(e+2)*(4*e+3)/6.
  • A062369 (program): Dirichlet convolution of n and tau^2(n).
  • A062370 (program): a(n) = Sum_{i|n,j|n} sigma(i)*sigma(j)/sigma(gcd(i,j)), where sigma(n) = sum of divisors of n.
  • A062371 (program): Numbers the product of whose nonzero digits is a perfect square.
  • A062378 (program): n divided by largest cubefree factor of n.
  • A062379 (program): n divided by largest 4th-power-free factor of n.
  • A062380 (program): a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.
  • A062383 (program): a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).
  • A062389 (program): a(n) = floor( (2n-1)*Pi/2 ).
  • A062392 (program): a(n) = n^4 - (n-1)^4 + (n-2)^4 - … 0^4.
  • A062393 (program): a(n) = n^5 - (n-1)^5 + (n-2)^5 - … +(-1)^n*0^5.
  • A062394 (program): a(n) = 6^n + 1.
  • A062395 (program): a(n) = 8^n + 1.
  • A062396 (program): a(n) = 9^n + 1.
  • A062397 (program): a(n) = 10^n + 1.
  • A062401 (program): a(n) = phi(sigma(n)).
  • A062402 (program): a(n) = sigma(phi(n)).
  • A062408 (program): Numbers k such that floor(Pi*k) is prime.
  • A062409 (program): Numbers k such that floor(e*k) is prime.
  • A062411 (program): a(n) = (-1)^(p-1)*(p-1)! + 1 where p = prime(n).
  • A062440 (program): a(n) = Sum_{k=1..n} (prime(k) - 1)^n.
  • A062457 (program): a(n) = prime(n)^n.
  • A062458 (program): Nearest integer to (n+1)^(n+1)/n^n.
  • A062481 (program): a(n) = n^prime(n).
  • A062501 (program): Number of distinct prime divisors of the nonprimes (including 1).
  • A062502 (program): Number of prime divisors (with repetition) of the nonprimes (including 1).
  • A062503 (program): Squarefree numbers squared.
  • A062507 (program): Table by antidiagonals related to partial sums and differences of Pell numbers (A000129).
  • A062508 (program): a(n) = 3^(2n)+7.
  • A062509 (program): a(n) = n^omega(n).
  • A062510 (program): a(n) = 2^n + (-1)^(n+1).
  • A062532 (program): Odd nonprimes squared, or A014076(n)^2.
  • A062533 (program): a(n) = A000010(A014076(n)).
  • A062534 (program): Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).
  • A062539 (program): Decimal expansion of the Lemniscate constant or Gauss’s constant.
  • A062544 (program): a(n) = n plus sum of previous three terms.
  • A062545 (program): Continued fraction for the 2nd Du Bois-Reymond constant.
  • A062546 (program): Decimal expansion of the 2nd Du Bois-Reymond constant.
  • A062547 (program): a(n) is least odd integer not a partial sum of 1, 3, …, a(n-1).
  • A062548 (program): Even integers that are not partial sums of A062547.
  • A062550 (program): a(n) = Sum_{k = 1..2n} floor(2n/k).
  • A062553 (program): Number of Abelian subgroups of the dihedral group with 2n elements.
  • A062557 (program): 2n-1 1’s followed by a 2.
  • A062558 (program): Number of nonisomorphic cyclic subgroups of dihedral group with 2n elements.
  • A062561 (program): a(n) = 3*binomial(2*n, n-1).
  • A062562 (program): a(n) = Sum_{k=1..n} mu(k)*sigma(k).
  • A062563 (program): a(n) = Sum_{k=1..n} d(k)* mu(k), where d(k) is the number of divisors function.
  • A062565 (program): Squarefree parts of 3-smooth numbers.
  • A062569 (program): a(n) = sigma(n!).
  • A062570 (program): a(n) = phi(2*n).
  • A062571 (program): a(n) = minimum over nonnegative integers m of the size of the largest subset of pairwise relatively prime numbers in {m+1, m+2, …, m+n}.
  • A062602 (program): Number of ways of writing n = p+c with p prime and c nonprime (1 or a composite number).
  • A062610 (program): Number of ways of writing n = c1 + c2 with c1 and c2 nonprimes [=1 or composite].
  • A062627 (program): a(n) = mu(n) * Catalan(n).
  • A062673 (program): Every divisor (except 1) contains the digit 6.
  • A062677 (program): Numbers with property that every divisor (except 1) contains the digit 8.
  • A062692 (program): Number of irreducible polynomials over F_2 of degree at most n.
  • A062707 (program): Table by antidiagonals of n*k*(k+1)/2.
  • A062708 (program): Write 0,1,2,3,4,… in a triangular spiral; then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0,2,…
  • A062709 (program): a(n) = 2^n + 3.
  • A062715 (program): Triangle T(i,j) (i >= -1, -1<=j<=i) whose (i,j)-th entry is 1 if j=-1 otherwise binomial(i,j)*2^(i-j).
  • A062717 (program): Numbers m such that 6*m+1 is a perfect square.
  • A062720 (program): If n is odd then 2*n else prime(n).
  • A062721 (program): Numbers k such that k is a product of two primes and k-2 is prime.
  • A062722 (program): a(n) = ceiling(n/3)*round(n/4).
  • A062723 (program): Least common multiple (LCM) of the first n+1 terms of A000792.
  • A062724 (program): a(n) = floor(tau^n) + 1, where tau = (1 + sqrt(5))/2.
  • A062725 (program): Write 0,1,2,3,4,… in a triangular spiral, then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0,7,…
  • A062727 (program): Sum of the divisors of n^n (A000312).
  • A062728 (program): Second 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n+7)/2.
  • A062730 (program): Rows of Pascal’s triangle which contain 3 terms in arithmetic progression.
  • A062731 (program): Sum of divisors of 2*n.
  • A062732 (program): Squares arising in A039770.
  • A062737 (program): Primes p such that 4p-1 is also prime.
  • A062739 (program): Odd powerful numbers.
  • A062741 (program): 3 times pentagonal numbers: 3*n*(3*n-1)/2.
  • A062747 (program): Row sums of (unsigned) staircase array A062746.
  • A062748 (program): Fourth column (r=3) of FS(3) staircase array A062745.
  • A062749 (program): Sixth column (r=5) of FS(3) staircase array A062745.
  • A062752 (program): Row sums of unsigned N(4) staircase array A062751.
  • A062753 (program): Multiples of 4 whose sum of digits is also a multiple of 4.
  • A062754 (program): a(n) = gcd(n, sigma(n+1)).
  • A062755 (program): a(n) = sigma_n(n^2): sum of n-th powers of divisors of n^2.
  • A062756 (program): Number of 1’s in ternary (base-3) expansion of n.
  • A062758 (program): Product of squares of divisors of n.
  • A062763 (program): a(n) is the greatest common divisor of (n-1)! and n^n.
  • A062765 (program): n*(n-1)*(n-3)*(n-5).
  • A062767 (program): Numbers k such that 3k+1, 3k+5 and 3k+7 are all prime.
  • A062768 (program): Multiples of 6 such that the sum of the digits is equal to 6.
  • A062770 (program): n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).
  • A062771 (program): Order of automorphism group of the group C_n X C_2 (where C_n is the cyclic group with n elements).
  • A062772 (program): Smallest prime larger than square of n-th prime.
  • A062773 (program): Index of the smallest prime which follows square of n-th prime.
  • A062774 (program): Inverse Moebius transform of PrimePi function.
  • A062776 (program): Greatest common divisor of (n+2)! and n^n.
  • A062777 (program): 2^n - mu(n).
  • A062779 (program): a(n) = 2*n*(2*n)!.
  • A062781 (program): Number of arithmetic progressions of four terms and any mean which can be extracted from the set of the first n positive integers.
  • A062783 (program): a(n) = 3*n*(4*n-1).
  • A062785 (program): Chowla’s function * sigma(n).
  • A062786 (program): Centered 10-gonal numbers.
  • A062789 (program): a(n) = gcd(n, phi(n) * (phi(n) + 1)).
  • A062790 (program): Moebius transform of the cototient function A051953.
  • A062796 (program): Inverse Moebius transform of f(n) = n^n (A000312).
  • A062797 (program): Inverse Moebius transform of f(x) = primorial(x) = A002110(x).
  • A062798 (program): Inverse Moebius transform of central binomial coefficients f[x]=C(c,[x/2])=A001405[x].
  • A062799 (program): Inverse Möbius transform of the numbers of distinct prime factors (A001221).
  • A062800 (program): Primes of form 100k + 1.
  • A062801 (program): Number of 2 X 2 non-singular integer matrices with entries from {0,…,n}.
  • A062803 (program): Number of solutions to x^2 == y^2 (mod n).
  • A062805 (program): a(n) = Sum_{i=1..n} i*n^(n-i).
  • A062806 (program): a(n) = Sum_{i=1..n} i*n^i.
  • A062807 (program): a(n) = Sum_{i=1..n} i*(n-i)^i.
  • A062808 (program): a(n) = Sum_{i=1..n} n^i * (n - i).
  • A062809 (program): a(n) = Sum_{i = 1..n} (n - i)^(1 + i).
  • A062810 (program): a(n) = Sum_{i=1..n} i^(n - i) + (n - i)^i.
  • A062811 (program): a(n) = Sum_{i=1..n} i^(n - i) + (i - n)^i.
  • A062812 (program): a(n) = Sum_{i=1..n} i^(n - i) + (-1)^(n - i)*(n - i)^i.
  • A062813 (program): a(n) = Sum_{i=0..n-1} i*n^i.
  • A062814 (program): a(n) = Sum_{i=0..n-1} i * (n - i)^(n - i).
  • A062815 (program): a(n) = Sum_{i=1..n} i^(i+1).
  • A062816 (program): a(n) = phi(n)*tau(n) - 2n = A000010(n)*A000005(n) - 2*n.
  • A062817 (program): a(n) = Sum_{i=0..n} i^(n - i)*(n - i)^i.
  • A062820 (program): Sum_{k=1..n} p(k)*mu(k).
  • A062821 (program): Number of divisors of totient of n.
  • A062822 (program): Sum of divisors of the squarefree numbers: sigma(A005117(n)).
  • A062824 (program): Ch(A005117(n)) where Ch(n) is Chowla’s function and A005117(n) are the squarefree numbers.
  • A062825 (program): Ch(n-th nonprime) where Ch(n) is Chowla’s function, cf. A048050.
  • A062828 (program): a(n) = gcd(2n, n(n+1)/2).
  • A062830 (program): a(n) = n - phi(n) + 1.
  • A062831 (program): Number of ways n can be expressed as the sum of a nonzero square and 1 or a prime.
  • A062833 (program): 2n! / number of divisors of n.
  • A062835 (program): a(1) = 0; for n > 1 a(n) = sum of divisors of n^2-1; or sigma(A005563(n-1)).
  • A062838 (program): Cubes of squarefree numbers.
  • A062839 (program): a(n) = floor((prime(n-1)+prime(n+1))/2).
  • A062849 (program): When expressed in base 2 and then interpreted in base 8, is a multiple of the original number.
  • A062870 (program): Number of permutations of degree n with greatest sum of distances.
  • A062871 (program): a(n) is the integer part of the geometric mean of n! and n^n.
  • A062872 (program): Nearest integer to geometric mean of n! and n^n.
  • A062873 (program): Nearest integer to arithmetic mean of n! and n^n.
  • A062874 (program): Integer part of arithmetic mean of n! and n^n.
  • A062875 (program): Records in A046112 (or A006339).
  • A062876 (program): Numbers of lattice points corresponding to incrementally largest circle radii in A062875.
  • A062877 (program): Apart from the initial term (0), each a(n) is representable as a sum of distinct odd-indexed Fibonacci numbers.
  • A062878 (program): a(n) is the position of A050614(n) in A062877.
  • A062879 (program): Integers whose Zeckendorf expansion does not contain ones at even positions.
  • A062880 (program): Zero together with the numbers which can be written as a sum of distinct odd powers of 2.
  • A062882 (program): a(n) = (1 - 2*cos(Pi/9))^n + (1 + 2*cos(Pi*2/9))^n + (1 + 2*cos(Pi*4/9))^n.
  • A062888 (program): Smallest palindromic multiple of n-th prime.
  • A062890 (program): Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.
  • A062897 (program): Number and its reversal are both multiples of 2.
  • A062898 (program): Number and its reversal are both multiples of 4.
  • A062899 (program): Number and its reversal are both multiples of 6.
  • A062900 (program): Number and its reversal are both multiples of 8.
  • A062901 (program): Number and its reversal are both multiples of 7.
  • A062902 (program): Number and its reversal are both multiples of 12.
  • A062903 (program): Numbers n such that n and its reversal are both multiples of 13.
  • A062905 (program): Numbers n such that n and its reversal are both multiples of 15.
  • A062908 (program): Non-palindromic number and its reversal are both even.
  • A062910 (program): Non-palindromic number and its reversal are both multiples of 6.
  • A062918 (program): Sum of the digit reversals of the first n natural numbers.
  • A062919 (program): “Reverse factorials”: product of the digit reversals of the numbers 1 through n.
  • A062938 (program): a(n) = n*(n+1)*(n+2)*(n+3)+1, which equals (n^2 +3*n + 1)^2.
  • A062940 (program): Number of squares (including 0) with n digits.
  • A062941 (program): Number of n-digit cubes (0 is included as a single-digit number).
  • A062947 (program): a(n) = binomial(n,floor(n/7)).
  • A062948 (program): H(A002808(n)) where H(n) is the half-totient function, H(n) = phi(n)/2: (A023022) and A002808(n) are the composites.
  • A062949 (program): Multiplicative with a(p^e) = ((e+1)*p^(e+1)-(e+2)*p^e+1)/(p-1).
  • A062950 (program): C(H(n)), where C(n) is Chowla’s function (A048050) and H(n) is the half-totient function (A023022).
  • A062951 (program): H(A005117(n)) where H(n) is the half-totient function and A005117(n) are the squarefree numbers.
  • A062952 (program): Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).
  • A062953 (program): Multiplicative with a(p^e) = -p.
  • A062954 (program): Least common multiple of Lucas numbers L(0), L(1), …, L(n).
  • A062955 (program): a(n) = phi(n^2) - phi(n) = (n-1) * phi(n).
  • A062956 (program): a(n) = h(n^2) - h(n), where h(n) is the half-totient function (A023022).
  • A062957 (program): C(n^2)-C(n), where C(n) is Chowla’s function (A048050).
  • A062958 (program): phi[n+1] < 2phi[n], phi[]=A000010.
  • A062960 (program): Number of divisors of (n!)^n (A036740).
  • A062961 (program): Number of divisors of n!^n! (A046882).
  • A062963 (program): Mu(n) * H(n) where H(n) is A023022.
  • A062965 (program): Positive numbers which are one less than a perfect square that is also another power.
  • A062966 (program): a(n) = C(3+n, n) + C(4+n, n) + C(5+n, n) + C(6+n, n).
  • A062967 (program): a(n) = 2*(sigma(n)-n-1)+1, where sigma = A000203, sum of divisors of n.
  • A062968 (program): n + 1 - d(n), where d(n) is the number of divisors function.
  • A062969 (program): d(n)-n-1 is prime.
  • A062970 (program): a(n) = 1 + Sum_{j=1..n} j^j.
  • A062971 (program): a(n) = (2*n)^n.
  • A062973 (program): Chowla function of n is not divisible by phi(n).
  • A062974 (program): omega[n+1] < 2*omega[n], where omega[n] is the number of distinct prime divisors of n.
  • A062981 (program): a(n) = n^phi(n).
  • A062988 (program): a(n) = binomial(n+6,5) - 1.
  • A062989 (program): a(n) = C(n+6, 6) - n - 1.
  • A062990 (program): Eighth column (r=7) of FS(5) staircase array A062985.
  • A062991 (program): Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).
  • A062992 (program): Row sums of unsigned triangle A062991.
  • A062993 (program): A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.
  • A062994 (program): Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.
  • A062997 (program): Sum of digits is strictly greater than product of digits.
  • A062998 (program): Numbers with property that sum of digits is less than or equal to product of digits.
  • A062999 (program): Numbers k with property that the sum of the digits of k is strictly less than the product of the digits of k.
  • A063003 (program): Difference between 3^n and the next larger power of 2.
  • A063004 (program): Difference between 2^n and the next larger power of 3.
  • A063005 (program): Difference between 2^n and the next smaller power of 3.
  • A063006 (program): Coefficients in a 10-adic square root of 1.
  • A063007 (program): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
  • A063008 (program): Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,…] with p1 >= p2 >= p3 >= … is encoded as 2^p1 * 3^p2 * 5^p3 * … .
  • A063009 (program): Write n in binary then square as if written in base 10.
  • A063010 (program): Carryless binary square of n; also Moser-de Bruijn sequence written in binary.
  • A063012 (program): Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.
  • A063013 (program): Numbers where k-th digit from right is either 0 or k.
  • A063014 (program): Number of solutions to n^2=b^2+c^2 [with c>=b>=0].
  • A063015 (program): Numbers k such that k + mu(k) is prime.
  • A063016 (program): a(n) is the product of Catalan(n) and (2^(n+1) - 1).
  • A063017 (program): a(n) = Catalan(n)*(3^(n+1) - 2^(n+1) + 1)/2.
  • A063019 (program): Reversion of y - y^2 + y^3 - y^4.
  • A063020 (program): Reversion of y - y^2 - y^3 + y^4.
  • A063021 (program): Reversion of y - y^2 - y^5.
  • A063026 (program): Reversion of y - y^2 + y^4 - y^5.
  • A063030 (program): Reversion of y - y^2 - y^4 + y^5.
  • A063033 (program): Reversion of y - y^2 + y^4.
  • A063038 (program): Floor(n*sqrt(n)) - d(n), where d(n) is the number of divisors function.
  • A063051 (program): ‘Reverse and Add!’ trajectory of 879.
  • A063070 (program): a(n) = phi(n) - d(n), where d(n) is the number of divisors function (A000005).
  • A063073 (program): Square of determinant of character table of the dihedral group with 2n elements.
  • A063077 (program): a(n) = phi(n^2 + 1) - 2n.
  • A063079 (program): Bisection of A001790.
  • A063081 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 13 ).
  • A063083 (program): Number of permutations of n elements with an odd number of fixed points.
  • A063084 (program): a(n) = pi(n-1)*n - pi(n)*(n-1), where pi() = A000720().
  • A063085 (program): a(n) = usigma(n) - (phi(n) + d(n)), where usigma(n) is the sum of the unitary divisors of n and d(n) is the number of divisors of n.
  • A063086 (program): a(n) = gcd(1 + prime(n+1), 1 + prime(n)).
  • A063087 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 19 ).
  • A063088 (program): a(n) = Sum_{k=1..n} phi(k) - Sum{k=1..n} d(k), where d() is the number of divisors function.
  • A063089 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 21 ).
  • A063090 (program): a(n)/(n*n!) is the average number of comparisons needed to find a node in a binary search tree containing n nodes inserted in a random order.
  • A063092 (program): a(0)=1, a(1)=2 and, for n>1, a(n) = a(n-1) + 11*a(n-2).
  • A063093 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 25 ).
  • A063094 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 26 ).
  • A063095 (program): Record prime gap among first n+1 primes.
  • A063097 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 29 ).
  • A063098 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 30 ).
  • A063099 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 31 ).
  • A063102 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 34 ).
  • A063105 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 37 ).
  • A063107 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 39 ).
  • A063108 (program): a(1) = 1; a(n+1) = a(n) + product of nonzero digits of a(n).
  • A063109 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 41 ).
  • A063110 (program): Dimension of the space of weight 2n cusp forms for Gamma_0(42).
  • A063111 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 43 ).
  • A063114 (program): n + product of nonzero digits of n.
  • A063116 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 48 ).
  • A063117 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 49 ).
  • A063118 (program): Dimension of the space of weight 2n cusp forms for Gamma_0(50).
  • A063120 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 52 ).
  • A063121 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 53 ).
  • A063122 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 54 ).
  • A063123 (program): Number of solutions (r,s), 0< r< s, to the equation 1/n = 1/r + 1/s + 1/(r*s).
  • A063124 (program): a(n) = # { i prime | prime(n) <= i < prime(n)*2 } (prime(n) = A000040, the prime enumeration).
  • A063125 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 57 ).
  • A063126 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 58 ).
  • A063128 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 60 ).
  • A063129 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 61 ).
  • A063130 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 62 ).
  • A063133 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 65 ).
  • A063134 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 66 ).
  • A063135 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 67 ).
  • A063136 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 68 ).
  • A063139 (program): Composite numbers which in base 3 contain their largest proper factor as a substring.
  • A063140 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 72 ).
  • A063141 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 73 ).
  • A063142 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 74 ).
  • A063143 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 75 ).
  • A063144 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 76 ).
  • A063146 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 78 ).
  • A063147 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 79 ).
  • A063148 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 80 ).
  • A063149 (program): Composite numbers which in base 5 contain their largest proper factor as a substring.
  • A063150 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 82 ).
  • A063151 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 83 ).
  • A063152 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 84 ).
  • A063153 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 85 ).
  • A063154 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 86 ).
  • A063155 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 87 ).
  • A063157 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 89 ).
  • A063158 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 90 ).
  • A063159 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 91 ).
  • A063160 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 92 ).
  • A063161 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 93 ).
  • A063162 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 94 ).
  • A063164 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 96 ).
  • A063165 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 97 ).
  • A063166 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 98 ).
  • A063168 (program): Dimension of the space of weight 2n cusp forms for Gamma_0( 100 ).
  • A063169 (program): a(n) = n*A001865(n).
  • A063170 (program): Schenker sums with n-th term.
  • A063171 (program): Dyck language interpreted as binary numbers in ascending order.
  • A063195 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 6 ).
  • A063196 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 7 ).
  • A063197 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 9 ).
  • A063198 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 10 ).
  • A063199 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 11 ).
  • A063200 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 15 ).
  • A063201 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 18 ).
  • A063202 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 22 ).
  • A063203 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 23 ).
  • A063204 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 25 ).
  • A063205 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 29 ).
  • A063206 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 31 ).
  • A063207 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 33 ).
  • A063208 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 36 ).
  • A063209 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 41 ).
  • A063210 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 42 ).
  • A063211 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 43 ).
  • A063212 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 44 ).
  • A063213 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0(45).
  • A063214 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 46 ).
  • A063215 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 47 ).
  • A063216 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 49 ).
  • A063217 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 50 ).
  • A063218 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 51 ).
  • A063219 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 53 ).
  • A063220 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 55 ).
  • A063221 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 57 ).
  • A063222 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 58 ).
  • A063223 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 59 ).
  • A063224 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 60 ).
  • A063225 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 62 ).
  • A063226 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).
  • A063227 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 66 ).
  • A063228 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 67 ).
  • A063229 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 69 ).
  • A063230 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 71 ).
  • A063231 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 75 ).
  • A063232 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 77 ).
  • A063233 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 79 ).
  • A063234 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 82 ).
  • A063235 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 83 ).
  • A063236 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 85 ).
  • A063237 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 86 ).
  • A063238 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 87 ).
  • A063240 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 89 ).
  • A063241 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0(90).
  • A063242 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 92 ).
  • A063244 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 94 ).
  • A063245 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 98 ).
  • A063246 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 99 ).
  • A063247 (program): Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 100 ).
  • A063249 (program): Doubly hexagonal numbers.
  • A063250 (program): Number of binary right-rotations (iterations of A038572) to reach fixed point.
  • A063258 (program): a(n) = binomial(n+5,4) - 1.
  • A063262 (program): Eighth column (k=7) of sextinomial array A063260.
  • A063263 (program): Ninth column (k=8) of sextinomial array A063260.
  • A063264 (program): Tenth column (k=9) of sextinomial array A063260.
  • A063267 (program): Eighth column (k=7) of septinomial array A063265.
  • A063270 (program): a(n) = 9^(2n) + 1.
  • A063273 (program): Number of times most common digit of primes appears in first n primes.
  • A063281 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 8 ).
  • A063289 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).
  • A063300 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 27 ).
  • A063305 (program): Dimension of the space S_n^{new}(Gamma_1(32)) of weight n cuspidal newforms for Gamma_1( 32 ).
  • A063321 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 48 ).
  • A063327 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 54 ).
  • A063337 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 64 ).
  • A063353 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 80 ).
  • A063354 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 81 ).
  • A063369 (program): Dimension of the space of weight n cuspidal newforms for Gamma_1( 96 ).
  • A063376 (program): a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1 + 2^n).
  • A063377 (program): Sophie Germain degree of n: number of iterations of n under f(k) = 2k+1 before we reach a number that is not a prime.
  • A063395 (program): T(2n,n) with T(n,m) as in A063394.
  • A063396 (program): T(3,n) with T(n,m) as in A063394.
  • A063401 (program): a(n) = a(n-1)*a(n-2)*a(n-3) with a(0)=1, a(1)=2, a(2)=2.
  • A063402 (program): a(0)=0; a(1)=1; a(2)=2; a(n)= a(n-1) + a(n-2)*a(n-3).
  • A063404 (program): a(0)=1; a(1)=1; a(2)=1; a(n) = a(n-1) + (1 + a(n-2))*(1 + a(n-3)).
  • A063416 (program): Multiples of 7 whose sum of digits is equal to 7.
  • A063417 (program): Ninth column (k=8) of septinomial array A063265.
  • A063418 (program): Tenth column (k=9) of septinomial array A063265.
  • A063425 (program): Unattainable numbers: integers not expressible as k + product of nonzero digits of k (A063114).
  • A063427 (program): a(n) is the smallest positive integer k such that n*k/(n+k) is an integer.
  • A063428 (program): a(n) is the smallest positive integer of the form n*k/(n+k).
  • A063434 (program): Integers n > 10577 such that the ‘Reverse and Add!’ trajectory of n joins the trajectory of 10577.
  • A063436 (program): Write 1,2,3,4,… counterclockwise in a hexagonal spiral around 0 starting left down, then a(n) is the sequence found by reading from 0 in the vertical upward direction.
  • A063438 (program): Floor((n+1)*Pi)-Floor(n*Pi).
  • A063439 (program): a(n) = phi(n)^phi(n).
  • A063440 (program): Number of divisors of n-th triangular number.
  • A063441 (program): a(n) = sigma(n) * mu(n).
  • A063445 (program): Moebius transform of f(x) = EulerPhi(x^2) function (A002618).
  • A063448 (program): Decimal expansion of Pi * sqrt(2).
  • A063451 (program): n * sigma(n)-1 is a prime.
  • A063452 (program): Numbers k such that k - mu(k) is prime.
  • A063453 (program): Multiplicative with a(p^e) = 1 - p^3.
  • A063459 (program): A Beatty sequence: a(n) = floor(n*(Pi - 1)).
  • A063460 (program): A Beatty sequence: a(n) = floor(n * (Pi-1)/(Pi-2)).
  • A063462 (program): n * last digit of n.
  • A063464 (program): omega(n) = omega(n+2), where omega(n) is the number of distinct prime divisors of n.
  • A063465 (program): Number k such that omega(k) = omega(k+3), where omega(k) is the number of distinct prime divisors of k.
  • A063468 (program): Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.
  • A063472 (program): Primes of the form 666*k - 1.
  • A063473 (program): M(2*n-1), where M(n) is Mertens’s function (A002321): Sum_{k=1..n} mu(k), where mu = Moebius function (A008683).
  • A063475 (program): Sum_{d | H(n)} d^2, where H(n) is the Half-Totient function (A023022).
  • A063476 (program): Sum_{d |C(n)} d^2, where C(n) is the Cototient function n - phi(n) (A051953).
  • A063479 (program): Omega(n+1)=2*omega(n), where omega is the number of distinct prime divisors.
  • A063481 (program): a(n) = 4^n + 8^n.
  • A063482 (program): p(n) * last digit of p(n) where p(n) is n-th prime.
  • A063483 (program): S[A002808(n)] where S[] is Boris Stechkin’s function (A055004) and A002808(n) are the composites.
  • A063484 (program): Omega(n+1) = 2*Omega(n), where Omega(n) is the number of prime divisors of n (with repetition).
  • A063487 (program): Number of distinct prime divisors of 2^(2^n)-1 (A051179).
  • A063488 (program): a(n) = (2*n-1)*(n^2 -n +2)/2.
  • A063489 (program): a(n) = (2*n-1)*(5*n^2-5*n+6)/6.
  • A063490 (program): a(n) = (2*n - 1)*(7*n^2 - 7*n + 6)/6.
  • A063491 (program): a(n) = (2*n - 1)*(3*n^2 - 3*n + 2)/2.
  • A063492 (program): a(n) = (2*n - 1)*(11*n^2 - 11*n + 6)/6.
  • A063493 (program): a(n) = (2*n-1)*(13*n^2-13*n+6)/6.
  • A063494 (program): a(n) = (2*n - 1)*(7*n^2 - 7*n + 3)/3.
  • A063495 (program): a(n) = (2*n-1)*(5*n^2-5*n+2)/2.
  • A063496 (program): a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)/3.
  • A063497 (program): Number of atoms in first n shells of type I hyperfullerene.
  • A063498 (program): Atoms in cluster of n layers around C_60.
  • A063506 (program): a(n) = phi(a(n-1)) * number of divisors of a(n-1), a(1)=3.
  • A063510 (program): a(1) = 1, a(n) = a(floor(square root(n))) + 1 for n > 1.
  • A063511 (program): a(n) = a(floor(square root(n))) * 2.
  • A063514 (program): a(n) = sigma(n) mod phi(n).
  • A063518 (program): Values of 17^n mod 23.
  • A063521 (program): a(n) = n*(7*n^2-4)/3.
  • A063522 (program): a(n) = n*(5*n^2 - 3)/2.
  • A063523 (program): a(n) = n*(8*n^2 - 5)/3.
  • A063524 (program): Characteristic function of 1.
  • A063531 (program): Numbers k such that sigma(k) + 1 is a square.
  • A063533 (program): Hypotenuses of special Pythagorean triples constructed from twin primes as follows: {u, w}={p,p+2}; side a=2p(p+2), side b=(p+2)^2-p^2 and the terms of sequence are values of c=a(n)=p^2+(p+2)^2=phi(a/2)+1+sigma(a/2)+1.
  • A063534 (program): C(n) = H(n) + d(n), where C(n) is Chowla’s function A048050, H(n) is the half-totient function A023022 and d(n) is the number of divisors function A000005.
  • A063535 (program): Primes prime(n) such that prime(n+1)^2 < prime(n)*prime(n+2).
  • A063538 (program): Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).
  • A063539 (program): Numbers n that are sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) < sqrt(n).
  • A063541 (program): Least number of empty triangles determined by n points in the plane.
  • A063542 (program): Least number of empty convex quadrilaterals (4-gons) determined by n points in the plane.
  • A063543 (program): a(n) = n - product of nonzero digits of n.
  • A063549 (program): Smallest number of crossing-free matchings on n points in the plane.
  • A063574 (program): Number of steps to reach an integer == 1 (mod 4) when iterating the map n -> 3n/2 if n even or (3n+1)/2 if n odd.
  • A063637 (program): Primes p such that p+2 is a semiprime.
  • A063638 (program): Primes p such that p-2 is a semiprime.
  • A063639 (program): Primes of the form p*q*r - 1, where p, q and r are primes (not necessarily distinct).
  • A063640 (program): Primes of form p*q*r + 1, where p, q and r are primes.
  • A063641 (program): Primes of form p*q*r - 2, where p, q and r are primes (not necessarily distinct).
  • A063642 (program): Primes of form p*q*r + 2, where p, q and r are primes (not necessarily distinct).
  • A063647 (program): Number of ways to write 1/n as a difference of exactly 2 unit fractions.
  • A063648 (program): Smallest c such that 1/n=1/c+1/b has integer solutions with c>b.
  • A063649 (program): Largest b such that 1/n=1/c+1/b has integer solutions with c>b.
  • A063655 (program): Smallest semiperimeter of integral rectangle with area n.
  • A063656 (program): Numbers k such that the truncated square root of k is equal to the rounded square root of k.
  • A063657 (program): Numbers with property that truncated square root is unequal to rounded square root.
  • A063658 (program): The number of integers m in [1..n] for which gcd(m,n) is divisible by a square greater than 1.
  • A063659 (program): The number of integers m in [1..n] for which gcd(m,n) is not divisible by a square greater than 1.
  • A063672 (program): Sequence A019320 in binary.
  • A063683 (program): Integers formed from the reduced residue sets of even numbers and Fibonacci numbers.
  • A063691 (program): Number of solutions to x^2 + y^2 + z^2 = n in positive integers.
  • A063694 (program): Remove odd-positioned bits from the binary expansion of n.
  • A063695 (program): Remove even-positioned bits from the binary expansion of n.
  • A063709 (program): Remainder when n^n is divided by n!.
  • A063711 (program): Table of bits required for product of n- and k-bit nonnegative numbers read by antidiagonals.
  • A063712 (program): Table of bits required for product of n- and k-bit positive numbers read by antidiagonals.
  • A063717 (program): a(n) is the greatest divisor of n^2 that is less than n.
  • A063718 (program): a(n) is the smallest divisor of n^2 that is greater than n.
  • A063724 (program): Consider problem of placing N queens on an n X n board so that each queen attacks precisely 4 others. Sequence gives maximal number of queens.
  • A063725 (program): Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n.
  • A063726 (program): a(n) = gcd(1 + Fibonacci(n+1), 1 + Fibonacci(n)).
  • A063727 (program): a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
  • A063730 (program): Number of solutions to w^2 + x^2 + y^2 + z^2 = n in positive integers.
  • A063732 (program): Numbers whose Lucas representation excludes L_0 = 2.
  • A063734 (program): Square abundant numbers.
  • A063735 (program): Square deficient numbers.
  • A063743 (program): Numbers n such that n and Omega(n) are relatively prime, where Omega(n) is the number of prime divisors of n (with repetition).
  • A063745 (program): Even numbers with an even number of prime factors (counted with multiplicity).
  • A063749 (program): a(n) = floor((A000005(n)*(n+1)/2) - A000203(n)).
  • A063752 (program): Numbers k such that cototient(k) is a square.
  • A063754 (program): Dirichlet convolution of totient and cototient.
  • A063755 (program): Squares k which are divisible by phi(k).
  • A063757 (program): G.f.: (1+3*x+2*x^2)/((1-x)*(1-2*x^2)).
  • A063758 (program): a(0)=1, a(n) = 2*Fibonacci(n+4) - 6.
  • A063759 (program): Spherical growth series for modular group.
  • A063762 (program): Sqrt(n)-rough nonprimes: largest prime factor of n (A006530) >= sqrt(n).
  • A063763 (program): Composite integers k such that largest prime factor of k > sqrt(k).
  • A063765 (program): Least integer m whose largest prime factor > m^(n/(n+1)).
  • A063772 (program): a(k^2 + i) = k + a(i) for k >= 0 and 0 <= i <= k * 2; a(0) = 0.
  • A063774 (program): Number of divisors of n^2 is a square.
  • A063776 (program): Number of subsets of {1,2,…,n} which sum to 0 modulo n.
  • A063782 (program): a(0) = 1, a(1) = 3; for n > 1, a(n) = 2*a(n-1) + 4*a(n-2).
  • A063787 (program): a(2^k) = k + 1 and a(2^k + i) = 1 + a(i) for k >= 0 and 0 < i < 2^k.
  • A063806 (program): Numbers with a prime number of proper divisors.
  • A063808 (program): Spherical growth series for Z as generated by {2, 3}.
  • A063812 (program): Growth series for fundamental group of orientable closed surface of genus 2.
  • A063823 (program): G.f.: (1-2*x^2-3*x^3)/((1-x^3)*(1-2*x))
  • A063826 (program): Let 1, 2, 3, 4 represent moves to the right, down, left and up; this sequence describes the movements in the clockwise square spiral (a.k.a. Ulam Spiral).
  • A063842 (program): Number of colorings of K_4 using at most n colors.
  • A063845 (program): a(n) = sigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448) and sigma(n) is the sum of the divisors (A000203).
  • A063871 (program): Trajectory of 3 under map n->7n-1 if n odd, n->n/2 if n even.
  • A063872 (program): Let m be the n-th positive integer such that phi(m) is divisible by m - phi(m). Then a(n) = phi(m)/(m - phi(m)).
  • A063880 (program): Numbers n such that sigma(n) = 2*usigma(n).
  • A063886 (program): Number of n-step walks on a line starting from the origin but not returning to it.
  • A063887 (program): Number of n-step walks on a square lattice starting from the origin but not returning to it at any stage.
  • A063896 (program): a(n) = 2^Fibonacci(n) - 1.
  • A063905 (program): Each prime p appears p times.
  • A063908 (program): Numbers k such that k and 2*k-3 are primes.
  • A063909 (program): Primes p such that 2*p - 5 is also prime.
  • A063910 (program): Primes p such that 2*p - 7 is also prime.
  • A063911 (program): Primes p such that 2*p - 9 is also prime.
  • A063912 (program): Primes p such that 2*p - 11 is also prime.
  • A063913 (program): Primes p such that 2*p - 13 is also prime.
  • A063914 (program): Odd numbers interlaced with numbers 3m+2.
  • A063915 (program): G.f.: (1 + Sum_{ i >= 0 } 2^i*x^(2^(i+1)-1)) / (1-x)^2.
  • A063916 (program): G.f.: (1 + Sum_{ i >= 0 } 2^i*x^(2^(i+1)-1)) / (1-x)^3.
  • A063918 (program): a(1) = 1 and - applying the sieve of Eratosthenes - for n > 1: a(n) = if n is prime then 0 else the first prime p which marks n as composite.
  • A063919 (program): Sum of proper unitary divisors (or unitary aliquot parts) of n, including 1.
  • A063920 (program): Numbers k such that k = 2*phi(k) + phi(phi(k)).
  • A063928 (program): Largest nonprime proper divisor of n (with a(1)=1).
  • A063929 (program): Radius of A-excircle of Pythagorean triangle with a = (n+1)^2 - m^2, b = 2*(n+1)*m and c = (n+1)^2 + m^2.
  • A063930 (program): Radius of B-excircle of Pythagorean triangle with a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2.
  • A063932 (program): Average of largest prime less than or equal to n and smallest prime greater than or equal to n.
  • A063933 (program): Difference between n and the average of largest prime less than or equal to n and smallest prime greater than or equal to n.
  • A063934 (program): Numbers which are either prime or the average of consecutive odd primes.
  • A063941 (program): a(n) = 17*39^n.
  • A063942 (program): Follow k with k-1 and k-2.
  • A063945 (program): Number of nonnegative integers with n digits.
  • A063946 (program): Write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.
  • A063955 (program): Sum of unitary prime divisors (A056169, A056171) of n!.
  • A063956 (program): Sum of unitary prime divisors (A056169, A034444) of n.
  • A063957 (program): Numbers not of the form round(m*sqrt(2)) for any integer m, i.e., complement of A022846.
  • A063958 (program): Sum of the non-unitary prime factors of n: sum of those prime factors for which the exponent exceeds 1.
  • A063960 (program): Sum of non-unitary prime divisors of n!: sum of those prime factors for which the exponent exceeds 1.
  • A063962 (program): Number of distinct prime divisors of n that are <= sqrt(n).
  • A063966 (program): Number of Abelian groups of order <= n.
  • A063967 (program): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  • A063974 (program): Number of terms in inverse set of usigma = sum of unitary divisors = A034448.
  • A063977 (program): Numbers which are sums of unitary divisors, the usigma values: their inverse usigma set is not empty; usigma[]=A034448().
  • A063978 (program): Sum_{i for which n - i*(i-1)/2 >= 0} binomial (n - i*(i-1)/2, i).
  • A063985 (program): Partial sums of cototient sequence A051953.
  • A063989 (program): Numbers with a prime number of prime divisors (counted with multiplicity).
  • A063992 (program): Numbers that are not factorials.
  • A063994 (program): a(n) = Product_{primes p dividing n } gcd(p-1, n-1).
  • A064000 (program): Unitary untouchable numbers of second kind: numbers n such that usigma(x) = n has no solution, where usigma(x) (A034448) is the sum of unitary divisors of x.
  • A064008 (program): a(n) = (10^n - 1)/9*prime(n).
  • A064009 (program): a(n) = Sum_{k=1..n} d(k)*prime(k), where d(k) = A001223.
  • A064017 (program): Number of ternary trees (A001764) with n nodes and maximal diameter.
  • A064024 (program): a(n) = value of k such that absolute difference of 2^n and 3^k is minimized.
  • A064027 (program): a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.
  • A064028 (program): Sum of the unitary divisors of n!.
  • A064037 (program): Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant.
  • A064038 (program): Numerator of average number of swaps needed to bubble sort a string of n distinct letters.
  • A064040 (program): Number of distinct prime divisors of n is a prime.
  • A064041 (program): Number of divisors of A064040(n).
  • A064043 (program): Number of length 3 walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.
  • A064046 (program): Number of length 6 walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.
  • A064051 (program): a(n) = 2*prime(n)^2 - prime(n+1)^2.
  • A064052 (program): Not sqrt(n)-smooth: some prime factor of n is > sqrt(n).
  • A064053 (program): Auxiliary sequence gamma(n) used to compute coefficients in series expansion of the mock theta function f(q) via A(n) = Sum_{r=0..n} p(r)*gamma(n-r), with p(r) the partition function A000041.
  • A064054 (program): Tenth column of trinomial coefficients.
  • A064055 (program): Ninth column of quadrinomial coefficients.
  • A064056 (program): Seventh column of quintinomial coefficients.
  • A064057 (program): Eighth column of quintinomial coefficients.
  • A064058 (program): Ninth column of quintinomial coefficients.
  • A064059 (program): Seventh column of Catalan triangle A009766.
  • A064061 (program): Eighth column of Catalan triangle A009766.
  • A064062 (program): Generalized Catalan numbers C(2; n).
  • A064063 (program): Generalized Catalan numbers C(3; n).
  • A064069 (program): Generalized Euler number c(8,n).
  • A064070 (program): Generalized Euler number c(9,n).
  • A064073 (program): Generalized tangent number d(8,n).
  • A064074 (program): Generalized tangent number d(9,n).
  • A064076 (program): Lesser of odd twin prime powers (greater = A064077).
  • A064077 (program): Greater of odd twin prime powers (lesser = A064076).
  • A064078 (program): Zsigmondy numbers for a = 2, b = 1: Zs(n, 2, 1) is the greatest divisor of 2^n - 1 (A000225) that is coprime to 2^m - 1 for all positive integers m < n.
  • A064079 (program): Zsigmondy numbers for a = 3, b = 1: Zs(n, 3, 1) is the greatest divisor of 3^n - 1^n (A024023) that is relatively prime to 3^m - 1^m for all positive integers m < n.
  • A064080 (program): Zsigmondy numbers for a = 4, b = 1: Zs(n, 4, 1) is the greatest divisor of 4^n - 1^n (A024036) that is relatively prime to 4^m - 1^m for all positive integers m < n.
  • A064081 (program): Zsigmondy numbers for a = 5, b = 1: Zs(n, 5, 1) is the greatest divisor of 5^n - 1^n (A024049) that is relatively prime to 5^m - 1^m for all positive integers m < n.
  • A064082 (program): Zsigmondy numbers for a = 6, b = 1: Zs(n, 6, 1) is the greatest divisor of 6^n - 1^n (A024062) that is relatively prime to 6^m - 1^m for all positive integers m < n.
  • A064083 (program): Zsigmondy numbers for a = 7, b = 1: Zs(n, 7, 1) is the greatest divisor of 7^n - 1^n (A024075) that is relatively prime to 7^m - 1^m for all positive integers m < n.
  • A064087 (program): Generalized Catalan numbers C(4; n).
  • A064088 (program): Generalized Catalan numbers C(5; n).
  • A064089 (program): Generalized Catalan numbers C(6; n).
  • A064090 (program): Generalized Catalan numbers C(7; n).
  • A064091 (program): Generalized Catalan numbers C(8; n).
  • A064092 (program): Generalized Catalan numbers C(9; n).
  • A064093 (program): Generalized Catalan numbers C(10; n).
  • A064096 (program): Fifth diagonal of triangle A064094.
  • A064097 (program): A quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.
  • A064098 (program): a(n+1) = (a(n)^2 + a(n-1)^2)/a(n-2), with a(1) = a(2) = a(3) = 1.
  • A064099 (program): a(n) = ceiling(log(3 + 2*n)/log(3)).
  • A064100 (program): a(n) = (100^n - 1)/99*n.
  • A064105 (program): 2nd column of 3rd-order Zeckendorf array.
  • A064106 (program): 3rd column of 3rd-order Zeckendorf array.
  • A064108 (program): a(n) = (20^n-1)/19.
  • A064138 (program): Sum of non-unitary divisors of n!.
  • A064139 (program): Sum of divisors of central binomial coefficient C(n, floor(n/2)).
  • A064140 (program): Sum of unitary divisors of central binomial coefficient C(n, floor(n/2)).
  • A064142 (program): Sum of all distinct primes dividing central binomial coefficient C(n, floor(n/2)).
  • A064143 (program): Sum of unitary prime divisors (A056169, A034444) of central binomial coefficient C(n, floor(n/2)).
  • A064146 (program): Sum of non-unitary prime divisors (A034444, A056169) of central binomial coefficient C(n,floor(n/2)) (A001405). If A001405(n) is squarefree (A046098) then a(n)=0.
  • A064148 (program): Numbers k such that mu(k) = mu(k+1), where mu is the Möbius function (A008683).
  • A064150 (program): Numbers divisible by the sum of their ternary digits.
  • A064161 (program): Least abundant number divisible by the n-th prime number.
  • A064162 (program): Least abundant number divisible by n.
  • A064167 (program): Product of numerator and denominator of the n-th harmonic number, 1 + 1/2 + 1/3 +…+ 1/n.
  • A064168 (program): Sum of numerator and denominator in n-th harmonic number, 1 + 1/2 + 1/3 +…+ 1/n.
  • A064169 (program): Numerator - denominator in n-th harmonic number, 1 + 1/2 + 1/3 + … + 1/n.
  • A064170 (program): a(1) = 1; a(n+1) = product of numerator and denominator in Sum_{k=1..n} 1/a(k).
  • A064179 (program): Infinitary version of Moebius function: infinitary MoebiusMu of n, equal to mu(n) iff mu(n) differs from zero, else 1 or -1 depending on whether the sum of the binary digits of the exponents in the prime decomposition of n is even or odd.
  • A064183 (program): Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = q(n) and A064526(n) = p(n).
  • A064184 (program): Denominator of sequence defined by recursion c(n)=1+c(n-2)/c(n-1), c(0)=0, c(1)=1.
  • A064188 (program): Sum_{ i = 0 .. floor(n/2)} binomial (n - i*(i-1)/2, i).
  • A064189 (program): Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if n < k, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1).
  • A064191 (program): Triangle T(n,k) (n >= 0, 0 <= k <= n) generalizing Motzkin numbers.
  • A064194 (program): a(2n) = 3*a(n), a(2n+1) = 2*a(n+1)+a(n), with a(1) = 1.
  • A064197 (program): a(n) = 27*(n-1)*(n-2)*(n-3)*(3*n-8)/2.
  • A064198 (program): a(n) = 3*(n-2)*(n-3)*(3*n^2-3*n-8)/2.
  • A064199 (program): a(n) = 9*(n-2)^2*(n^2-2*n-1)/2.
  • A064200 (program): a(n) = 12*n*(n-1).
  • A064201 (program): 9 times octagonal numbers: a(n) = 9n(3n-2).
  • A064202 (program): a(n) = n*(n+1)*(n+2)*(2*n^3 + 6*n^2 + 7*n - 3)/36.
  • A064203 (program): n*(n^2 - 1)*(n+2)*(2*n^5 + 14*n^4 + 49*n^3 + 91*n^2 + 90*n + 18)/324.
  • A064212 (program): a(n) = sigma(n) + usigma(n), or A000203(n) + A034448(n).
  • A064216 (program): Replace each p^e with prevprime(p)^e in the prime factorization of odd numbers; inverse of sequence A048673 considered as a permutation of the natural numbers.
  • A064222 (program): a(0) = 0; a(n) = DecimalDigitsSortedDecreasing(a(n-1) + 1) for n > 0.
  • A064223 (program): a(1) = 1; a(n+1) = a(n) + number of decimal digits of a(n) for n > 0.
  • A064225 (program): (9*n^2+5*n+2)/2.
  • A064226 (program): a(n) = (9*n^2 + 13*n + 6) / 2.
  • A064232 (program): a(n) = n^(n+2) mod (n+1)^(n+1).
  • A064235 (program): The smallest power of 3 that is greater than or equal to n.
  • A064263 (program): a(n) = 11*n mod 30.
  • A064264 (program): a(n) = 19*n mod 30.
  • A064265 (program): Paschal regular in Julian calendar for a year with Golden Number n.
  • A064266 (program): Lune of Jan 01 in Julian calendar for a year with Golden Number n.
  • A064267 (program): Clavis terminorum in Julian calendar for a year with Golden Number n.
  • A064268 (program): a(n) = (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7). a(1) = … = a(7) = 1. Somos-7 variation.
  • A064270 (program): Primes of the form prime(k) - k; or primes arising in A014689.
  • A064272 (program): Number of representations of n as the sum of a prime number and a nonzero square.
  • A064273 (program): Permutation of nonnegative integers: a(n) = A013928(A019565(n)).
  • A064276 (program): Number of 2 X 2 singular integer matrices with elements from {0,…,n} up to row and column permutation.
  • A064279 (program): Number of ordered pairs a,b of elements in the cyclic group C_n such that the subgroup generated by the pair a,b is a proper subgroup of C_n.
  • A064282 (program): Triangle read by rows: T(n,k) = binomial(3n+3, k)*(n-k+1)/(n+1).
  • A064299 (program): a(n) = B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108).
  • A064302 (program): Sixth diagonal of triangle A064094.
  • A064303 (program): Seventh diagonal of triangle A064094.
  • A064304 (program): Eighth diagonal of triangle A064094.
  • A064305 (program): Ninth diagonal of triangle A064094.
  • A064306 (program): Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.
  • A064310 (program): Generalized Catalan numbers C(-1; n).
  • A064311 (program): Generalized Catalan numbers C(-2; n).
  • A064312 (program): a(n) = B(n)*P(n), where B(n) are Bell numbers (A000110) and P(n) are numbers of arrangements of a set of n elements (A000522).
  • A064318 (program): a(n) satisfies a(n)! <= n^n < (a(n)+1)!.
  • A064321 (program): n*(n-1)^3*(n-2)^3*(n-3).
  • A064322 (program): Triply triangular numbers.
  • A064323 (program): a(n) = a(n-1)+ceiling(a(n-2)/2) with a(0)=0, a(1)=1.
  • A064324 (program): a(n) = a(n-1) + floor(a(n-2)/2) with a(0)=1, a(1)=2.
  • A064325 (program): Generalized Catalan numbers C(-3; n).
  • A064326 (program): Generalized Catalan numbers C(-4; n).
  • A064327 (program): Generalized Catalan numbers C(-5; n).
  • A064328 (program): Generalized Catalan numbers C(-6; n).
  • A064329 (program): Generalized Catalan numbers C(-7; n).
  • A064330 (program): Generalized Catalan numbers C(-8; n).
  • A064331 (program): Generalized Catalan numbers C(-9; n).
  • A064332 (program): Generalized Catalan numbers C(-10; n).
  • A064333 (program): Generalized Catalan numbers C(-11; n).
  • A064335 (program): a(n) = 6*(2*n)!/(n+2).
  • A064340 (program): Generalized Catalan numbers C(2,2; n).
  • A064350 (program): a(n) = (3*n)!/n!.
  • A064352 (program): a(n) = (3*n)!/(2*n)!.
  • A064353 (program): Kolakoski-(1,3) sequence: the alphabet is {1,3}, and a(n) is the length of the n-th run.
  • A064355 (program): Number of subsets of {1,2,..n} that sum to 1 mod n.
  • A064359 (program): Inverse of sequence A052331 considered as a permutation of the natural numbers.
  • A064363 (program): Number of 2 X 2 regular integer matrices with elements from {0,…,n} up to row and column permutation.
  • A064366 (program): Special binomial coefficient: a(n) = C(sigma(n), phi(n)).
  • A064367 (program): a(n) = 2^n mod prime(n), or 2^n = k*prime(n) + a(n) with integer k.
  • A064368 (program): Number of 2 X 2 symmetric singular matrices with entries from {0,…,n}.
  • A064378 (program): a(0) = 2, a(n) = 2^(n+1)*(n-1)! (n >= 1).
  • A064382 (program): Number of ways to put numbers 1, 2, …, n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing or decreasing.
  • A064385 (program): a(n) = 2*5^n - 3.
  • A064400 (program): Number of ordered pairs a,b of elements in the dihedral group D_2n such that the subgroup generated by the pair a,b is the entire group D_2n.
  • A064405 (program): Number of even entries (A048967) minus the number of odd entries (A001316) in row n of Pascal’s triangle (A007318).
  • A064406 (program): The accumulation of the number of even entries (A048967) over the number of odd entries (A001316) in row n of Pascal’s triangle (A007318).
  • A064407 (program): Even numbers not the sum of a pair of the lesser of the twin primes.
  • A064409 (program): Even numbers not the sum of a pair of twin primes, one the lesser and the other the greater.
  • A064412 (program): At stage 1, start with a unit equilateral equiangular triangle. At each successive stage add 3*(n-1) new triangles around outside with edge-to-edge contacts. Sequence gives number of triangles (regardless of size) at n-th stage.
  • A064415 (program): a(1) = 0, a(n) = iter(n) if n is even, a(n) = iter(n)-1 if n is odd, where iter(n) = A003434(n) = smallest number of iterations of Euler totient function phi needed to reach 1.
  • A064427 (program): (Number of primes <= n - 1) + n.
  • A064429 (program): a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).
  • A064433 (program): Number of iterations of A064455 to reach 2 (or 1 in the case of 1).
  • A064434 (program): a(n) = (2*a(n-1) + 1) mod n.
  • A064437 (program): a(1)=1, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
  • A064438 (program): Numbers which are divisible by the sum of their quaternary digits.
  • A064446 (program): a(n) = gcd(n!, n^n, lcm(1, 2, …, n)), or gcd(n^n, lcm(1, 2, …, n)).
  • A064447 (program): a(n) = EulerPhi(n^n).
  • A064448 (program): a(n) = gcd(n^n, EulerPhi(n^n)).
  • A064455 (program): a(2n) = 3n, a(2n-1) = n.
  • A064458 (program): Highest power of 11 dividing n!.
  • A064459 (program): a(n) = Sum_{k>=1} floor(n/12^k).
  • A064460 (program): Number of distinct nonsquarefree entries in n-th row of Pascal’s triangle.
  • A064464 (program): Binary order (cf. A029837) of the number of parts if 3^n is partitioned into parts of size 2^n as far as possible and into parts of size 1^n (cf. A060692).
  • A064478 (program): If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(0) = 1, a(1)=2.
  • A064481 (program): Divisible by the sum of the digits of its base-5 representation.
  • A064485 (program): Number of ordered pairs a,b of elements in the dihedral group D_2n such that the subgroup generated by the pair a,b is a proper subgroup of D_2n.
  • A064487 (program): Order of twisted Suzuki group Sz(2^(2*n + 1)), also known as the group 2B2(2^(2*n + 1)).
  • A064488 (program): A Beatty sequence: Floor[n*c], where c = A064648 is the sum of the reciprocals of primorials.
  • A064491 (program): a(1) = 1, a(n+1) = a(n) + tau(a(n)), where tau(n) (A000005) is the number of divisors of n.
  • A064494 (program): Shotgun (or Schrotschuss) numbers: limit of the recursion B(k) =Tk, where B(1) = (1,2,3,4,5,…) and T[k] is the Transformation that permutes the entries k(2i-1) and k(2i) for all positive integers i.
  • A064497 (program): a(n) = prime(n) * Fibonacci(n).
  • A064506 (program): a(n) = Max { k | k*(k+1)/2 <= n*(n+1)/2 - k*(k+1)/2 }.
  • A064520 (program): a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - … + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).
  • A064524 (program): Number of noncubes <= n.
  • A064525 (program): Smallest Fibonacci number with a prime number of decimal digits.
  • A064526 (program): Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n).
  • A064529 (program): Number of connected components remaining when n-th letter of English alphabet is cut from a piece of paper.
  • A064530 (program): Number of holes in n-th capital letter of English alphabet.
  • A064534 (program): If p >= 11 is prime, n is a power of one of the primes in this sequence but n is not a power of p, then the equation x^p + y^p = n*z^p has no solution in integers x,y,z.
  • A064535 (program): a(n) = (2^prime(n)-2)/prime(n); a(0) = 0 by convention.
  • A064536 (program): a(n) = (4^n mod 3^n) mod 2^n.
  • A064542 (program): a(n) = Max { k | k! <= n! / k! } where m! = A000142(m), factorial.
  • A064546 (program): Remainder when (n!)^2 is divided by n^n.
  • A064547 (program): Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n.
  • A064548 (program): Numbers n for which the sum of the binary digits (or count of 1-bits) equals the sum of the prime exponents of n+1 (or the factor-count of n+1).
  • A064549 (program): a(n) = n * Product_{primes p|n} p.
  • A064551 (program): Ado [Simone Caramel]’s Fibonacci function: define the Fibonacci sequence by f(0) = 1, f(1) = 1, f(n) = f(n-1)+f(n-2); then a(0) = 1, a(n) = a(n-1) + 2*(f(n)-n), n > 0.
  • A064553 (program): a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.
  • A064557 (program): a(n) = # { p | A064553(k) = p prime and k <= n}.
  • A064558 (program): a(n) = A064553(A064553(n)).
  • A064559 (program): Number of iterations in A064553 to reach a fixed point.
  • A064560 (program): Numbers n such that reciprocal of n terminates with an infinite repetition of digit 1. Multiples of 10 are omitted.
  • A064562 (program): Reciprocal of n terminates with an infinite repetition of digit 3. Multiples of 10 are omitted.
  • A064570 (program): Binomial transform of (2n)!.
  • A064571 (program): Binomial transform of (3n)!.
  • A064578 (program): Inverse permutation to A057027.
  • A064583 (program): a(n) = n^4*(n^4+1)*(n^2-1).
  • A064587 (program): a(n) = n^6*(n^4 + n^2 + 1)*(n^3 - 1)*(n - 1).
  • A064594 (program): Nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a multiple of n; i.e., n divides sigma(n) - usigma(n).
  • A064597 (program): Nonunitary abundant numbers: the sum of the nonunitary divisors of n is larger than n; i.e., sigma(n) - usigma(n) > n.
  • A064601 (program): a(n) = # { p | A064558(k) = p prime and k <= n}.
  • A064602 (program): Partial sums of A001157: Sum_{j=1..n} sigma_2(j).
  • A064603 (program): Partial sums of A001158: Sum_{j=1..n} sigma_3(j).
  • A064604 (program): Partial sums of A001159: Sum_{j=1..n} sigma_4(j).
  • A064608 (program): Partial sums of A034444: sum of number of unitary divisors from 1 to n.
  • A064609 (program): Partial sums of A034448: sum of unitary divisors from 1 to n.
  • A064613 (program): Second binomial transform of the Catalan numbers.
  • A064614 (program): Exchange 2 and 3 in the prime factorization of n.
  • A064615 (program): Numbers of the form m * 6^k for k >= 0 and m > 0 with gcd(m, 6) = 1.
  • A064616 (program): (10^n-1)*(91/81)-n*10^n/9.
  • A064617 (program): a(n) = (10^n-1)*(80/81)+n/9.
  • A064618 (program): Stirling transform of (n!)^2.
  • A064628 (program): Floor(4^n / 3^n).
  • A064629 (program): a(n) = 4^n mod 3^n.
  • A064630 (program): Number of parts if 4^n is partitioned into parts of size 3^n as far as possible into parts of size 2^n as far as possible and into parts of size 1^n.
  • A064631 (program): 4^n = x*3^n+y*2^n+z*1^n, so 4^n equals the sum of a(n)=x+y+z pieces of like powers (=length of right side of solution of this Diophantine equation). Length of solutions obtained with “greedy algorithm” are given in A064630[n]. Here the binary order [A029837] of the length of those solutions is displayed, which “on the average” nearly equals n.
  • A064633 (program): a(n) = 3^n*n!*(n+2)!/2!.
  • A064635 (program): Even numbers not appearing in A064466. a(n) = A064466(A064634(n)) + 2 for n > 0.
  • A064641 (program): Unidirectional ‘Delannoy’ variation of the Boustrophedon transform applied to all 1’s sequence: construct an array in which the first element of each row is 1 and subsequent entries are given by T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k) + T(n-2,k-1). The last number in row n gives a(n).
  • A064646 (program): Numerators of partial sums of reciprocals of primorial numbers.
  • A064647 (program): Denominators of partial sums of reciprocals of primorial numbers.
  • A064650 (program): a(n) = floor(a(n-1)/2) + a(n-2) with a(0)=1, a(1)=2.
  • A064651 (program): a(n) = ceiling(a(n-1)/2) + a(n-2) with a(0)=0 and a(1)=1.
  • A064671 (program): Number of n-digit base 4 biquanimous numbers (with leading 0’s allowed, but not all-0 string).
  • A064677 (program): a(n) = sigma(n) - D(n) - pi(n), where D(n)=A001223, pi(n)= A000720.
  • A064680 (program): Halve every even number, double every odd number.
  • A064685 (program): Length of orbit of 2n+1 in the 3x+1 problem.
  • A064686 (program): a(n) = number of n-digit base-3 biquams.
  • A064694 (program): Add column entries of the table with rows (1,2,0,0…), (0,3,4,5,0,0…), (0,0,6,7,8,9,0,0…), (0,0,0,10,11,12,13,14,0,0…), …
  • A064700 (program): Numbers k that are divisible by the multiplicative digital root of k.
  • A064706 (program): Square of permutation defined by A003188.
  • A064707 (program): Inverse square of permutation defined by A003188.
  • A064717 (program): A Beatty sequence for 2^i + 2^(-i) where i = sqrt(-1).
  • A064718 (program): A Beatty sequence for 2^i + 2^-i where i = sqrt(-1).
  • A064719 (program): A Beatty sequence for 3^i + 3^-i + 1.
  • A064722 (program): a(1) = 0; for n >= 2, a(n) = n - (largest prime <= n).
  • A064723 (program): (L(p)-1)/p where L() are the Lucas numbers (A000032) and p runs through the primes.
  • A064724 (program): A Beatty sequence for 2^sqrt(2).
  • A064726 (program): Sum of primes dividing the partitions of n into distinct parts (with repetition).
  • A064727 (program): Number of pairs x,y such that 0 < x <= y < n and x+y = n and x*y = kn for some k.
  • A064733 (program): Final digits of A005165(2n) for large n, read from right.
  • A064734 (program): Final digits of A005165(2n+1) for large n, read from right.
  • A064739 (program): Primes p such that Fibonacci(p)-1 is divisible by p.
  • A064746 (program): a(n) = n*8^n + 1.
  • A064747 (program): a(n) = n*9^n + 1.
  • A064748 (program): a(n) = n*10^n + 1.
  • A064749 (program): a(n) = n*11^n + 1.
  • A064750 (program): a(n) = n*12^n + 1.
  • A064751 (program): a(n) = n*5^n - 1.
  • A064752 (program): a(n) = n*6^n - 1.
  • A064753 (program): a(n) = n*7^n - 1.
  • A064754 (program): a(n) = n*8^n - 1.
  • A064755 (program): a(n) = n*9^n - 1.
  • A064756 (program): a(n) = n*10^n - 1.
  • A064757 (program): a(n) = n*11^n - 1.
  • A064758 (program): a(n) = n*12^n - 1.
  • A064760 (program): Variant of A002034 with initial term 0.
  • A064761 (program): a(n) = 15*n^2.
  • A064762 (program): a(n) = 21*n^2.
  • A064763 (program): a(n) = 28*n^2.
  • A064766 (program): Fill a triangular array by rows by writing numbers 1, then 1 up to 2*3/2, then 1 up to 3*4/2, then 1 up to 4*5/2 and so on from 1 up to the n-th triangular number. The final elements of the rows form the sequence.
  • A064767 (program): Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).
  • A064770 (program): Replace each digit of n with the floor of its square root.
  • A064775 (program): Card{ k<=n, k such that all prime divisors of k are <= sqrt(k) }.
  • A064778 (program): Largest m such that 1..m all divide n!.
  • A064780 (program): Number of times n occurs in A000195.
  • A064784 (program): Difference between n-th triangular number t(n) and the largest square <= t(n).
  • A064786 (program): Inverse permutation to A054084.
  • A064788 (program): Inverse permutation to A060736.
  • A064789 (program): Inverse permutation to A057028.
  • A064796 (program): Largest integer m such that every permutation (p_1, …, p_n) of (1, …, n) satisfies p_i * p_{i+1} >= m for some i, 1 <= i <= n, where p_{n+1} = p_1.
  • A064799 (program): Sum of n-th prime number and n-th composite number.
  • A064800 (program): n plus the number of its prime factors: a(n) = n + A001222(n).
  • A064801 (program): Take 1, skip 2, take 2, skip 3, take 3, etc.
  • A064806 (program): a(n) = n + digital root of n.
  • A064808 (program): a(n) is the (n+1)st (n+2)-gonal number.
  • A064811 (program): a(n) = Sum_{k=1..n} binomial(prime(n),k).
  • A064813 (program): a(n) = binomial(composite(n), n), where composite = A002808, composite numbers.
  • A064814 (program): Greatest common divisor of n and the n-th composite number.
  • A064815 (program): Related to enumeration of finite automata.
  • A064816 (program): Numbers which are the sums of two positive triangular numbers (A000217) in exactly two different ways.
  • A064819 (program): a(n) = p(1)*p(2)*…*p(n) - p(n+1)^2, where p(i) = i-th prime.
  • A064824 (program): Same as A065191 but with B_[1]=( i mod 10, i=0,1,2,3..).
  • A064830 (program): a(n) = gcd(n, prime(n)^2 - 1).
  • A064831 (program): Partial sums of A001654, or sum of the areas of the first n Fibonacci rectangles.
  • A064835 (program): If n mod 2 = 0 then a(n) = n^4/4 - 2*n^2 + 3*n; otherwise, a(n) = n^4/4 - 2*n^2 + 3*n - 5/4.
  • A064836 (program): a(n) = A064835(n)/2.
  • A064840 (program): a(n) = tau(n)*sigma(n).
  • A064842 (program): Maximal value of Sum_{i=1..n} (p(i) - p(i+1))^2, where p(n+1) = p(1), as p ranges over all permutations of {1, 2, …, n}.
  • A064843 (program): A064842/2.
  • A064847 (program): Sequence a(n) such that there is a sequence b(n) with a(1) = b(1) = 1, a(n+1) = a(n) * b(n) and b(n+1) = a(n) + b(n) for n >= 1.
  • A064853 (program): Lemniscate constant.
  • A064854 (program): a(n) = ((5^n mod 4^n) mod 3^n) mod 2^n.
  • A064855 (program): a(n) = (((6^n mod 5^n) mod 4^n) mod 3^n) mod 2^n.
  • A064857 (program): Numerators of partial sums of reciprocals of lcm(1..n) = A003418(n).
  • A064858 (program): Denominators of partial sums of reciprocals of lcm(1..n) = A003418(n).
  • A064861 (program): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  • A064865 (program): Fill a triangular array by rows by writing numbers 1, then 1 up to 2^2, then 1 up to 3^2, then 1 up to 4^2 and so on. The final elements of the rows form the sequence.
  • A064866 (program): Write numbers 1, then 1 up to 2^2, then 1 up to 3^2, then 1 up to 4^2 and so on.
  • A064873 (program): First of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = a(n)^2 + A064874(n)^2 + A064875(n)^2 + A064876(n)^2.
  • A064881 (program): Eisenstein array Ei(1,2).
  • A064882 (program): Eisenstein array Ei(2,1).
  • A064883 (program): Eisenstein array Ei(1,3).
  • A064884 (program): Eisenstein array Ei(3,1).
  • A064885 (program): Eisenstein array Ei(3,2).
  • A064886 (program): Eisenstein array Ei(2,3).
  • A064894 (program): Binary dilution of n. GCD of exponents in binary expansion of n.
  • A064899 (program): Numbers that are of the form pq where p and q are distinct primes that are the orders of non-Abelian groups.
  • A064911 (program): If n is semiprime (or 2-almost prime) then 1 else 0.
  • A064916 (program): a(n) = n/lpf(n) + lpf(n) - 1, where lpf = A020639 = least prime factor.
  • A064917 (program): a(n) is the result of beginning with n and iterating k -> A064916(k) until a prime is reached.
  • A064918 (program): a(n) is the number of iterations of k -> A064916(k) to reach a prime, starting at n.
  • A064919 (program): a(n) = Min { k | A064916(k) = n }.
  • A064920 (program): a(n) = n/gpf(n) + gpf(n) - 1, where gpf = A006530 = greatest prime factor.
  • A064921 (program): Iterate A064920 until a prime is reached.
  • A064922 (program): Number of iterations in A064920 to reach a prime.
  • A064924 (program): If n is prime then a(n) = n; for the subsequent nonprime positions a(n + k) = (k+1)*n; then at the next prime position a new subsequence begins.
  • A064939 (program): a(n) = Sum_{i=1..omega(n)} i*p_i, where {p_i}, i=1..omega(n) is the increasing sequence of prime divisors of n, where omega is the number of distinct prime factors of n (A001221).
  • A064944 (program): a(n) = Sum_{i|n, j|n, j >= i} j.
  • A064945 (program): a(n) = Sum_{i|n, j|n, j >= i} i.
  • A064946 (program): a(n) = Sum_{i|n, j|n, j>i} j.
  • A064947 (program): a(n) = Sum_{i|n, j|n, j>i} i.
  • A064948 (program): a(n) = Sum_{i|n, j|n} max(i,j).
  • A064949 (program): a(n) = Sum_{i|n, j|n} min(i,j).
  • A064950 (program): a(n) = Sum_{i|n, j|n} lcm(i,j).
  • A064951 (program): Sum of lcm(x, y) for 1 <= x, y <= n.
  • A064969 (program): Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).
  • A064971 (program): a(n) = n*usigma(n), where usigma(n) is the sum of unitary divisors of n (A034448).
  • A064987 (program): a(n) = n*sigma(n).
  • A064988 (program): Multiplicative with a(p^e) = prime(p)^e.
  • A064989 (program): Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.
  • A064990 (program): If A_k denotes the first 3^k terms, then A_0 = 0, A_{k+1} = A_k A_k B_k, where B_k is obtained from A_k by interchanging 0’s and 1’s.
  • A064992 (program): a(n) = usigma(n+1) - usigma(n), where usigma(n) is the sum of unitary divisors of n (A034448).
  • A064994 (program): A Beatty sequence from Khintchine’s constant (A002210).
  • A064995 (program): A Beatty sequence from Khintchine’s constant (A002210).
  • A064996 (program): A Beatty sequence: [Pi^2 -8].
  • A064997 (program): A Beatty sequence: [Pi^2 -8].
  • A064999 (program): Partial sums of sequence (essentially A002378): 1, 2, 6, 12, 20, 30, 42, 56, 72, 90, …
  • A065001 (program): a(n) = (presumed) number of palindromes in the ‘Reverse and Add!’ trajectory of n, or -1 if this number is not finite.
  • A065005 (program): Integers for which the periodic part of the continued fraction for the square root of n begins with 2.
  • A065018 (program): a(n) = Sum_{d|n} sigma(d)^2.
  • A065031 (program): In the decimal expansion of n, replace each odd digit with 1 and each even digit with 2.
  • A065033 (program): 1 appears three times, other numbers twice.
  • A065034 (program): a(n) = Lucas(2*n) + 1.
  • A065035 (program): a(n+1) = a(n)^2 + 3*a(n) + 1.
  • A065036 (program): Product of the cube of a prime (A030078) and a different prime.
  • A065037 (program): Inverse permutation to A036552.
  • A065039 (program): If n in base 10 is d_1 d_2 … d_k then a(n) = d_1 + d_1d_2 + d_1d_2d_3 + … + d_1…d_k.
  • A065040 (program): Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k).
  • A065043 (program): Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.
  • A065065 (program): Number of noncrossing connected graphs with nodes on a circle having n edges.
  • A065073 (program): n-th prime + sum of digits of n-th prime.
  • A065075 (program): Sum of digits of the sum of the preceding numbers.
  • A065076 (program): a(0) = 0, a(1) = 1, a(n) = (sum of digits of a(n-1)) + a(n-2).
  • A065081 (program): Alternating bit sum (A065359) for n-th prime p: replace 2^k with (-1)^k in binary expansion of p.
  • A065087 (program): a(n) = A000166(n)*binomial(n+1,2).
  • A065088 (program): a(n) = A000166(n)*binomial(n,2).
  • A065090 (program): Natural numbers which are not odd primes: composites plus 1 and 2.
  • A065091 (program): Odd primes.
  • A065093 (program): Convolution of A000010 with itself.
  • A065094 (program): a(1) = 1, a(n+1) is the sum of a(n) and floor( arithmetic mean of a(1) … a(n) ).
  • A065095 (program): a(1) = 1, a(n+1) is the sum of a(n) and ceiling( arithmetic mean of a(1) … a(n) ).
  • A065096 (program): Sums of lists produced by a variant of the iteration that produces the Catalan numbers: start with 0 and at each iteration replace each integer k with the list 0,1,…,k-1,k,k+1,k,k-1,…,1,0 and let a(n) be the sum of the resulting (flattened) list after n iterations.
  • A065097 (program): a(n) = ((2n+1) + (2n-1) - 1)!/((2n+1)!*(2n-1)!).
  • A065100 (program): a(n+2) = 9*a(n+1) - a(n), a(0) = 3, a(1) = 27.
  • A065101 (program): a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 3, c = 2.
  • A065102 (program): a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 2, c = 3.
  • A065109 (program): Triangle T(n,k) of coefficients relating to Bezier curve continuity.
  • A065113 (program): Sum of the squares of the a(n)-th and the (a(n)+1)st triangular numbers (A000217) is a perfect square.
  • A065118 (program): Numbers which are 19 times the sum of their digits.
  • A065119 (program): n-th cyclotomic polynomial is a trinomial.
  • A065120 (program): Highest power of 2 dividing A057335(n).
  • A065124 (program): a(n) = (sum of digits of a(n-2)) + a(n-1); a(0) = 0 and a(1) = 1.
  • A065128 (program): Number of invertible n X n matrices mod 4 (i.e., over the ring Z_4).
  • A065130 (program): a(n) = A005228(n) - A000217(n).
  • A065133 (program): Remainder when n-th prime is divided by the number of primes not exceeding n.
  • A065134 (program): Remainder when n is divided by the number of primes not exceeding n.
  • A065137 (program): Sum of digits of n plus sum of cubes of digits of n.
  • A065140 (program): a(n) = 2^n*(2*n)!.
  • A065141 (program): a(n) = (n+1)*2^n*(2*n)!.
  • A065142 (program): a(n) = 2^n*(n+1)*(3*n)!.
  • A065143 (program): a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.
  • A065151 (program): a(n) = prime(1 + A064722(n)).
  • A065152 (program): Cototient(totient(n)) - totient(cototient(n)).
  • A065153 (program): Numbers for which the cototient of the totient is equal to the totient of the cototient.
  • A065154 (program): Numbers for which the cototient of the totient is strictly less than the totient of the cototient.
  • A065155 (program): Numbers whose cototient of totient is strictly greater than totient of cototient.
  • A065159 (program): Binary string self-substitutions: a(n) is obtained by substituting the binary expansion of n for each 1-bit in the binary expansion of n.
  • A065160 (program): Reduced binary string self-substitutions: a(n) is obtained by substituting n for each 1-bit in the binary expansion of n, then dividing by n.
  • A065164 (program): Permutation t->t+1 of Z, folded to N.
  • A065165 (program): Permutation t->t+2 of Z, folded to N.
  • A065166 (program): Permutation t->t+3 of Z, folded to N.
  • A065168 (program): Permutation t->t-1 of Z, folded to N.
  • A065169 (program): Permutation t->t-2 of Z, folded to N.
  • A065170 (program): Permutation t->t-3 of Z, folded to N.
  • A065171 (program): Permutation of Z, folded to N, corresponding to the site swap pattern …26120123456… which ascends infinitely after t=0.
  • A065172 (program): Inverse permutation to A065171.
  • A065173 (program): Site swap sequence that rises infinitely after t=0. The associated delta sequence p(t)-t for the permutation of Z: A065171.
  • A065176 (program): Site swap sequence associated with the permutation A065174 of Z.
  • A065179 (program): Number of site swap patterns with 3 balls and exact period n.
  • A065186 (program): a(1)=1, a(2)=3, a(3)=5, a(4)=2, a(5)=4; for n > 5, a(n) = a(n-5) + 5.
  • A065187 (program): “Greedy Dragons” permutation of the natural numbers, inverse of A065186.
  • A065190 (program): Self-inverse permutation of the positive integers: 1 is fixed, followed by an infinite number of adjacent transpositions (n n+1).
  • A065191 (program): Limit of the recursion B_[k] = Tk, where B_[1] = (1,2,3,4,5,…) and T[k] is the transformation that permutes the entries k(2i-1) + j and k(2i) + j for all j = 0,..,k-1 and positive integers i.
  • A065200 (program): Numbers of the form n = m * p^k, with p prime, k >= 0, m squarefree and p > any prime factor of m.
  • A065201 (program): Numbers having a non-maximal prime-factor with exponent greater than 1.
  • A065202 (program): Characteristic function of A065201: a(n) = if A065201(k) = n for some k then 1 else 0.
  • A065220 (program): a(n) = Fibonacci(n) - n.
  • A065221 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the pentagonal numbers. The final elements of the rows form a(n).
  • A065222 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The final elements of the rows form a(n).
  • A065223 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the heptagonal numbers (A000566). The final elements of the rows form a(n).
  • A065224 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the octagonal numbers. The final elements of the rows form a(n).
  • A065225 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the 9-gonal (nonagonal) numbers. The final elements of the rows form a(n).
  • A065226 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the decagonal numbers. The final elements of the rows form a(n).
  • A065227 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the triangular numbers. The first elements of the rows form a(n).
  • A065228 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the square numbers. The first elements of the rows form a(n).
  • A065229 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the pentagonal numbers. The first elements of the rows form a(n).
  • A065230 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The first elements of the rows form a(n).
  • A065231 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the heptagonal numbers (A000566). The first elements of the rows form a(n).
  • A065232 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the octagonal numbers. The first elements of the rows form a(n).
  • A065233 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where the b(n) are the nonzero 9-gonal (nonagonal) numbers 1, 9, 24, 46, … (A001106). The initial elements of the rows form a(n).
  • A065234 (program): Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the decagonal numbers. The first elements of the rows form a(n).
  • A065236 (program): a(n) = (4*n)!*(n+1)!/(2*n)!.
  • A065244 (program): Primes of form p = 2 + Sum_{k = 1..m} k^2 for some m.
  • A065245 (program): Positive numbers k such that 2 + Sum_{j = 1..k} j^2 is a prime.
  • A065251 (program): Simple quasi-periodic sequence consisting of the terms 1, 0 and -1.
  • A065252 (program): The sequence A065251 reduced modulo 3 (i.e., replace every -1 with 2).
  • A065256 (program): Quintal Queens permutation of N: halve or multiply by 3 (mod 5) each digit (0->0, 1->3, 2->1, 3->4, 4->2) of the base 5 representation of n.
  • A065257 (program): Quintal Queens permutation of N: double (mod 5) each digit (0->0, 1->2, 2->4, 3->1, 4->3) of the base-5 representation of n-1, add one.
  • A065258 (program): Quintal Queens permutation of N: halve or multiply by 3 (mod 5) each digit (0->0, 1->3, 2->1, 3->4, 4->2) of the base 5 representation of n-1, add one.
  • A065259 (program): A057114 conjugated with A059893, inverse of A065260.
  • A065260 (program): A057115 conjugated with A059893, inverse of A065259.
  • A065261 (program): The siteswap sequence (the deltas p[i]-i, i in ]-inf,+inf[, folded from Z to N, mapping 0->1, 1->2, -1->3, 2->4, -2->5, etc.) for A065260.
  • A065262 (program): The nonpositive side (-1, -2, -3, …) of the site swap sequence A065261. The bisection of odd terms of A065261.
  • A065268 (program): The bisection of odd terms (the <= 0 half of Z) of A065267.
  • A065274 (program): The bisection of odd terms (the <= 0 half of Z) of A065273.
  • A065280 (program): The bisection of odd terms (the <= 0 half of Z) of A065279.
  • A065286 (program): The bisection of odd terms (the <= 0 half of Z) of A065285.
  • A065293 (program): Number of values of s, 0 <= s <= n-1, such that 2^s mod n = s.
  • A065294 (program): Values of k such that A065293(k) = 0.
  • A065295 (program): Number of values of s, 0 < s <= n-1, such that s^s == s (mod n).
  • A065300 (program): Numbers n such that sum of divisors is a squarefree number.
  • A065301 (program): Both n and the sum of its divisors are squarefree numbers.
  • A065305 (program): Triangular array giving means of two odd primes: T(n,k) = (n-th prime + k-th prime)/2, n >= k >= 2.
  • A065308 (program): Prime(n - PrimePi(n)).
  • A065309 (program): a(n) = prime(n) - prime(n - pi(n)).
  • A065310 (program): Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).
  • A065311 (program): Primes which occur exactly twice in the sequence of a(n) = prime(n) - prime(n - pi(n)) = A065308(n).
  • A065312 (program): Primes which occur exactly once in A065308 (prime(n - pi(n))).
  • A065313 (program): a(n) = Pi(n*Pi(n)).
  • A065328 (program): a(n) is the number of primes less than or equal to prime(n) - n.
  • A065330 (program): a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.
  • A065331 (program): Largest 3-smooth divisor of n.
  • A065332 (program): 3-smooth numbers in their natural position, gaps filled with 0.
  • A065333 (program): Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).
  • A065334 (program): 2-exponents to represent 3-smooth numbers (A065332).
  • A065335 (program): 3-exponents to represent 3-smooth numbers (A065332).
  • A065338 (program): a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0.
  • A065339 (program): Number of primes congruent to 3 modulo 4 dividing n (with multiplicity).
  • A065340 (program): Third diagonal of triangle in A046740.
  • A065342 (program): Triangle of sum of two primes: prime(n)+prime(k) with n >= k >= 1.
  • A065344 (program): a(n) = Mod( binomial(2*n,n), (n+1)*(n+2) ).
  • A065345 (program): a(n) = Mod( binomial(2*n,n), (n+1)*(n+2)*(n+3) ).
  • A065346 (program): a(n) = Mod( binomial(2*n, n), (n+1)*(n+2)*(n+3)*(n+4) ).
  • A065347 (program): Positions of zeros in A065344, i.e., binomial(2n,n) mod ((n+1)*(n+2)) = 0.
  • A065348 (program): Positions of zeros in A065345.
  • A065349 (program): Positions of zeros in A065346.
  • A065350 (program): Mod( binomial(2*n, n), (n+1)*(n+1) ).
  • A065355 (program): a(n) = n! - Sum_{k=0..n-1} k!.
  • A065356 (program): Final digits of A065355(n) (in reverse order) for sufficiently large n.
  • A065357 (program): a(n) = (-1)^pi(n) where pi(n) is the number of primes <= n.
  • A065358 (program): The Jacob’s Ladder sequence: a(n) = Sum_{k=1..n} (-1)^pi(k), where pi = A000720.
  • A065359 (program): Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n.
  • A065360 (program): Alternating sum of “negabits”. Replace (-2)^k with (-1)^k in negabinary expansion of n.
  • A065361 (program): Rebase n from 3 to 2. Replace 3^k with 2^k in ternary expansion of n.
  • A065362 (program): Rebase n from 4 to 2. Replace 4^k with 2^k in quaternary expansion of n.
  • A065363 (program): Sum of balanced ternary digits in n. Replace 3^k with 1 in balanced ternary expansion of n.
  • A065364 (program): Alternating sum of balanced ternary digits in n. Replace 3^k with (-1)^k in balanced ternary expansion of n.
  • A065365 (program): Replace 3^k with 2^k in balanced ternary expansion of n.
  • A065366 (program): Replace 3^k with (-2)^k in balanced ternary expansion of n.
  • A065367 (program): Replace 3^k with (-3)^k in balanced ternary expansion of n.
  • A065368 (program): Alternating sum of ternary digits in n. Replace 3^k with (-1)^k in ternary expansion of n.
  • A065369 (program): Replace 3^k with (-3)^k in ternary expansion of n.
  • A065371 (program): a(1) = 1, a(prime(i)) = prime(i) - i for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.
  • A065373 (program): Number of iterations of A065371 starting at n until 1 is reached.
  • A065376 (program): Primes of the form p + k^2, p prime and k > 0.
  • A065382 (program): Number of primes between n(n+1)/2 (exclusive) and (n+1)(n+2)/2 (inclusive).
  • A065383 (program): a(n) = smallest prime >= n*(n + 1)/2.
  • A065384 (program): Largest prime <= n * (n + 1) / 2.
  • A065387 (program): a(n) = sigma(n) + phi(n).
  • A065398 (program): Fibonacci numbers whose digits sum to a prime.
  • A065408 (program): Squares whose digits sum to a prime.
  • A065420 (program): Triangle T(n,k) = binomial(n+2,k+1)*(binomial(n+2,k+1)-1), n >=0, 0 <= k <= n.
  • A065423 (program): Number of ordered length 2 compositions of n with at least one even summand.
  • A065424 (program): Catalan-like formula: a(n) = binomial(6m, 3m+1)/(9m+6).
  • A065430 (program): Order of commutator subgroup of GL(2,Z_n) (invertible 2 X 2 matrices mod n: A000252).
  • A065432 (program): Triangle related to Catalan triangle: recurrence related to A033877 (Schroeder numbers).
  • A065438 (program): Complement of A065039.
  • A065440 (program): a(n) = (n-1)^n.
  • A065447 (program): Concatenation of 1, 00, 111, 0000, …, n 1’s (if n is odd) or n 0’s (if n is even).
  • A065450 (program): Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.
  • A065451 (program): a(n) = Fibonacci(phi(n)), a(0) = 0.
  • A065454 (program): Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).
  • A065455 (program): Number of (binary) bit strings of length n in which no even block of 0’s is followed by an odd block of 1’s.
  • A065456 (program): Number of functions on n labeled nodes whose representation as a digraph has two components.
  • A065457 (program): Period of the flip-riffle shuffle function on a deck of 2n cards.
  • A065475 (program): Natural numbers excluding 2.
  • A065482 (program): a(n) = round( 2^n/n ).
  • A065494 (program): Number of (binary) bit strings in which no even length block of 0’s is followed by an even length block of 1’s.
  • A065495 (program): Number of (binary) bit strings of length n in which an odd length block of 0’s is followed by an odd length block of 1’s.
  • A065496 (program): Numbers n such that sigma(n) is a nontrivial power, i.e., sigma(n) = a^b where a and b are greater than 1.
  • A065497 (program): Number of (binary) bit strings of length n having at least one even length block of 0’s followed by an even length block of 1’s.
  • A065500 (program): Number of distinct functions from a set with n^n elements to itself that can be defined naturally (in n) by typed lambda-calculus expressions.
  • A065502 (program): Positive numbers divisible by 2 or 5; 1/n not purely periodic after decimal point.
  • A065503 (program): Let p(k) denote k-th prime; consider solutions (x,y) of Diophantine equation p(x+1)-6p(y)=1 (*), where p(x) belongs to A060213 and p(m)=(p(n)+1)/6; sequence gives values of x.
  • A065504 (program): a(n+1) = a(n) + n + the number of a(k)’s <= n, 1 <= k <= n and a(1) = 1.
  • A065506 (program): Number of (binary) bit strings of length n having an even length block of 0’s followed by an odd length block of 1’s.
  • A065508 (program): Primes p such that p^2 - p + 1 is prime.
  • A065512 (program): Numbers n such that sigma(n) + 1 is prime.
  • A065513 (program): Number of endofunctions of [n] with a cycle a->b->c->a and for all x in [n], some iterate f^k(x)=a.
  • A065514 (program): Largest prime power < prime(n).
  • A065515 (program): Number of prime powers <= n.
  • A065516 (program): Differences between products of 2 primes.
  • A065517 (program): Numerator of n/(sum of the digits of n).
  • A065521 (program): a(n) = floor(prime(n) / n) * n - prime(n) mod n.
  • A065530 (program): If n is odd then a(n) = n, else a(n) = n*(n+2).
  • A065532 (program): a(n) = 48*n^2 - 1.
  • A065535 (program): Number of strongly perfect lattices in dimension n.
  • A065540 (program): a(n) is smallest prime > 3*a(n-1), a(1) = 3.
  • A065554 (program): Numbers n such that floor((3/2)^(n+1))/floor((3/2)^n) = 3/2.
  • A065558 (program): Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the maximal degree of an irreducible representation of G_n.
  • A065562 (program): a(n) = b(n)-th highest positive integer not equal to any a(k), 1 <= k < n, where {b(n)} = 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, … (sequence A002260).
  • A065563 (program): Product of three consecutive Fibonacci numbers.
  • A065564 (program): Numbers k such that floor((4/3)^(k+1))/floor((4/3)^k) = 4/3.
  • A065565 (program): a(n) = floor((5/4)^n).
  • A065566 (program): Numbers k such that floor((5/4)^(k+1))/floor((5/4)^k) = 5/4.
  • A065568 (program): Sum over all subsets of {1,..,n} of the GCD of the subset.
  • A065583 (program): Sum of numbers which in base n have (n-1) distinct nonzero digits.
  • A065595 (program): a(n) = (sum of first n primes)^2 - sum of squares of first n primes.
  • A065597 (program): a(0)=0, a(1)=1, a(2)=1; for n >= 3, a(n) = 2*a(n-1)*a(n-2) - a(n-3).
  • A065598 (program): a(0)=0, a(1)=1, a(2)=2; for n >= 3, a(n) = 2*a(n-1)*a(n-2) - a(n-3).
  • A065599 (program): If n odd, a(n) = n^2 else a(n) = n.
  • A065600 (program): Triangle T(n,k) giving number of Dyck paths of length 2n with exactly k hills (0 <= k <= n).
  • A065601 (program): Number of Dyck paths of length 2n with exactly 1 hill.
  • A065602 (program): Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.
  • A065603 (program): Transposition diameter: maximal number of moves in an optimal sorting of n objects by moving blocks.
  • A065608 (program): Sum of divisors of n minus the number of divisors of n.
  • A065619 (program): Expansion of e.g.f. x * (tan(x) + sec(x)).
  • A065620 (program): a(0)=0; thereafter a(2n) = 2a(n), a(2n+1) = -2a(n) + 1.
  • A065621 (program): Reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n.
  • A065651 (program): Sum_{k=1..n} (-1)^tau(k)=n-2*floor(sqrt(n)).
  • A065652 (program): a(0) = 0 and a(n+1) = if a(n) - 1 is new and > 0 then a(n) - 1 else a(n)*a(n) + 1 for n >= 0.
  • A065653 (program): a(0) = 0, a(1) = 1, a(n) = a(n-2)*a(n-2) + 2 for n > 1.
  • A065654 (program): Fixed points for A065652, a permutation of the natural numbers.
  • A065677 (program): Maximal Diffy_length for quadruples of numbers <= n.
  • A065678 (program): Minimum value t such that all quadruples of Diffy_length >= n have a maximal value >= t.
  • A065679 (program): If n is even, a(n) = n^2 else a(n) = n.
  • A065680 (program): Number of primes <= prime(n) which begin with a 1.
  • A065681 (program): Number of primes <= prime(n) which begin with a 2.
  • A065684 (program): Number of primes <= prime(n) which begin with a 5.
  • A065692 (program): Braided power sequence: this is b(n+1)=3b(n)+2d(n)-c(n), A065693 is c(n+1)=3c(n)+2b(n)-d(n) and A065694 is d(n+1)=3d(n)+2c(n)-b(n), starting with b(0)=0, c(0)=1 and d(0)=2.
  • A065693 (program): Braided power sequence: A065692 is b(n+1)=3b(n)+2d(n)-c(n), this is c(n+1)=3c(n)+2b(n)-d(n) and A065694 is d(n+1)=3d(n)+2c(n)-b(n), starting with b(0)=0, c(0)=1 and d(0)=2.
  • A065694 (program): Braided power sequence: A065692 is b(n+1)=3b(n)+2d(n)-c(n), A065693 is c(n+1)=3c(n)+2b(n)-d(n) and this is d(n+1)=3d(n)+2c(n)-b(n), starting with b(0)=0, c(0)=1 and d(0)=2.
  • A065704 (program): Number of squares or twice squares dividing n.
  • A065705 (program): a(n) = Lucas(10*n).
  • A065707 (program): Bessel polynomial {y_n}’(-2).
  • A065710 (program): Number of 2’s in decimal expansion of 2^n.
  • A065712 (program): Number of 1’s in decimal expansion of 2^n.
  • A065713 (program): Sum of digits of 4^n.
  • A065714 (program): Number of 3’s in decimal expansion of 2^n.
  • A065715 (program): Number of 4’s in decimal expansion of 2^n.
  • A065716 (program): Number of 5’s in decimal expansion of 2^n.
  • A065717 (program): Number of 6’s in decimal expansion of 2^n.
  • A065718 (program): Number of 7’s in decimal expansion of 2^n.
  • A065719 (program): Number of 8’s in decimal expansion of 2^n.
  • A065726 (program): Primes p whose base-8 expansion is also the decimal expansion of a prime.
  • A065730 (program): Largest square <= n-th prime.
  • A065731 (program): Greatest perfect square that does not exceed n!.
  • A065732 (program): Largest square <= 2^n.
  • A065733 (program): Largest square <= n^3.
  • A065734 (program): Largest square <= sigma(n).
  • A065735 (program): Largest square <= product of first n primes.
  • A065736 (program): Largest square <= 10^n.
  • A065737 (program): Largest square <= binomial(n,2).
  • A065738 (program): Largest square <= binomial(2n,n).
  • A065739 (program): Largest square <= sum of first n squares.
  • A065740 (program): Largest square <= n^n.
  • A065741 (program): Largest square <= sum of squares of divisors of n.
  • A065744 (program): Number of 9’s in the decimal expansion of 2^n.
  • A065745 (program): Sum of squares and twice squares dividing n.
  • A065760 (program): Concatenation of increasing number of alternating digits in base 2, starting with 1.
  • A065761 (program): Concatenation of increasing number of alternating digits in base 2, starting with 0.
  • A065762 (program): a(n) = (sum of first n primes)^2 + sum of (squares of first n primes).
  • A065764 (program): Sum of divisors of square numbers.
  • A065765 (program): Sum of divisors of twice square numbers.
  • A065766 (program): Sum of divisors of twice a square number, divided by three.
  • A065795 (program): Number of subsets of {1,2,…,n} that contain the average of their elements.
  • A065796 (program): Alternating sum of digits of n^2.
  • A065801 (program): Least k such that n^k > 2^n.
  • A065803 (program): a(n) = (sigma_2(n) mod 2) * (sigma_2(n) mod 5). Residue-product modulo 2 and 5 of sum of square of divisors.
  • A065805 (program): a(n) = Sum_{j=0..n} sigma(j,n).
  • A065814 (program): a(n) = tau(n)^2 - tau(n^2) = A000005(n)^2 - A000005(n^2).
  • A065816 (program): Numbers k such that the alternating sum of digits of k^2 is 0.
  • A065826 (program): Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.
  • A065827 (program): Sum of squares of divisors of square numbers.
  • A065855 (program): Number of composites <= n.
  • A065858 (program): m-th composite number c(m) = A002808(m), where m is the n-th prime number: a(n) = A002808(A000040(n)).
  • A065859 (program): Remainder when the n-th prime is divided by the n-th composite number.
  • A065860 (program): Remainder when the n-th composite number is divided by n.
  • A065861 (program): Remainder when the n-th composite number is divided by pi(n), the number of primes not exceeding n.
  • A065862 (program): Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.
  • A065863 (program): Remainder when n-th prime is divided by the number of nonprimes not exceeding n.
  • A065864 (program): Remainder when n is divided by the number of nonprimes not exceeding n.
  • A065865 (program): a(n) is the least k such that nk - 1 and nk + 1 are both composite.
  • A065866 (program): a(n) = n! * Catalan(n+1).
  • A065870 (program): n-th prime - n-th semiprime.
  • A065871 (program): Numbers k such that usigma(k) + 1 is a prime.
  • A065872 (program): Numbers k such that usigma(k) - 1 is a prime (cf. A034448).
  • A065874 (program): a(n) = (7^(n+1) - (-6)^(n+1))/13.
  • A065877 (program): Non-Niven (or non-Harshad) numbers: numbers which are not a multiple of the sum of their digits.
  • A065878 (program): Numbers which are not an integer multiple of their number of binary 1’s.
  • A065881 (program): Ultimate modulo 10: right-hand nonzero digit of n.
  • A065882 (program): Ultimate modulo 4: right-hand nonzero digit of n when written in base 4.
  • A065883 (program): Remove factors of 4 from n (i.e., write n in base 4, drop final zeros, then rewrite in decimal).
  • A065886 (program): Smallest square divisible by n!.
  • A065887 (program): Smallest number whose square is divisible by n!.
  • A065888 (program): a(n) = number of endofunctions on [n] with a 4-cycle a->b->c->d->a and for any x in [n], some iterate f^k(x) = a.
  • A065889 (program): a(n) = number of unicyclic connected simple graphs whose cycle has length 4.
  • A065890 (program): Number of composites less than the n-th prime.
  • A065893 (program): Which composite number is the square of n? Index of n^2 in A002808.
  • A065896 (program): Number of composites <= 2*n.
  • A065897 (program): The a(n)-th composite number is twice the n-th prime.
  • A065915 (program): Numerator of sigma(8*n^2)/sigma(4*n^2).
  • A065916 (program): Denominator of sigma(8*n^2)/sigma(4*n^2).
  • A065917 (program): Boundaries of primorial intervals [1,3]; [3,9],[9,15]; [15,45], etc.
  • A065919 (program): Bessel polynomial y_n(4).
  • A065920 (program): Bessel polynomial {y_n}‘(2).
  • A065921 (program): Bessel polynomial {y_n}‘(3).
  • A065922 (program): Bessel polynomial {y_n}‘(4).
  • A065923 (program): Bessel polynomial y_n(-3).
  • A065928 (program): (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy)) = t(2) = 3, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.
  • A065929 (program): (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy)) = t(3) = 6, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.
  • A065930 (program): (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy)) = t(4) = 10, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.
  • A065931 (program): Triangle of coefficients of Bessel polynomials {y_n(x)}’.
  • A065941 (program): T(n,k) = binomial(n-floor((k+1)/2), floor(k/2)). Triangle read by rows, for 0 <= k <= n.
  • A065942 (program): Central column of triangle A065941.
  • A065943 (program): Triangle of coefficients of Bessel polynomials {y_n(x)}’’.
  • A065944 (program): Bessel polynomial {y_n}’’(-1).
  • A065945 (program): Bessel polynomial {y_n}’‘(2).
  • A065946 (program): Bessel polynomial {y_n}’’(-2).
  • A065947 (program): Bessel polynomial {y_n}’‘(3).
  • A065948 (program): Bessel polynomial {y_n}’’(-3).
  • A065949 (program): Bessel polynomial {y_n}’’‘(0).
  • A065953 (program): Denominator of (-1)^n*n!*(E(n,2)-E(n,1)*E(n-1,1)) where E(n,x) = Sum_{k=0..n} (-1)^k*x^k/k!.
  • A065958 (program): a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).
  • A065959 (program): a(n) = n^3*Product_{distinct primes p dividing n} (1+1/p^3).
  • A065960 (program): a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).
  • A065961 (program): a(n) = (3*n - 1)!*n/2.
  • A065962 (program): a(1) = 1, a(n) = a(n - 1) + pi(a(n - 1)) + 1.
  • A065966 (program): Numbers k such that phi(k) / 2 is prime.
  • A065967 (program): a(n) = n * Sum_{primes p dividing n} (1 + 1/p).
  • A065968 (program): a(n) = n * Sum_{primes p dividing n} (1 - 1/p).
  • A065969 (program): a(n) = n^2 * Sum_{primes p dividing n} (1 + 1/p^2).
  • A065970 (program): a(n) = n^2 * Sum_{primes p dividing n} (1 - 1/p^2).
  • A065974 (program): Numerators in expansion of (exp(x)-1)^3.
  • A065975 (program): Denominators in expansion of (exp(x)-1)^3.
  • A065979 (program): Binomial transform of A002024.
  • A065980 (program): Inverse binomial transform of [1^1,2^2,3^3,…], shifted right by one index.
  • A065981 (program): Best approximation of the remainder in the zeta(4) series using the remainder in the zeta(3) series.
  • A065982 (program): a(n) = (n+1)*binomial(2*n,n) - 2^(2*n-1).
  • A065985 (program): Numbers n such that d(n) / 2 is prime, where d(n) = number of divisors of n.
  • A065994 (program): a(n) = prime(prime(n) - n).
  • A065995 (program): a(n) = prime(prime(n) + n).
  • A065996 (program): a(n) = prime(prime(n) mod n).
  • A065998 (program): Concatenate n and number of divisors of n.
  • A065999 (program): Sum of digits of 9^n.
  • A066001 (program): Sum of digits of 5^n.
  • A066002 (program): Sum of digits of 6^n.
  • A066003 (program): Sum of digits of 7^n.
  • A066004 (program): Sum of digits of 8^n.
  • A066005 (program): Sum of digits of 11^n.
  • A066006 (program): Sum of digits of 12^n.
  • A066007 (program): a(n) is that n-digit number m which minimizes m/(sum of digits of m); in case of a tie pick the smallest.
  • A066014 (program): Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105682.
  • A066022 (program): Number of digits in n^n.
  • A066023 (program): (a(n)^7+1)/(n^7+1) is the smallest integer > 1.
  • A066030 (program): Card{i<=n, floor(n/i)=0 mod(i)}.
  • A066033 (program): Alternating sum of primes: a(1) = A000040(1) = 2 and a(n) = a(n-1) + A000040(n)*(-1)^n for n > 1.
  • A066039 (program): Largest prime less than or equal to the sum of first n primes (A007504).
  • A066043 (program): a(1) = 1; for m > 0, a(2m) = 2m, a(2m+1) = 4m+2.
  • A066046 (program): a(1) = 1; a(2) = 2; a(3) = 3; a(n+3) = a(n+2)*a(n+1) + a(n+1)*a(n) + a(n)*a(n+2).
  • A066047 (program): Numbers k that divide A001045(k-1).
  • A066048 (program): Product of smallest and greatest prime factors of n.
  • A066049 (program): Numbers n such that 2*n^2 - 1 is a prime.
  • A066050 (program): Average of divisors of n (sigma(n)/d(n)) is greater than average of divisors for all k < n.
  • A066052 (program): Number of permutations in the symmetric group S_n with order >= 3.
  • A066063 (program): Size of the smallest subset S of T={0,1,2,…,n} such that each element of T is the sum of two elements of S.
  • A066066 (program): a(n) = prime(2*n) - 2*prime(n).
  • A066067 (program): Number of binary strings u of any length with property that length(u) + number of 0’s in u <= n (only one of a string and its reversal are counted).
  • A066068 (program): a(n) = n^n + n.
  • A066069 (program): a(n) is the smallest positive integer m such that n divides (n + m)^m.
  • A066070 (program): a(1) = 1; for m > 0, a(2m) = 2(2m+1), a(2m+1) = 2m+1.
  • A066071 (program): Nonprime numbers n such that phi(n) + 1 is prime.
  • A066072 (program): Prime numbers arising in A066071.
  • A066073 (program): Composite numbers n such that sigma(n) - 1 is prime.
  • A066074 (program): Primes arising in A066073.
  • A066075 (program): Number of solutions x to prime(n) = sigma(x) - 1, where prime(n) is the n-th prime.
  • A066077 (program): a(n) is the number of x such that sigma(x)-1 is 0 or one of the first n-1 primes.
  • A066084 (program): a(n) = (n!)^2 + n! + n.
  • A066086 (program): Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).
  • A066088 (program): Number of distinct prime factors of sigma_2(n) = A001157(n), the sum of squares of divisors of n.
  • A066090 (program): a(n) = binomial(sigma(n), n).
  • A066096 (program): Duplicate values in A060143.
  • A066098 (program): Sum of digits of primorial(n) (A002110).
  • A066102 (program): Number of distinct prime factors of Sigma_4[n], the sum of 4th powers of divisors of n.
  • A066104 (program): a(2n) = 2n, a(2n+1) = 4(n+1).
  • A066106 (program): a(2n) = (2n)(2n+2); a(2n+1) = 4n + 4.
  • A066107 (program): a(0) = 0; for n > 0, a(2n+1) = (2n+1)*(2n+3); a(2n) = 2n + 2.
  • A066108 (program): Sum n^d over all divisors of n.
  • A066114 (program): a(0) = 1; for n > 0, a(n) = (n!*(3*n+1))/2.
  • A066116 (program): a(n) = prime(n-2)*prime(n-1)^2*prime(n).
  • A066118 (program): a(n) = n!*(3*n-1)/2.
  • A066136 (program): Primes are replaced by their local sequence number in A000040, while composites are replaced by their sequence number in A002808; (a kind of eigen- or home-indexing).
  • A066138 (program): a(n) = 10^(2n) + 10^n + 1.
  • A066141 (program): a(n) = n^(n-1) + n + 1.
  • A066142 (program): a(n) = (n!)^2 + n! + 1.
  • A066143 (program): a(n) = n! + n^2 + n.
  • A066148 (program): Primes with an even number of 0’s in binary expansion.
  • A066149 (program): Primes with an odd number of 0’s in binary expansion.
  • A066161 (program): Let p = n-th prime; sequence gives ((p-2)!-1)/p.
  • A066164 (program): Sum of interior angles in an n-sided polygon in degrees.
  • A066168 (program): a(n) = least k such that phi(k) > sigma(n).
  • A066169 (program): Least k such that phi(k) >= n.
  • A066170 (program): Triangle read by rows: T(n,k) = (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n, n >= 0.
  • A066178 (program): Number of binary bit strings of length n with no block of 8 or more 0’s. Nonzero heptanacci numbers, A122189.
  • A066179 (program): Primes p such that (p-1)/2 and (p-3)/4 are also prime.
  • A066181 (program): Permutation of the integers with cycle form {1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}, …
  • A066182 (program): Permutation of the integers with cycle form {1}, {3, 2}, {6, 5, 4}, {10, 9, 8, 7}, …
  • A066186 (program): Sum of all parts of all partitions of n.
  • A066189 (program): Sum of all partitions of n into distinct parts.
  • A066190 (program): Numbers k such that the sum of the even aliquot parts of k divides k.
  • A066191 (program): Numbers n such that the sum of the odd aliquot parts of n divides n.
  • A066194 (program): A permutation of the integers (a fractal sequence): a(n) = A006068(n-1) + 1.
  • A066205 (program): a(n) = Product_{k=1..n} prime(2k-1), where prime(k) is k-th prime.
  • A066206 (program): a(n) = Product_{k=1..n} prime(2k), where prime(k) is the k-th prime.
  • A066207 (program): All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.
  • A066208 (program): All primes that divide n are of the form prime(2k-1), where prime(k) is k-th prime.
  • A066209 (program): A053041(n)-10^(n-1).
  • A066210 (program): a(n) = ((2*n)^(2*n+2) - 1)/(4*n^2 - 1).
  • A066211 (program): a(n) = Sum_{j=0..n} (2*n)!/(2*n-j)!.
  • A066221 (program): Bisection of A001189.
  • A066222 (program): Bisection of A001189.
  • A066223 (program): Bisection of A000085.
  • A066224 (program): Bisection of A000085.
  • A066237 (program): First differences give A052849.
  • A066241 (program): 1 + number of anti-divisors of n.
  • A066246 (program): 0 unless n is a composite number A002808(k) when a(n) = k.
  • A066247 (program): Characteristic function of composite numbers: 1 if n is composite else 0.
  • A066248 (program): a(n) = if n+1 is prime then A049084(n+1)*2 else A066246(n+1)*2 - 1.
  • A066250 (program): a(n) = if n+1 is prime then A049084(n+1)*2 - 1 else A066246(n+1)*2.
  • A066252 (program): a(n) = A066248(A066248(n)).
  • A066258 (program): a(n) = Fibonacci(n)^2 * Fibonacci(n+1).
  • A066259 (program): a(n) = Fibonacci(n)*Fibonacci(n+1)^2.
  • A066266 (program): Product of first n primorials + 1.
  • A066268 (program): Product of first n primorials - 1.
  • A066270 (program): Multiples of 24 whose digits also sum to 24.
  • A066272 (program): Number of anti-divisors of n.
  • A066274 (program): Number of endofunctions of [n] such that 1 is not a fixed point.
  • A066275 (program): Number of endofunctions of [n] such that some element is fixed, but 1 is not fixed.
  • A066279 (program): a(n) = n^n + n + 1.
  • A066280 (program): a(n) = 1^n + 2^(n+1) + 3^(n+2).
  • A066283 (program): Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 6.
  • A066285 (program): a(n) is the minimal difference between primes p and q whose sum is 2n.
  • A066288 (program): Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24.
  • A066293 (program): a(n) = A000203(n)^2 - A001157(n) = sigma(n)^2 - sigma_2(n).
  • A066294 (program): a(n) = A000203(n)^2 - A001157(n) - 2n = sigma(n)^2 - sigma_2(n) - 2n.
  • A066298 (program): a(n) = googol (mod n).
  • A066300 (program): Number of n X n matrices with exactly 2 1’s in each row, other entries 0.
  • A066301 (program): a(n) = 0 if n is squarefree, otherwise 1 + a(n/rad(n)) where rad = A007947 (squarefree kernel).
  • A066308 (program): a(n) = (sum of digits of n) * (product of digits of n).
  • A066311 (program): All distinct primes dividing n are consecutive.
  • A066318 (program): Number of necklaces with n labeled beads of 2 colors.
  • A066319 (program): A labeled structure simultaneously a tree and a cycle.
  • A066324 (program): Number of endofunctions on n labeled points constructed from k rooted trees.
  • A066325 (program): Coefficients of unitary Hermite polynomials He_n(x).
  • A066328 (program): a(n) = sum of indices of distinct prime factors of n; here, index(i-th prime) = i.
  • A066332 (program): a(1)=1; for n > 0, a(n+1) = rad(a(n))*n where rad=A007947.
  • A066333 (program): a(n) = min(x : x^2 + n^2 = 0 mod (x+n-1)).
  • A066335 (program): Binary string which equals n when 1’s and 2’s bits have negative weights.
  • A066339 (program): Number of primes p of the form 4m+1 with p <= n.
  • A066341 (program): Sum of distinct terms in n-th row of Fermat’s triangle.
  • A066342 (program): Number of triangulations of the cyclic polytope C(n, n-4).
  • A066343 (program): Beatty sequence for log_2(10).
  • A066344 (program): Beatty sequence for log_5(10).
  • A066353 (program): 1 + partial sums of A032378.
  • A066356 (program): Numerator of sequence defined by recursion c(n) = 1 + c(n-2) / c(n-1), c(0) = 0, c(1) = 1.
  • A066357 (program): Number of ordered (i.e., planar) trees on 2n edges with every subtree at the root having an even number of edges.
  • A066368 (program): a(n) = (n+2)*2^(n-1) - 2*n.
  • A066370 (program): Quadruply triangular numbers.
  • A066373 (program): a(n) = (3*n-2)*2^(n-3).
  • A066374 (program): (3*n+4)*2^(n-3)-(2*n-1).
  • A066375 (program): a(n) = 6*binomial(n,4) + 3*binomial(n,3) + 4*binomial(n,2) - n + 2.
  • A066377 (program): Number of numbers m <= n such that floor(sqrt(m)) divides m.
  • A066380 (program): a(n) = Sum_{k=0..n} binomial(3*n,k).
  • A066381 (program): a(n) = Sum_{k=0..n} binomial(4*n,k).
  • A066382 (program): a(n) = Sum_{k=0..n} binomial(n^2,k).
  • A066383 (program): a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).
  • A066384 (program): a(n) = Sum_{k=0..n} binomial(2^n,k).
  • A066393 (program): Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.
  • A066394 (program): Coordination sequence for ReO_3 net with respect to oxygen atom O_1.
  • A066398 (program): Reversion of g.f. (with constant term included) for partition numbers.
  • A066399 (program): From reversion of e.g.f. for squares.
  • A066406 (program): a(n) = 2^n*(3^n-3).
  • A066415 (program): Denominators of coefficients in series expansion of -512*(1+x)^3/(x-8)^3.
  • A066417 (program): Sum of anti-divisors of n.
  • A066422 (program): a(n) = least k such that phi^(k)(n) + 1 is prime, where phi^(k) denotes application of phi k times.
  • A066423 (program): Composite numbers n such that the product of proper divisors of the n does not equal n.
  • A066429 (program): a(n) = 7^n mod n^7.
  • A066430 (program): a(n) = 8^n mod n^8.
  • A066431 (program): a(n) = 9^n mod n^9.
  • A066432 (program): a(n) = 10^n mod n^10.
  • A066436 (program): Primes of the form 2*n^2 - 1.
  • A066438 (program): a(n) = 7^n mod n.
  • A066439 (program): a(n) = 8^n mod n.
  • A066440 (program): a(n) = 9^n mod n.
  • A066441 (program): a(n) = 11^n mod n.
  • A066442 (program): a(n) = 12^n mod n.
  • A066443 (program): Number of distinct paths of length 2n+1 along edges of a unit cube between two fixed adjacent vertices.
  • A066444 (program): a(n) = 11^n mod n^11.
  • A066445 (program): a(n) = 12^n mod n^12.
  • A066446 (program): Number of unordered divisor pairs of n.
  • A066449 (program): Binomial(n, phi(n)), where phi(n) is the Euler totient function.
  • A066451 (program): a(n) is the number of integers k > 0 such that (n*k+1)/(k^2+1) is an integer.
  • A066455 (program): 6*binomial(n,4)+5*binomial(n,2)-4*n+5.
  • A066456 (program): Upper bound on number of regular triangulations of cyclic polytope C(n, n-4).
  • A066459 (program): Product of factorials of the digits of n.
  • A066461 (program): a(3) = 5; a(n) = min(x>1,x^2+x*n+n^2 = 1 mod(x+n)).
  • A066468 (program): Numbers having just three anti-divisors.
  • A066470 (program): Numbers having just five anti-divisors.
  • A066473 (program): Numbers having just seven anti-divisors.
  • A066475 (program): Numbers having just nine anti-divisors.
  • A066477 (program): Numbers having just eleven anti-divisors.
  • A066479 (program): a(n) = min( x : x^3+n^3+x^2+n^2+x+n=1 mod(x+n)).
  • A066481 (program): Largest anti-divisor of n.
  • A066482 (program): The smallest anti-divisor of n.
  • A066486 (program): a(n) = min( x : x^3 + n^3 == 0 mod (x+n-1) ).
  • A066490 (program): Number of primes of the form 4m+3 that are <= n.
  • A066492 (program): a(n) = A056524(n)/11.
  • A066498 (program): Numbers k such that 3 divides phi(k).
  • A066499 (program): Numbers k such that phi(k) == 2 (mod 4).
  • A066500 (program): Numbers k such that 5 divides phi(k).
  • A066501 (program): Numbers k such that x^6 == 1 (mod(k)) has no solution 1 < x < k-1.
  • A066502 (program): Numbers k such that 7 divides phi(k).
  • A066503 (program): a(n) = n - squarefree kernel of n, A007947.
  • A066504 (program): Sum of n/p^k over all maximal prime-power divisors of n.
  • A066507 (program): Numbers k such that there is a solution to x^2 == 2^k (mod k).
  • A066518 (program): Anti-divisor class sums of n.
  • A066519 (program): Gaps between successive numbers with an anti-divisor class sum of zero.
  • A066520 (program): Number of primes of the form 4m+3 <= n minus number of primes of the form 4m+1 <= n.
  • A066524 (program): a(n) = n*(2^n - 1).
  • A066526 (program): a(n) = binomial(Fibonacci(n), Fibonacci(n-1)).
  • A066530 (program): Expansion of (1+x+x^3)/((1-x)*(1-x^4)).
  • A066532 (program): If n is odd a(n) = 1, if n is even a(n) = 2^(n-1).
  • A066534 (program): Total number of walks with length > 0 in the Hasse diagram of a Boolean algebra of order n.
  • A066535 (program): Number of ways of writing n as a sum of n squares.
  • A066536 (program): Number of ways of writing n as a sum of n+1 squares.
  • A066542 (program): Nonnegative integers all of whose anti-divisors are either 2 or odd.
  • A066557 (program): Largest n-digit multiple of n.
  • A066558 (program): a(n) = A066557(n)/n.
  • A066559 (program): a(n) = ceiling(10^(n-1)/n).
  • A066560 (program): Smallest composite number divisible by n.
  • A066561 (program): a(n) is the smallest triangular number divisible by n.
  • A066568 (program): a(n) = n - sum of digits of n.
  • A066570 (program): Product of numbers <= n that have a prime factor in common with n.
  • A066577 (program): a(n) = floor(n/(product of nonzero digits of n)).
  • A066578 (program): a(n) = floor(n/(sum of digits of n)).
  • A066586 (program): Number of normal subgroups of the group of n X n signed permutations matrices (described in sequence A066051).
  • A066588 (program): The sum of the digits of n^n.
  • A066600 (program): Sum of the digits in the n-th row of Pascal’s triangle.
  • A066601 (program): a(n) = 3^n mod n.
  • A066602 (program): a(n) = 4^n mod n.
  • A066603 (program): a(n) = 5^n mod n.
  • A066604 (program): a(n) = 6^n mod n.
  • A066606 (program): a(n) = 2^n mod n^2.
  • A066607 (program): a(n) = 3^n mod n^3.
  • A066608 (program): a(n) = 4^n mod n^4.
  • A066609 (program): a(n) = remainder when 5^n is divided by n^5.
  • A066610 (program): a(n) = remainder when 6^n is divided by n^6.
  • A066611 (program): a(1) = 1; a(n) = remainder when n^n is divided by (n-1)^(n-1) for n > 1.
  • A066616 (program): a(1) = 1; a(n) = n*a(n-1) if n does not divide a(n-1), otherwise a(n) = a(n-1).
  • A066620 (program): Number of unordered triples of distinct pairwise coprime divisors of n.
  • A066628 (program): a(n) = n - the largest Fibonacci number <= n.
  • A066629 (program): a(n) = 2*Fibonacci(n+2) + ((-1)^n - 3)/2.
  • A066635 (program): Distance from n to closest square different from n.
  • A066638 (program): Smallest power of a squarefree number that is a multiple of n.
  • A066639 (program): Number of partitions of n with floor(n/2) parts.
  • A066642 (program): a(n) = floor(n^(n/2)).
  • A066643 (program): a(n) = floor(Pi*n^2).
  • A066644 (program): a(n) = floor(surface area of a sphere with radius n).
  • A066645 (program): a(n) = floor( (4/3)*Pi*n^3 ).
  • A066647 (program): Squares of the form a^2 + b^3 with a, b > 0.
  • A066650 (program): Numbers not of the form a^2 + b^3 with a, b > 0.
  • A066651 (program): Primes of the form 2*s + 1, where s is a squarefree number (A005117).
  • A066652 (program): Primes of the form 2*s - 1, where s is a squarefree number (A005117).
  • A066656 (program): a(n) = A000031(n) - A001037(n).
  • A066660 (program): Number of divisors of 2n excluding 1.
  • A066665 (program): a(n) = #{(x,y) | 0<=y<=x<=n and x+y is prime}.
  • A066667 (program): Coefficient triangle of generalized Laguerre polynomials (a=1).
  • A066668 (program): Signed row sums of A066667.
  • A066669 (program): Numbers n such that phi(n) = 2^k*prime for some k >= 0.
  • A066670 (program): Primes arising in A066669: the only odd prime divisor of phi(A066669(n)).
  • A066674 (program): Least number m such that phi(m) = A000010(m) is divisible by the n-th prime.
  • A066675 (program): a(n) = A066674(n)-1 divided by the n-th prime.
  • A066686 (program): Array T(i,j) read by antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).
  • A066691 (program): Value of tau(2n-1) when tau(2n-1) = tau(2n+1).
  • A066692 (program): Odd n such that tau(n) = tau(n+2), where tau(n) = A000005(n) is the number of divisors of n.
  • A066704 (program): Triangle with a(n,k) = C(n,floor(n/k)) with n>=k>=1.
  • A066710 (program): RATS: Reverse Add Then Sort the digits applied to previous term, starting with 3.
  • A066711 (program): RATS: Reverse Add Then Sort the digits applied to previous term, starting with 9.
  • A066713 (program): RATS(2^n): Reverse Add the digits of 2^n, Then Sort: a(n) = A036839(2^n).
  • A066714 (program): Coordination sequence for ReO_3 net with respect to Re atom.
  • A066715 (program): a(n) = gcd(2n+1, sigma(2n+1)).
  • A066722 (program): Numbers that can be expressed as the sum of two primes in exactly six ways.
  • A066728 (program): a(n) is the number of integers of the form (n+k+n*k)/(n-k) for k = 1,2,…,n-1.
  • A066729 (program): a(n) = Product_{d|n, d<n} d if n is composite, n otherwise.
  • A066733 (program): Numbers such that the nonzero product of the digits of its square is also a square.
  • A066743 (program): a(n) is the number of integers of the form (n^2+1)/(k^2+1), where k = 1,2,3,…,n.
  • A066744 (program): a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +3 for 0, +1 for 1.
  • A066745 (program): Least number of applications of f(k) = k(k+1)+1 to n to yield a prime, if this number exists; 0 otherwise.
  • A066746 (program): Conjectured values of a(n) defined by a(n) = least number of applications of f(k) = k^2 + 1 to n to yield a prime, if this number exists; = -1 otherwise.
  • A066750 (program): Greatest common divisor of n and its digit sum.
  • A066752 (program): a(n) = gcd(prime(n)+1, n+1).
  • A066755 (program): Numbers n such that n^2 + 1 is not divisible by k^2 + 1 for any k in [1,n-1].
  • A066759 (program): a(n) = least multiple of n differing from a prime by at most 1, if such a multiple exists; = 0 otherwise.
  • A066760 (program): Sum_{1<=k<=n, k is not a divisor of n and k is not coprime to n} k.
  • A066761 (program): Number of positive integers of the form (n^2+k^2)/(n-k) for k=1,2,3,4,….,n-1.
  • A066767 (program): a(n) = Sum_{k=1..n} sigma(k)*2^(n-k) where sigma(k) = A000203(k) is the sum of divisors of k.
  • A066768 (program): Sum_{d|n} binomial(2*d-2,d-1).
  • A066769 (program): a(n) = Sum_{d|n} d*Fibonacci(n/d).
  • A066770 (program): a(n) = 5^n*sin(2n*arctan(1/2)) or numerator of tan(2n*arctan(1/2)).
  • A066771 (program): 5^n cos(2n arctan(1/2)) or denominator of tan(2n arctan(1/2)).
  • A066774 (program): A066728(a(n))=3.
  • A066778 (program): a(n) = Sum_{i=1..n} floor((3/2)^i).
  • A066779 (program): Sum of squarefree numbers <= n.
  • A066780 (program): a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.
  • A066781 (program): a(n) = 2^phi(n).
  • A066787 (program): a(n) = gcd(2^n + 1, n^2 + 1).
  • A066788 (program): a(n) = gcd(phi(n), n^2 + 1).
  • A066791 (program): a(n) = phi(n^2 + n + 1).
  • A066792 (program): a(n) = phi(n^3 + n^2 + n + 1).
  • A066796 (program): a(n) = Sum_{i=1..n} binomial(2*i,i).
  • A066797 (program): a(n) = Sum_{i=1..n} binomial(4*i,2*i).
  • A066798 (program): a(n) = Sum_{i=1..n} binomial(6*i,3*i).
  • A066802 (program): a(n) = binomial(6*n,3*n).
  • A066803 (program): a(n) = gcd(2^n + 1, 3^n + 1).
  • A066804 (program): Sum of diagonal elements and those below it for a square matrix of integers, starting with 1.
  • A066809 (program): a(n) = sum of modular offsets: mod[n+c,b]-(mod[n,b]+c) for c<=b<=n.
  • A066810 (program): Expansion of x^2/((1-3*x)*(1-2*x)^2).
  • A066813 (program): a(n) = lcm(phi(n), phi(n+1)).
  • A066819 (program): Sum of the first n Sophie Germain primes.
  • A066822 (program): The fourth column of A038622, triangular array that counts rooted polyominoes.
  • A066827 (program): a(n) = gcd(2^((n*(n+1)/2)) + 1, 2^n + 1).
  • A066829 (program): Parity of Omega(n): a(n) = 1 if n is the product of an odd number of primes; 0 if product of even number of primes.
  • A066830 (program): a(n) = lcm(n+1, n-1).
  • A066838 (program): Product of primes < n that do not divide n.
  • A066839 (program): a(n) = sum of positive divisors k of n with k <= sqrt(n).
  • A066840 (program): Sum of positive integers k where k <= n/2 and gcd(k,n) = 1.
  • A066841 (program): a(n) = Product{k|n} k^(n/k); product is over the positive divisors of n.
  • A066842 (program): a(n) = Product_{k|n} k^k; product is over the positive divisors, k, of n.
  • A066843 (program): a(n) = Product_{k=1..n} d(k); d(k) is the number of positive divisors of k.
  • A066846 (program): Numbers of the form a^a + b^b, a >= b > 0.
  • A066847 (program): Integers of the form m! + n!, m and n = positive integers.
  • A066857 (program): Smallest number k such that n! - k is a square
  • A066859 (program): Product of sums of divisors and non-divisors.
  • A066869 (program): Sum of the first n safe primes.
  • A066872 (program): p^2 + 1 as p runs through the primes.
  • A066879 (program): n such that there are as many 1’s as 0’s in the base 2 expansion of Floor(n/2).
  • A066880 (program): Biased numbers: n such that all terms of the sequence f(n), f(f(n)), f(f(f(n))), …, 1, where f(k) = Floor(k/2), are odd.
  • A066885 (program): a(n) = (prime(n)^2 + 1)/2.
  • A066886 (program): Sum of the elements in any transversal of a prime(n) X prime(n) array containing the numbers from 1 to prime(n)^2 in standard order.
  • A066888 (program): Number of primes p between triangular numbers T(n) < p <= T(n+1).
  • A066903 (program): Primes in A006577.
  • A066906 (program): Places n where A006577(n) is a prime number.
  • A066908 (program): n^n minus largest factorial less than or equal to n^n.
  • A066909 (program): (product of primes < n that do not divide n) (mod n).
  • A066910 (program): a(1) = 1; a(n+1) = (sum{k=1 to n} a(k) ) (mod n).
  • A066911 (program): Sum of primes < n that do not divide n.
  • A066912 (program): Fourth column of the Eulerian triangle A008292 in square array format.
  • A066913 (program): (sum of primes < n that do not divide n) (mod n).
  • A066915 (program): a(n) = n^phi(n) + 1.
  • A066916 (program): a(n) = n^phi(n) - 1.
  • A066921 (program): a(n) = lcm(Omega(n), omega(n)).
  • A066922 (program): a(n) = gcd(Omega(n), omega(n)).
  • A066927 (program): Least k such that between p and 2p, for all primes > 3, there is always a number that is twice a square, i.e.; a k such that p < 2k^2 < 2p.
  • A066932 (program): a(n) is the denominator of b(n) where b(n)=1/b(n-1)+1/b(n-2) with b(1)=1 and b(2)=2.
  • A066935 (program): bigomega(n+1)==0 (mod bigomega(n)) where bigomega(n)=A001222(n) is the number of prime divisors of n (counted with multiplicity).
  • A066940 (program): Numbers n such that gcd(prime(n+1) + 1, prime(n) + 1) = 2.
  • A066949 (program): Take the sum of the previous two terms, subtract n if this sum is greater than n.
  • A066959 (program): Bigomega(n^n) where bigomega(x) is the number of prime factors in x (counted with multiplicity).
  • A066971 (program): a(n) = sigma(sigma(sigma(n))).
  • A066973 (program): a(n) = phi(binomial(2n, n)).
  • A066975 (program): a(n) = gcd(binomial(2n,n), 2^n + 1).
  • A066976 (program): a(0) = 1; for n>0: a(n) = sum{a(i)*n^i : 0<=i<n}.
  • A066977 (program): a(n) = gcd(prime(n^2) + 1, prime(n) + 1).
  • A066978 (program): a(n) = gcd(prime(2*n)+1, prime(n)+1).
  • A066979 (program): a(n) = floor(n!/2^n).
  • A066982 (program): a(n) = Lucas(n+1) - (n+1).
  • A066983 (program): a(n+2) = a(n+1) + a(n) + (-1)^n, with a(1) = a(2) = 1.
  • A066984 (program): a(n) = gcd(prime(n+1) - 1, prime(n) + 1).
  • A066989 (program): a(n) = (n!)^3 * Sum_{i=1..n} 1/i^3.
  • A066990 (program): In canonical prime factorization of n replace even exponents with 2 and odd exponents with 1.
  • A066991 (program): Square array read by antidiagonals of number of ways of dividing nk labeled items into k unlabeled orders with n items in each order.
  • A066997 (program): Survivor number for 2nd-order Josephus problem.
  • A066998 (program): a(0)=0; a(n) = n^2*a(n-1) + 1.
  • A066999 (program): a(n) = 3^n * Sum_{i=1..n} i^3/3^i.
  • A067002 (program): Numerator of Sum_{k=0..n} 2^(k-2*n) * binomial(2*n-2*k,n-k) * binomial(n+k,n).
  • A067016 (program): Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = max_i (a(i)+a(n-i)).
  • A067018 (program): Start with a(0)=1, a(1)=4, a(2)=3, a(3)=2; for n>=3, a(n+1) = mex_i (nim-sum a(i)+a(n-i)), where mex means smallest nonnegative missing number.
  • A067019 (program): Odd numbers with an odd number of prime factors (counted with multiplicity).
  • A067028 (program): Numbers with a composite number of prime factors (counted with multiplicity).
  • A067029 (program): Exponent of least prime factor in prime factorization of n, a(1)=0.
  • A067037 (program): a(n) = n^m where m = floor(Sum_{k=1..n} 1/k).
  • A067040 (program): a(n) = n^(sum of digits of n).
  • A067041 (program): a(n) = n^(product of digits of n).
  • A067044 (program): Smallest positive k such that k*n contains only even digits.
  • A067046 (program): a(n) = lcm(n, n+1, n+2)/6.
  • A067047 (program): a(n) = lcm(n, n+1, n+2, n+3)/12.
  • A067048 (program): a(n) = lcm(n, n+1, n+2, n+3, n+4) / 60.
  • A067053 (program): Floor[ Sum_{1..n} 1/i ]^n.
  • A067056 (program): a(n) = (1)*(2 + 3 + 4 + … + n) + (1 + 2)*(3 + 4 + 5 + … + n) + (1 + 2 + 3)*(4 + 5 + 6 + … + n) + … + (1 + 2 + 3 + … + n-1)*n.
  • A067057 (program): Let A(n) = {1,2,3,…n}. Let B(r) and C(n-r) be two subsets of A(n) having r and n-r elements respectively, such that B(r) U C(n-r) = A(n) and B and C are disjoint; then a(n) = sum of the products of all combination sums of elements of B and C for r =1 to n-1.
  • A067060 (program): A permutation of the positive integers in groups of four such that any two consecutive numbers differ by at least 2.
  • A067061 (program): A permutation of the natural numbers.
  • A067066 (program): Number of Gnutella users reachable with given connections and hops.
  • A067067 (program): Product of nonzero digits of n! (A000142).
  • A067068 (program): a(n) = n* - 2^n, where n* (A003418) = least common multiple of the numbers [1,…,n].
  • A067076 (program): Numbers k such that 2*k + 3 is a prime.
  • A067078 (program): a(1) = 1, a(2) = 2, a(n) = (n-1)*a(n-1) - (n-2)*a(n-2).
  • A067080 (program): If n = ab…def in decimal notation then the left digitorial function Ld(n) = ab…def*ab…de*ab…d*…*ab*a.
  • A067082 (program): If n = abc…def in decimal notation then the right digit sum function = abc…def + bc…def + c…def + … + def + ef + f.
  • A067085 (program): a(n) = floor(Sum_{k=1..n} 1/k^(1/2)).
  • A067087 (program): Concatenation of n-th prime and its reverse.
  • A067095 (program): a(n) = floor(X/Y) where X is the concatenation in increasing order of the first n even numbers and Y is that of the first n odd numbers.
  • A067096 (program): Floor[X/Y] where X = concatenation in increasing order of first n even numbers and Y = that of first n natural numbers.
  • A067102 (program): Floor[ X/Y] where X = concatenation of the squares and Y = concatenation of natural numbers.
  • A067103 (program): a(n) = floor(X/Y), where X = concatenation of cubes and Y = concatenation of natural numbers.
  • A067104 (program): a(n) = floor[ X/Y], where X = concatenation of first n factorials and Y = concatenation of first n natural numbers.
  • A067111 (program): Floor[ Product of first n primes / Sum of first n primes].
  • A067114 (program): Let N = 24681012141618202224262830…, the concatenation of the even numbers. Then a(n) = sum of first n digits of N.
  • A067116 (program): Floor(decimal concatenation of first n natural numbers/their sum).
  • A067117 (program): a(n) = floor((concatenation of first n natural numbers)/ (n!)).
  • A067119 (program): a(n) = floor[X/Y] where X = concatenation of first n even numbers in increasing order and Y = n-th triangular number.
  • A067120 (program): a(n) =floor[X/Y] where X= concatenation of first n ODD numbers in increasing order and Y = n-th triangular number.
  • A067121 (program): a(n) = floor[X/Y] where X = the concatenation of the first n even numbers in increasing order and Y = their sum.
  • A067122 (program): Floor[X/Y] where X = concatenation of first n odd numbers in increasing order (A019519) and Y = their sum (A000290 = n^2).
  • A067123 (program): Floor[X/Y] where X = concatenation of first n cubes in increasing order and Y = concatenation of first n squares.
  • A067124 (program): Floor [sqrt{concatenation of first n squares}].
  • A067126 (program): Numbers for which phi(n) >= phi(k) for all k = 1 to n-1.
  • A067132 (program): Number of elements in the largest set of divisors of n which are in geometric progression.
  • A067133 (program): n is a term if the phi(n) numbers in [0,n-1] and coprime to n form an arithmetic progression.
  • A067147 (program): Triangle of coefficients for expressing x^n in terms of Hermite polynomials.
  • A067148 (program): Fibonacci-like sequences. a(n) is the number of pairs of integers (n,i), 0<i<n, with Property F: i and n are consecutive terms i=b(j-1) and n=b(j), for some j>2, of a sequence {b(k)} satisfying b(1)=1, b(2)>0 and b(k)=b(k-1)+b(k-2) for all k>2.
  • A067161 (program): a(n) = prime(sigma(n)).
  • A067175 (program): Number of digits in the n-th primorial (A002110).
  • A067176 (program): A triangle of generalized Stirling numbers: sum of consecutive terms in the harmonic sequence multiplied by the product of their denominators.
  • A067186 (program): Numbers n such that C(n) = (n^2 + n + 2)/2 is prime.
  • A067187 (program): Numbers that can be expressed as the sum of two primes in exactly one way.
  • A067189 (program): Numbers that can be expressed as the sum of two primes in exactly three ways.
  • A067190 (program): Numbers that can be expressed as the sum of two primes in exactly four ways.
  • A067191 (program): Numbers that can be expressed as the sum of two primes in exactly five ways.
  • A067197 (program): Numbers k such that k*(k+1)/2 is not squarefree.
  • A067200 (program): Numbers n such that n^3 + 2 is prime.
  • A067201 (program): Numbers k such that k^2 + 2 is prime.
  • A067239 (program): a(0)=1, a(n) = 8n*(2n-1).
  • A067240 (program): If n = Product_{i} p_i^e_i, a(n) = Sum_{i} (p_i - 1)*p_i^(e_i - 1).
  • A067251 (program): Numbers with no trailing zeros in decimal representation.
  • A067259 (program): Cubefree numbers which are not squarefree.
  • A067268 (program): Numbers k such that k and k^2+1 have the same number of distinct prime factors.
  • A067272 (program): a(n) = 2*10^(n-1)-1.
  • A067273 (program): a(n) = n*(a(n-1)*2+1), a(0) = 0.
  • A067274 (program): Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of x^2+bx+c=0 are integers.
  • A067275 (program): Number of Fibonacci numbers A000045(k), k <= 10^n, which end in 4.
  • A067292 (program): a(n)=prime(n)-n*tau(n) where tau(n) is the number of divisors of n.
  • A067294 (program): Third column of triangle A028364.
  • A067295 (program): Fourth column of triangle A028364.
  • A067299 (program): Second column of triangle A067298.
  • A067315 (program): Central binomial coefficient C(n, n/2) is not divisible by n.
  • A067318 (program): Total number of transpositions in all permutations of n letters.
  • A067324 (program): Third column of triangle A067323.
  • A067325 (program): Fourth column of triangle A067323.
  • A067326 (program): Fifth column of triangle A067323.
  • A067331 (program): Convolution of Fibonacci F(n+1), n >= 0, with F(n+3), n >= 0.
  • A067332 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+4), n>=0.
  • A067333 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+5), n>=0.
  • A067334 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+6), n>=0.
  • A067336 (program): a(0)=1, a(1)=2, a(n) = a(n-1)*9/2 - Catalan(n-1) where Catalan(n) = binomial(2n,n)/(n+1) = A000108(n).
  • A067337 (program): Triangle where T(n,k)=2*T(n,k-1)+C(n-1,k)-C(n-1,k-1) and n>=k>=0.
  • A067342 (program): Sum of decimal digits of sum of divisors of n.
  • A067348 (program): Even numbers n such that binomial(n, [n/2]) is divisible by n.
  • A067349 (program): Number of divisors of sigma(n)+phi(n).
  • A067350 (program): Numbers n such that sigma(n)+phi(n) has exactly 4 divisors.
  • A067352 (program): Divide the natural numbers in sets of consecutive numbers starting with {1,2} as the first set. The number of elements of the n-th set is equal to the sum of the n-1 final numbers in the (n-1)st set. The number of elements of the n-th set gives a(n).
  • A067353 (program): Divide the natural numbers in sets of consecutive numbers starting with {1,2} as the first set. The number of elements of the n-th set is equal to the sum of the n-1 final numbers in the (n-1)st set. The final number of the n-th set gives a(n).
  • A067358 (program): Imaginary part of (5+12i)^n.
  • A067359 (program): Real part of (5 + 12i)^n.
  • A067360 (program): a(n) = 17^n sin(2n arctan(1/4)) or numerator of tan(2n arctan(1/4)).
  • A067361 (program): a(n) = 17^n*cos(2*n*arctan(1/4)) or denominator of tan(2*n*arctan(1/4)).
  • A067368 (program): a(n) is the smallest positive even integer that cannot be expressed as the product of two or three previous terms (not necessarily distinct).
  • A067369 (program): Weight of the alternating group (A_n) in transpositions.
  • A067370 (program): The weight of the periphery of the alternating group, denoted v(P_N).
  • A067371 (program): Arithmetic derivatives of 3-smooth numbers.
  • A067389 (program): a(n) = 3*n^3 + 2*n^2 + n.
  • A067391 (program): a(n) is the least common multiple of numbers in {1,2,3,…,n-1} which do not divide n.
  • A067392 (program): Sum of numbers <= n which have common prime factors with n.
  • A067395 (program): First differences of A067368.
  • A067396 (program): a(n) is the position of the n-th occurrence of the pair “2,2” in A067395 (the first difference sequence of A067368).
  • A067397 (program): Maximal power of 3 that divides n-th Catalan number.
  • A067402 (program): Triangle with columns built from certain power sequences.
  • A067403 (program): Third column of triangle A067402.
  • A067404 (program): Fourth column of triangle A067402.
  • A067405 (program): Fifth column of triangle A067402.
  • A067406 (program): Sixth column of triangle A067402.
  • A067407 (program): Seventh column of triangle A067402.
  • A067408 (program): Eighth column of triangle A067402.
  • A067409 (program): Ninth column of triangle A067402.
  • A067410 (program): Triangle with columns built from certain power sequences.
  • A067411 (program): Third column of triangle A067410 and second column of A067417.
  • A067412 (program): Fourth column of triangle A067410.
  • A067413 (program): Sixth column of triangle A067410.
  • A067414 (program): Seventh column of triangle A067410.
  • A067415 (program): Eighth column of triangle A067410.
  • A067416 (program): Ninth column of triangle A067410.
  • A067417 (program): Triangle with columns built from certain power sequences.
  • A067419 (program): Fourth column of triangle A067417.
  • A067420 (program): Fifth column of triangle A067417.
  • A067421 (program): Sixth column of triangle A067417.
  • A067422 (program): Seventh column of triangle A067417.
  • A067423 (program): Eighth column of triangle A067417.
  • A067424 (program): Ninth column of triangle A067417.
  • A067425 (program): Triangle with columns built from certain power sequences.
  • A067426 (program): Sixth column of triangle A067425.
  • A067427 (program): Seventh column of triangle A067425.
  • A067428 (program): Eighth column of triangle A067425.
  • A067429 (program): Ninth column of triangle A067425.
  • A067430 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+7), n>=0.
  • A067431 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+8), n>=0.
  • A067434 (program): Number of distinct prime factors in binomial(2*n,n).
  • A067435 (program): a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.
  • A067436 (program): a(n) = sum of all the remainders when n-th even number is divided by even numbers < 2n.
  • A067439 (program): a(n) = sum of all the remainders when n is divided by positive integers less than and coprime to n.
  • A067440 (program): Sum(i(n)), where i(n) is the smallest integer with i(n)^m=n for some m.
  • A067455 (program): n! divided by the product of the decimal digits of n.
  • A067456 (program): Floor( sqrt( sum of the decimal digits of n squared)).
  • A067459 (program): Sum of the remainders when n^2 is divided by squares less than n.
  • A067460 (program): mu(prime(n)-1)+1.
  • A067461 (program): mu(prime(n)+1)+1.
  • A067462 (program): a(n) = (1! + 2! + … + (n-1)!) mod n.
  • A067466 (program): Primes p such that p-1 has 2 distinct prime factors.
  • A067467 (program): Primes p such that p-1 has 3 distinct prime factors.
  • A067469 (program): Numbers k such that 2 is the first digit of 2^k.
  • A067470 (program): Smallest n-digit n-th power.
  • A067471 (program): n-th root of A067470(n).
  • A067472 (program): Smallest n-digit square starting with 2.
  • A067473 (program): Smallest n-digit square starting with 3.
  • A067474 (program): Smallest n-digit square starting with 4.
  • A067475 (program): Smallest n-digit square starting with 5.
  • A067476 (program): Smallest n-digit square starting with 6.
  • A067477 (program): Smallest n-digit square starting with 7.
  • A067478 (program): Smallest n-digit square starting with 8.
  • A067480 (program): Powers of 2 with initial digit 2.
  • A067481 (program): Powers of 3 with initial digit 3.
  • A067482 (program): Powers of 4 with initial digit 4.
  • A067488 (program): Powers of 2 with initial digit 1.
  • A067490 (program): Powers of 4 with initial digit 1.
  • A067491 (program): Powers of 5 with initial digit 1.
  • A067495 (program): Powers of 9 having initial digit 1.
  • A067497 (program): Smallest power of 2 with n+1 digits (n>=0). Also numbers k such that 1 is the first digit of 2^k.
  • A067498 (program): Maximum number of reflections for a ray of light which reflects at n points (reflecting more than once at most or all points).
  • A067499 (program): Powers of 2 with digit sum also a power of 2.
  • A067500 (program): Powers of 3 with digit sum also a power of 3.
  • A067507 (program): Powers of 2 with even digit sum.
  • A067510 (program): Powers of 6 with digit sum divisible by 6.
  • A067513 (program): Number of divisors d of n such that d+1 is prime.
  • A067514 (program): Number of distinct primes of the form floor(n/k) for 1 <= k <= n.
  • A067515 (program): Fibonacci numbers with index = digit sum.
  • A067520 (program): Triangular numbers whose index is a multiple of the sum of the digits.
  • A067521 (program): Numbers n such that the square root of n is an integer and a multiple of the sum of the digits of n.
  • A067525 (program): Define I(n) = number obtained by incrementing each digit from 0 to 8 of n by 1. A ‘9’ is replaced by a ‘0’. Sequence gives digitriangular numbers n*I(n)/2.
  • A067531 (program): Numbers n such that n - number of divisors of n is a prime.
  • A067532 (program): Numbers n such that n + number of divisors is a prime.
  • A067534 (program): a(n) = 4^n * sum_{i=1,n} i^4/4^i.
  • A067535 (program): Smallest squarefree number >= n.
  • A067541 (program): phi(n*(n+1)/2)/phi(n) where phi is the Euler totient function A000010(n).
  • A067546 (program): Determinant of an n X n matrix whose diagonal are the first n nonprime numbers and all other elements are 1’s.
  • A067550 (program): a(n) = (n-1)!(n+2)!/(3*2^n).
  • A067552 (program): a(n) = SumOfDigits(n)^2 - SumOfDigits(n^2), where SumOfDigits = A007953.
  • A067556 (program): Terms in the decimal expansion of 1/(7*2^n) before the block of decimals 142857 (the period of 1/7) appears.
  • A067558 (program): Sum of squares of proper divisors of n.
  • A067563 (program): Product of n-th prime number and n-th composite number.
  • A067574 (program): Array T(i,j) read by ascending antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).
  • A067585 (program): Binary representation of a(n) is obtained thus: replace every digit in the binary representation of n with “1” if the sum of its neighbors is 1 and with “0” otherwise.
  • A067586 (program): Number of 0’s in the binary expansion of A066884(n+1).
  • A067589 (program): Numbers k such that A067588(k) is an odd number.
  • A067590 (program): Number of partitions of n into odious numbers (A000069).
  • A067591 (program): Number of partitions of n into evil numbers (A001969).
  • A067602 (program): 5^n reduced modulo 3^n.
  • A067611 (program): Numbers of the form 6xy +- x +- y, where x, y are positive integers.
  • A067612 (program): Numbers n such that sigma(n) = 3*phi(sigma(n)).
  • A067614 (program): a(n) is the second partial quotient in the simple continued fraction for sqrt(prime(n)).
  • A067621 (program): Let t = coefficient of x^(2n+1) in expansion of sin(x)/(1-x^2); a(n)=denominator(t)-numerator(t).
  • A067622 (program): Consider the power series (x + 1)^(1/3) = 1 + x/3-x^2/9 + 5x^3/81 + …; sequence gives numerators of coefficients.
  • A067623 (program): Consider the power series (x+1)^(1/3)=1+x/3-x^2/9+5x^3/81+…; sequence gives denominators of coefficients.
  • A067624 (program): a(n) = 2^(2*n)*(2*n)!.
  • A067626 (program): a(n) = 2^(2n+1)*(2n+1)!.
  • A067628 (program): Minimal perimeter of polyiamond with n triangles.
  • A067630 (program): Denominators in power series for cos(x)*cosh(x).
  • A067634 (program): a(1) = 1; string of digits of a(n)^2 is a substring of the string of digits of a(n+1)^2.
  • A067636 (program): Row 1 of table in A067640.
  • A067653 (program): Denominators of the coefficients in exp(x/(1-x)) power series.
  • A067654 (program): Numerators of the coefficients in power series expansion of exp(2x/(1-x)).
  • A067655 (program): Denominators of the coefficients in exp(2x/(1-x)) power series.
  • A067656 (program): Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.
  • A067659 (program): Number of partitions of n into distinct parts such that number of parts is odd.
  • A067661 (program): Number of partitions of n into distinct parts such that number of parts is even.
  • A067666 (program): Sum of squares of prime factors of n (counted with multiplicity).
  • A067683 (program): Phi(n)*sigma(n)+1 is prime.
  • A067687 (program): Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).
  • A067692 (program): a(n) = Sum_{0 < d <= t <= n, d|n, t|n} d*t.
  • A067699 (program): Number of comparisons made in a version of the sorting algorithm QuickSort for an array of size n with n identical elements.
  • A067703 (program): Terms in the decimal expansion of 1/(7*5^n) before the block of decimals 142857 (the period of 1/7) appears.
  • A067705 (program): a(n) = 11*n^2 + 22*n.
  • A067707 (program): a(n) = 3*n^2 + 12*n.
  • A067722 (program): Least positive integer k such that n*(n + k) is a perfect square.
  • A067724 (program): a(n) = 5*n^2 + 10*n.
  • A067725 (program): a(n) = 3*n^2 + 6*n.
  • A067726 (program): a(n) = 6*n^2 + 12*n.
  • A067727 (program): a(n) = 7*n^2 + 14*n.
  • A067728 (program): a(n) = 2*n^2 + 8*n.
  • A067731 (program): Maximum number of distinct parts in a self-conjugate partition of n, or 0 if n=2.
  • A067736 (program): Decimal expansion of exp(3/2).
  • A067742 (program): Number of middle divisors of n, i.e., divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).
  • A067743 (program): Number of divisors of n not in the half-open interval [sqrt(n/2), sqrt(n*2)).
  • A067745 (program): Numerator of ((3*n - 2)/(n^(2*n - 1)*(2*n - 1)*4^(n - 1))).
  • A067749 (program): Numbers k such that k and 3^k end with the same two digits.
  • A067755 (program): Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.
  • A067756 (program): Prime hypotenuses of Pythagorean triangles with a prime leg.
  • A067760 (program): a(n) = least positive k such that (2n+1)+2^k is prime, or 0 if no such k exists.
  • A067761 (program): Positive integers divisible by 5 but not by 7.
  • A067762 (program): Numbers n such that bigomega(sigma(n))=bigomega(n).
  • A067763 (program): Square array read by antidiagonals of base n numbers written as 122…222 with k 2’s (and a suitable interpretation for n=0, 1 or 2).
  • A067764 (program): Numerators of the coefficients in exp(x/(1-x)) power series.
  • A067770 (program): a(n) = Catalan(n) mod (n+2).
  • A067771 (program): Number of vertices in Sierpiński triangle of order n.
  • A067774 (program): Primes p such that p+2 is not a prime.
  • A067775 (program): Primes p such that p + 4 is composite.
  • A067782 (program): Minimal delay time for an n-element sorting network.
  • A067792 (program): a(n) is the least prime >= sigma(n).
  • A067800 (program): Nonprime n such that phi(n) > n/2.
  • A067802 (program): Triangle with T(n,k)=C(2n+1,n-k)^2*(2k+1)/(2n+1).
  • A067804 (program): Triangle read by rows: T(n,k) is the number of walks (each step +-1) of length 2n which have a cumulative value of 0 last at step 2k.
  • A067812 (program): Nonprime n such that 2n+1 is prime.
  • A067815 (program): a(n) = gcd(n, floor(sqrt(n))).
  • A067819 (program): Sum of the divisors of binomial(2n,n).
  • A067824 (program): a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).
  • A067829 (program): Primes p such that sigma(p-2) < p.
  • A067830 (program): Primes p such that sigma(p-4) < p.
  • A067831 (program): Primes p such that sigma(p-6) < p.
  • A067832 (program): Primes p such that sigma(p-6) > p.
  • A067833 (program): Primes p such that sigma(p-4) > p.
  • A067842 (program): Expansion of 1/Product_{k=1..infinity} (1-x^A007097(k)).
  • A067844 (program): Numbers k such that k and 2^k end with the same digit.
  • A067849 (program): a(n) = max{k: f(n),…,f^k(n) are prime}, where f(m) = 2m+1 and f^k denotes composition of f with itself k times.
  • A067850 (program): Highest power of 2 not exceeding n!.
  • A067856 (program): Sum_{n >= 1} a(n)/n^s = 1/(Sum_{n >= 1} (-1)^(n + 1)/n^s).
  • A067862 (program): Numbers k that divide the sum of digits of 3^k.
  • A067865 (program): Numbers n such that n and 2^n end with the same two digits.
  • A067866 (program): Numbers n such that n and 2^n end with the same three digits.
  • A067867 (program): Numbers n such that n and 2^n end with the same 4 digits.
  • A067868 (program): a(n) = a(n-1) + a(floor(n/2))^2 for n > 0, a(0) = 1.
  • A067869 (program): Numbers n such that n and 2^n end with the same 5 digits.
  • A067870 (program): Numbers k such that k and 3^k end with the same digit.
  • A067872 (program): Least m > 0 for which m*n^2 + 1 is a square.
  • A067874 (program): Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.
  • A067894 (program): Write 0, 1, …, n in binary and add as if they were decimal numbers.
  • A067895 (program): Write 2^n, 2^n+1, 2^n+2, …, 2^(n+1)-1 in binary and add as if they were decimal numbers.
  • A067896 (program): Trajectory of 41 under map x -> x/2 if x even, x-> 3x+3 if x odd.
  • A067897 (program): a(n) = A000085(n) - (1 + Sum_{j=1..n-1} A000085(j)).
  • A067900 (program): a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8.
  • A067902 (program): a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.
  • A067911 (program): Product of gcd(k,n) for 1 <= k <= n.
  • A067934 (program): Let rep(k) = (10^k - 1)/9 be the k-th repunit number = 11111..1111 with k 1 digits, then sequence gives values of k such that rep(k) == 1 (mod k).
  • A067935 (program): Let rep(n) be the n-th repunit number, sequence gives values of n such that : rep(n)==rep(2) (mod n).
  • A067936 (program): Let rep(n) be the n-th repunit number, sequence gives values of n such that : rep(n) == rep(3) (mod n).
  • A067946 (program): Numbers k that divide 5^k - 1.
  • A067947 (program): Numbers k such that k divides 7^k - 1.
  • A067955 (program): Number of dissections of a convex polygon by nonintersecting diagonals into polygons with even number of sides and having a total number of n edges (sides and diagonals).
  • A067956 (program): Number of nodes in virtual, “optimal”, chordal graphs of diameter 4 and degree n+1.
  • A067961 (program): Number of binary arrangements without adjacent 1’s on n X n torus connected n-s.
  • A067962 (program): a(n) = F(n+2)*(Product_{i=1..n+1} F(i))^2 where F(i)=A000045(i) is the i-th Fibonacci number.
  • A067966 (program): Number of binary arrangements without adjacent 1’s on n X n array connected n-s.
  • A067969 (program): Number of nodes in virtual, “optimal”, chordal graphs of diameter 5, degree =n+1.
  • A067970 (program): First differences of A014076, the odd nonprimes.
  • A067977 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+9), n>=0.
  • A067978 (program): Convolution of Fibonacci F(n+1), n>=0, with F(n+10), n>=0.
  • A067980 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+2), n>=0.
  • A067981 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+3), n>=0.
  • A067982 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+4), n>=0.
  • A067983 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+5), n>=0.
  • A067984 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+6), n>=0.
  • A067985 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+7), n>=0.
  • A067986 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+8), n>=0.
  • A067987 (program): Convolution of L(n+1) := A000204(n+1) (Lucas), n>=0, with L(n+9), n>=0.
  • A067988 (program): Row sums of triangle A067330; also of triangle A067418.
  • A067989 (program): Row sums of triangle A067979; also of triangle A067990.
  • A067994 (program): Hermite numbers.
  • A067998 (program): a(n) = n^2 - 2*n.
  • A068010 (program): Number of subsets of {1,2,3,…,n} that sum to 0 mod 3.
  • A068011 (program): Number of subsets of {1,2,3,…,n} that sum to 0 mod 5.
  • A068012 (program): Number of subsets of {1,2,3,…,n} that sum to 0 mod 6.
  • A068015 (program): Gaps between non-twin primes.
  • A068018 (program): Number of fixed points in all 132- and 213-avoiding permutations of {1,2,…,n} (these are permutations with runs consisting of consecutive integers).
  • A068028 (program): Decimal expansion of 22/7.
  • A068031 (program): Number of subsets of {1,2,3,…,n} that sum to 0 mod 10.
  • A068033 (program): Number of subsets of {1,2,3,…,n} that sum to 0 mod 12.
  • A068037 (program): Number of subsets of {1,2,3,…,n} that sum to 0 mod 16.
  • A068049 (program): The first term greater than one on each row of A068009. a(n) = A068009[n, A002024[n]].
  • A068050 (program): Number of values of k, 1<=k<=n, for which floor(n/k) is prime.
  • A068061 (program): Palindromic numbers j that are not of the form k + reverse(k) for any k.
  • A068067 (program): Number of integers m, 0 < m <= n, such that n divides m(m+1)/2.
  • A068068 (program): Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.
  • A068073 (program): Period 4 sequence [ 1, 2, 3, 2, …].
  • A068074 (program): a(n) = Sum_{d|n} (-1)^d*2^omega(n/d) where omega(x) is the number of distinct prime factors in the factorization of x.
  • A068076 (program): Number of positive integers < n with the same number of 1’s in their binary expansions as n.
  • A068079 (program): Decimal expansion of 355 / 113.
  • A068080 (program): Integers n such that n + phi(n) is a prime.
  • A068082 (program): a(1) = 1, a(n) = smallest triangular number of the form k*a(n-1) + 1 for some positive integer k.
  • A068083 (program): a(1) = 1, a(n) is the smallest Fibonacci number of the form k*a(n-1) + 1 with k>0.
  • A068085 (program): Numbers k such that k and 10*k are both triangular numbers.
  • A068087 (program): a(n) = n^(2*n-2).
  • A068092 (program): Index of smallest triangular number with n digits.
  • A068093 (program): Smallest n-digit triangular number.
  • A068094 (program): Number of n-digit triangular numbers.
  • A068096 (program): a(n) = F(L(n)) where F(n) = n-th Fibonacci number and L(n) = n-th Lucas number.
  • A068098 (program): a(n) = Lucas(Fibonacci(n)).
  • A068102 (program): a(n) = n! * 2^n * Sum_{i=1..n} 1/(i*2^i).
  • A068106 (program): Euler’s difference table: triangle read by rows, formed by starting with factorial numbers (A000142) and repeatedly taking differences. T(n,n) = n!, T(n,k) = T(n,k+1) - T(n-1,k).
  • A068107 (program): a(n) = n!*Sum_{k=1..n} mu(k)/k!, where mu(k) is the Moebius function.
  • A068108 (program): a(1) = 1; a(n+1) = sum{k|n k<=sqrt(n)} a(k) where sum is over the positive divisors k of n with k <= sqrt(n).
  • A068110 (program): Denominators of coefficients in J0(i*sqrt(x))^2 power series where J0 denotes the ordinary Bessel function of order 0.
  • A068111 (program): Numerators of coefficients in J0(i*sqrt(x))^2, where J0 denotes the ordinary Bessel function of order 0.
  • A068113 (program): Numerator of coefficient of (-x^2)^n in F(x)*F(-x) where F(x)=sum(k>=0,x^k/(k!)^3).
  • A068129 (program): Triangular numbers with sum of digits = 10.
  • A068130 (program): Triangular numbers with sum of digits = 15.
  • A068131 (program): Triangular numbers with sum of digits = 21.
  • A068156 (program): G.f.: (x+2)*(x+1)/((x-1)*(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.
  • A068158 (program): a(n) = floor[ n!/(R(n))! ], where R(n) = Digit reversal of n (A004086).
  • A068159 (program): a(n) = floor[ n/R(n) ], where R(n) (A004086) = Digit reversal of n.
  • A068179 (program): Product_{i=1..3} (i+x) / Product_(i=1..3} (i-x) = Sum_{n>=0} (a(n)/b(n))*x^n.
  • A068181 (program): a(n)=-1/b(2n) where 1/(e^y-e^(y/3))= sum(i=-1,inf,b(i)*y^i).
  • A068191 (program): Numbers n such that A067734(n)=0; complement of A002473; at least one prime-factor of n is larger than 7, it has 2 decimal digits.
  • A068199 (program): One of a family of sequences that interpolates between the Bell numbers and the factorials.
  • A068203 (program): Chebyshev T-polynomials T(n,15) with Diophantine property.
  • A068204 (program): Let (x_n, y_n) be n-th solution to the Pell equation x^2 = 14*y^2 + 1. Sequence gives {y_n}.
  • A068205 (program): Denominator of S(n)/Pi^n, where S(n) = Sum((4k+1)^(-n),k,-inf,+inf).
  • A068211 (program): Largest prime factor of Euler totient function phi(n).
  • A068212 (program): a(n) = phi(n) divided by its largest prime factor.
  • A068217 (program): Denominators of coefficients in 3*log(sqrt(1+x)) power series.
  • A068219 (program): Denominators of coefficients in log(1+x)*(1+x)^(1/3) power series.
  • A068227 (program): The “genity” sequence of the primes, i.e., a(n) = g(p) = ((p mod 4) + (p mod 6))/2, where p is the n-th prime.
  • A068228 (program): Primes congruent to 1 (mod 12).
  • A068229 (program): Primes congruent to 7 (mod 12).
  • A068231 (program): Primes congruent to 11 mod 12.
  • A068236 (program): First differences of (n+1)^5-n^5.
  • A068237 (program): Numerators of arithmetic derivative of 1/n: -A003415(n)/n^2.
  • A068238 (program): Denominators of arithmetic derivative of 1/n: -A003415(n)/n^2.
  • A068239 (program): 1/2 the number of colorings of a 3 X 3 square array with n colors.
  • A068244 (program): 1/6 the number of colorings of a 3 X 3 rhombic- or staggered- hexagonal array with n colors.
  • A068250 (program): 1/24 the number of colorings of a 3 X 3 octagonal array with n colors.
  • A068293 (program): a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).
  • A068310 (program): n^2 - 1 divided by its largest square divisor.
  • A068311 (program): Arithmetic derivative of n!.
  • A068312 (program): Arithmetic derivative of triangular numbers.
  • A068318 (program): Sum of prime factors of n-th semiprime.
  • A068319 (program): a(n) = if n <= lpf(n)^2 then lpf(n) else a(lpf(n) + n/lpf(n)), where lpf = least prime factor, A020639.
  • A068327 (program): Arithmetic derivative of n^n.
  • A068328 (program): Arithmetic derivative of squarefree numbers.
  • A068333 (program): Product(n/k - k) where the product is over the divisors k of n and where 1 <= k <= sqrt(n).
  • A068336 (program): a(1) = 1; a(n+1) = 1 + sum{k|n} a(k), sum is over the positive divisors, k, of n.
  • A068340 (program): Sum_{k=1..n} mu(k)*k, where mu(k) is the Moebius function.
  • A068344 (program): Square array read by antidiagonals of T(n,k) = sign(n-k).
  • A068346 (program): a(n) = n’’ = second arithmetic derivative of n.
  • A068354 (program): Numbers n such that sigma(n)*tau(n)>prime(2*n) where sigma(n) is the sum of divisors of n and tau(n) the number of divisors of n = A000005(n).
  • A068377 (program): Engel expansion of sinh(1).
  • A068379 (program): Engel expansion of sinh(1/2).
  • A068380 (program): Engel expansion of sinh(1/3).
  • A068383 (program): Numbers k such that k divides 11^k - 1.
  • A068389 (program): Differences between the primes generating the n-th prime power.
  • A068395 (program): a(n) = n-th prime minus its sum of digits.
  • A068396 (program): n-th prime minus its reversal.
  • A068397 (program): a(n) = Lucas(n) + (-1)^n + 1.
  • A068398 (program): Number of digits in (2^n)*(n!).
  • A068406 (program): Numbers n such that n and 2n+1 have the same number of prime divisors.
  • A068409 (program): a(n) = binomial(binomial(2*n,n),n).
  • A068424 (program): Triangle of falling factorials, read by rows: T(n, k) = n*(n-1)*…*(n-k+1), n > 0, 1 <= k <= n.
  • A068425 (program): a(n) = floor(2^n*Pi).
  • A068426 (program): Expansion of log(2) in base 2.
  • A068427 (program): Expansion of zeta(2) in base 2.
  • A068432 (program): Expansion of golden ratio (1 + sqrt(5))/2 in base 2.
  • A068433 (program): Expansion of log(3) in base 2.
  • A068434 (program): Expansion of log(5) in base 2.
  • A068443 (program): Triangular numbers which are the product of two primes.
  • A068444 (program): a(0) = 10; for n>0, a(n) = n*a(n-1)-n-2.
  • A068475 (program): a(n) = Sum_{m=0..n} m*n^(m-1).
  • A068477 (program): a(n) is the digital sum of 1^n + 2^n + … + n^n.
  • A068485 (program): One-sixtieth of the even leg of Pythagorean triangles whose other sides are both primes (other than 3, 5 or 13).
  • A068486 (program): Smallest prime equal to n^2 + m^2 with n >= m.
  • A068491 (program): Expansion of Molien series for a certain 4-D group of order 96.
  • A068494 (program): a(n) = n mod phi(n).
  • A068496 (program): n! reduced mod 2^n.
  • A068497 (program): Primes p such that 2*p+1 and 2*p-1 are composites.
  • A068499 (program): Numbers m such that m! reduced modulo (m+1) is not zero.
  • A068501 (program): Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.
  • A068503 (program): Highest power of 3 dividing prime(n)-1.
  • A068504 (program): Highest power of 2 dividing prime(n)+1.
  • A068508 (program): a(n) = round((a(n-1) + a(n-2))/a(n-3)) starting with a(1)=a(2)=a(3)=1.
  • A068509 (program): a(n) = maximum length of a subset in {1,..,n} whose integers have pairwise LCM not exceeding n.
  • A068510 (program): a(n) = lcm(1,…,n) - (product of primes <= n).
  • A068511 (program): (Product of primes <= n) - 2^(n-1).
  • A068512 (program): Numerators of arithmetic derivative of n/2: A003415(n)/2 - n/4; denominators: A010685.
  • A068519 (program): If n is prime then a(n) = n, else a(n) = prime(n).
  • A068522 (program): In base 10 notation replace digits of n with their squared values (Version 2).
  • A068527 (program): Difference between smallest square >= n and n.
  • A068548 (program): Coefficients of (-x^(2n-6)) in Chebyshev polynomial of degree 2n.
  • A068550 (program): a(n) = lcm{1, …, 2n} / binomial(2n, n).
  • A068551 (program): a(n) = 4^n - binomial(2*n,n).
  • A068552 (program): 2n*binomial(2n,n) - 4^n.
  • A068553 (program): a(n) = lcm(1,2,…,2n) / (n*binomial(2n, n)).
  • A068554 (program): a(n) = n*binomial(2n, n) - 4^n.
  • A068555 (program): Triangle read by rows in which row n contains (2i)!*(2j)!/(i!*j!*(i+j)!) for i+j=n, i=0..n.
  • A068561 (program): Numerators of coefficients in (1+x)^(1/3)-(1-x)^(1/3) power series.
  • A068562 (program): Denominators of coefficients in (1+x)^(1/3)-(1-x)^(1/3) power series.
  • A068563 (program): Numbers n such that 2^n == 4^n (mod n).
  • A068565 (program): Denominators of Sum_{k=1..n} 1/(k * 2^k).
  • A068566 (program): Numerator of Sum_{k=1..n} 1/(k * 2^k).
  • A068601 (program): a(n) = n^3 - 1.
  • A068605 (program): Number of functions from [1,2,…,n] to [1,2,…,n] such that the image contains exactly two elements.
  • A068606 (program): Square table by antidiagonals of T(n,k)=n*k*(n+k+1).
  • A068607 (program): Triangle of T(n,k)=n*k*(n+k+1) with n>=k>=0.
  • A068625 (program): Reduced root factorial of n: product of the smallest integer root of numbers from 1 to n.
  • A068626 (program): a(3n) = a(3n-1) = 3*n^2, a(3n-2) = 3*n^2 - 3*n + 1.
  • A068628 (program): Numbers occurring twice in A068627.
  • A068634 (program): a(n) = lcm(n, R(n)), where R(n) (A004086) = digit reversal of n.
  • A068636 (program): a(n) = Min(n, R(n)), where R(n) (A004086) = digit reversal of n.
  • A068637 (program): a(n) = Max(n, R(n)), where R(n) (A004086) = digit reversal of n.
  • A068639 (program): a(0) = 0, a(n) = a(n-1) + (-1)^p(n) for n >= 1, where p(n) = highest power of 2 dividing n.
  • A068640 (program): Define f(n) = 2n+1, a(n) = largest prime of the form f(f(f(…(n))). If no such prime exists then a(n) = 0.
  • A068657 (program): Successive left concatenation of floor(k/2) beginning with n until we reach 1.
  • A068670 (program): Number of digits in the concatenation of first n primes.
  • A068690 (program): Primes such that all digits (except perhaps the least significant digit) are even.
  • A068704 (program): a(n) = smallest prime obtained as the concatenation of n^k, n^(k-1), n^(k-2), …, n^2, n, 1 for some k >= 1; or 0 if no such prime exists.
  • A068719 (program): Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).
  • A068720 (program): Arithmetic derivative of squares: a(n) = 2*n*A003415(n).
  • A068721 (program): Arithmetic derivative of cubes: a(n) = 3*n^2*A003415(n).
  • A068722 (program): Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n.
  • A068762 (program): Alternating sum sigma(1)-sigma(2)+sigma(3)-sigma(4)+…+((-1)^(n+1))*sigma(n).
  • A068764 (program): Generalized Catalan numbers.
  • A068765 (program): Generalized Catalan numbers.
  • A068766 (program): Generalized Catalan numbers.
  • A068767 (program): Generalized Catalan numbers.
  • A068768 (program): Generalized Catalan numbers.
  • A068769 (program): Generalized Catalan numbers.
  • A068770 (program): Generalized Catalan numbers.
  • A068771 (program): Generalized Catalan numbers.
  • A068772 (program): Generalized Catalan numbers.
  • A068773 (program): Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + … + ((-1)^(n+1))*phi(n).
  • A068780 (program): Composite numbers n such that n+1 is also composite.
  • A068781 (program): Lesser of two consecutive numbers each divisible by a square.
  • A068792 (program): a(n) = (n-1)*n^(n-2) + Sum_{i=1..n} (n-i)*(n^(n-i-1) + n^(n+i-3)).
  • A068794 (program): In prime factorization of n replace all primes with the least prime factor of n; a(1)=1.
  • A068795 (program): In prime factorization of n replace all primes with the greatest prime factor of n; a(1)=1.
  • A068811 (program): Numbers n such that n and its 10’s complement are both primes, i.e., n and 10^k - n where k is the number of digits in n, are primes.
  • A068819 (program): n!/((n+1)*(n+2)*…*(n+k)) where k is largest value that gives an integer quotient.
  • A068822 (program): a(n) = gcd(n,c(n)), where c(n) is the 10’s complement of n.
  • A068823 (program): a(n) = lcm(n, c(n)), where c(n) is the 10’s complement of n.
  • A068824 (program): a(n) = n*c(n), where c(n) is the 10’s complement of n.
  • A068828 (program): Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).
  • A068866 (program): Numbers n such that A068865(n) = n.
  • A068869 (program): Smallest number k such that n! + k is a square.
  • A068875 (program): Expansion of (1 + x*C)*C, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108.
  • A068878 (program): sin(x)+exp(x)-1-2x=sum(n>=1,x^(2n)/a(n)).
  • A068900 (program): Digit reversal of n = 11 times digit reversal of n/11.
  • A068901 (program): Least number that when added to the n-th prime gives a multiple of n.
  • A068902 (program): Least multiple of n not less than the n-th prime.
  • A068903 (program): Binomial(tau(n),omega(n)), where tau(n) is the number of divisors of n (A000005) and omega the number of distinct prime factors (A001221).
  • A068904 (program): a(n) = binomial(sigma(n),tau(n)), where sigma(n) is the sum and tau(n) the number of divisors of n (A000203, A000005).
  • A068905 (program): Binomial(sigma(n),omega(n)), where sigma(n) is the sum of divisors of n (A000203) and omega the number of distinct prime factors (A001221).
  • A068906 (program): Square array read by ascending antidiagonals of partitions of k modulo n.
  • A068907 (program): Number of partitions of n modulo 3.
  • A068908 (program): Number of partitions of n modulo 5.
  • A068909 (program): Number of partitions of n modulo 7.
  • A068911 (program): Number of n step walks (each step +-1 starting from 0) which are never more than 2 or less than -2.
  • A068912 (program): Number of n step walks (each step +/-1 starting from 0) which are never more than 3 or less than -3.
  • A068915 (program): a(n) = n if n<2; a(n) = |a(n/2)-a(n/2-1)| if n is even, and a(n) = a((n-1)/2) + a((n-1)/2+1) if n is odd.
  • A068917 (program): Numerators of coefficients in 1/sin(x) - 1/sinh(x) power series.
  • A068921 (program): Number of ways to tile a 2 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.
  • A068922 (program): Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.
  • A068924 (program): Number of ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.
  • A068928 (program): Number of incongruent ways to tile a 3 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.
  • A068938 (program): Numbers having the sum of distinct prime factors greater than the sum of exponents in prime factorization, A008472(n)>A001222(n).
  • A068951 (program): Scan the primes, record digit-sum if it is itself prime.
  • A068952 (program): Squares in A068949.
  • A068953 (program): Number of bases B (2 <= B <= n) such that every digit of n in base B is 0 or 1.
  • A068958 (program): Smallest value of k such that p | (10^k + 1), where p is the n-th prime; or 0 if no such k exists.
  • A068961 (program): Powers of 2 with exactly two 2’s in their decimal digits.
  • A068963 (program): a(n) = Sum_{d|n} phi(d^3).
  • A068970 (program): a(n) = Sum_{d|n} phi(d^4).
  • A068976 (program): a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.
  • A068980 (program): Number of partitions of n into nonzero tetrahedral numbers.
  • A068981 (program): Arithmetic derivative of n*prime(n).
  • A068983 (program): a(n) = Sum_{k=0..n} (k^k-k!).
  • A068984 (program): a(n) = Sum_{d|n} d*tau(d)^2.
  • A068985 (program): Decimal expansion of 1/e.
  • A068989 (program): Squares which when reversed are primes (ignore leading zeros).
  • A068993 (program): Numbers k such that A062799(k) = 4.
  • A068995 (program): Integer parts of the square roots of the schizophrenic numbers (A014824).
  • A068996 (program): Decimal expansion of 1 - 1/e.
  • A069003 (program): Smallest integer d such that n^2 + d^2 is a prime number.
  • A069005 (program): Let M = 4 X 4 matrix with rows /1,1,1,1/1,1,1,0/1,1,0,0/1,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n)) = M^n*A where A is the vector (1,1,1,1); then a(n)=z(n).
  • A069006 (program): Let M denote the 5 X 5 matrix with rows /1,1,1,1,1/1,1,1,1,0/1,1,1,0,0/1,1,0,0,0/1,0,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n),u(n)) = M^n*A where A is the vector (1,1,1,1,1); then a(n) = t(n).
  • A069007 (program): Let M denote the 6 X 6 matrix with rows /1,1,1,1,1,1/1,1,1,1,1,0/1,1,1,1,0,0/1,1,1,0,0,0/1,1,0,0,0,0/1,0,0,0,0,0/ and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = y(n).
  • A069008 (program): Let M denote the 6 X 6 matrix with rows /1,1,1,1,1,1/1,1,1,1,1,0/1,1,1,1,0,0/1,1,1,0,0,0/1,1,0,0,0,0/1,0,0,0,0,0/ and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = z(n).
  • A069009 (program): Let M denote the 6 X 6 matrix with rows / 1,1,1,1,1,1 / 1,1,1,1,1,0 / 1,1,1,1,0,0 / 1,1,1,0,0,0 / 1,1,0,0,0,0 / 1,0,0,0,0,0 / and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = t(n).
  • A069010 (program): Number of runs of 1’s in the binary representation of n.
  • A069011 (program): Triangle with T(n,k) = n^2 + k^2.
  • A069013 (program): a(1)=a(2)=a(3)=1; for n > 3, a(n) = floor(a(n-3) + a(n-2)/a(n-1)).
  • A069015 (program): a(n) = n! * 3^n * Sum_{i=1..n} 1/(i * 3^i).
  • A069017 (program): Triangular numbers of the form k^2 + k + 1.
  • A069027 (program): Powers of 2 with strictly increasing sum of digits.
  • A069029 (program): Powers of 4 with strictly increasing sum of digits.
  • A069036 (program): Smallest multiple of 2 with digit sum 2^n.
  • A069038 (program): Expansion of x*(1+x)^4/(1-x)^6.
  • A069039 (program): Expansion of x(1+x)^5/(1-x)^7.
  • A069041 (program): Fibonacci numbers with at most two distinct digits.
  • A069045 (program): Denominator(sum(i=1,n,1/i^4))/denominator(sum(i=1,n,1/i)).
  • A069052 (program): Denominator of Sum_{i = 1..n} 1/i^5.
  • A069059 (program): Numbers k such that k and sigma(k) are not relatively prime.
  • A069071 (program): a(n) = (2n+1) * ((2n+1)^4 + 4).
  • A069072 (program): a(n) = (2n+1)*(2n+2)*(2n+3).
  • A069073 (program): a(n) = n*(4n^2 - 1)^2.
  • A069074 (program): a(n) = (2*n+2)*(2*n+3)*(2*n+4) = 24*A000330(n+1).
  • A069075 (program): a(n) = (4*n^2 - 1)^2.
  • A069076 (program): a(n) = (4*n^2 - 1)^3.
  • A069078 (program): a(n) = n*(4*n^4 + 1).
  • A069079 (program): a(n) = (2n+1)(2n+2)(2n+4)(2n+5).
  • A069080 (program): a(n) = (2n+1)*(2n+2)*(2n+6)*(2n+7).
  • A069081 (program): Numbers n such that sigma(n)/tau(n) has denominator 2.
  • A069082 (program): Numbers n such that sigma(n)/tau(n) has denominator 3.
  • A069088 (program): a(n) = Sum_{d|n} core(d) where d are the divisors of n and where core(d) is the squarefree part of d: the smallest number such that d*core(d) is a square.
  • A069091 (program): Jordan function J_6(n).
  • A069092 (program): Jordan function J_7(n).
  • A069093 (program): Jordan function J_8(n).
  • A069094 (program): Jordan function J_9(n).
  • A069095 (program): Jordan function J_10(n).
  • A069097 (program): Moebius transform of A064987, n*sigma(n).
  • A069099 (program): Centered heptagonal numbers.
  • A069102 (program): a(1) = 1; a(2) = 1; a(n) = Prime[n-1] + Prime[n-2] if n > 2.
  • A069104 (program): Numbers m such that m divides Fibonacci(m+1).
  • A069113 (program): Squarefree part of C(2n,n), the central binomial numbers: the smallest number such that a(n)*C(2n,n) is a square.
  • A069114 (program): Squarefree part of prime(n)-1 : the smallest number such that a(n)*(prime(n)-1) is a square.
  • A069115 (program): Squarefree part of prime(n)+1 : the smallest number such that a(n)*(prime(n)+1) is a square.
  • A069121 (program): a(n) = n^4*binomial(2n,n).
  • A069123 (program): Triangle formed as follows: For n-th row, n >= 0, record the A000041(n) partitions of n; for each partition, write down number of ways to arrange the parts.
  • A069125 (program): a(n) = (11*n^2 - 11*n + 2)/2.
  • A069126 (program): Centered 13-gonal numbers.
  • A069127 (program): Centered 14-gonal numbers.
  • A069128 (program): Centered 15-gonal numbers: a(n) = (15*n^2 - 15*n + 2)/2.
  • A069129 (program): Centered 16-gonal numbers.
  • A069130 (program): Centered 17-gonal numbers: (17*n^2 - 17*n + 2)/2.
  • A069131 (program): Centered 18-gonal numbers.
  • A069132 (program): Centered 19-gonal numbers.
  • A069133 (program): Centered 20-gonal (or icosagonal) numbers.
  • A069134 (program): (n!*(3*n))^2.
  • A069135 (program): a(n) = (n!*(n+1)!)^2.
  • A069136 (program): Numbers that are not the sum of 5 nonnegative cubes.
  • A069140 (program): a(n) = (4n-1)*4n*(4n+1).
  • A069141 (program): n^2*(n+1)!/(n^tau(n)) where tau(n) is the number of divisors of n.
  • A069152 (program): a(n)=(n-1)!-n^tau(n)/n^2.
  • A069153 (program): a(n) = Sum_{d|n} d*(d-1)/2.
  • A069157 (program): Number of positive divisors of n that are divisible by the smallest prime that divides n.
  • A069158 (program): a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).
  • A069159 (program): a(n) = d(1) - d(2) + d(3) - d(4) + … + (-1)^(n+1) d(n), where d(k) denotes the k-th term of the digit sequence 3, 1, 4, 1, 5, 9,…. of Pi.
  • A069161 (program): Numbers n such that no group of order n can be a central factor.
  • A069162 (program): a(1)=1, a(2)=2, a(n+2)=(a(n+1)+a(n))/2 if a(n+1)+a(n) is even, a(n+2)=(3*(a(n+1)+a(n))+1)/2 otherwise.
  • A069170 (program): Values of phi(k)*Sum_{d|k} 1/phi(d) for nonprimes k.
  • A069171 (program): Numbers k such that gcd(k, 2^k-1) = 3.
  • A069173 (program): Centered 22-gonal numbers.
  • A069174 (program): Centered 23-gonal numbers.
  • A069177 (program): Maximal power of 2 that divides Phi(n), or the size of the Sylow 2-subgroup of the group of units mod n.
  • A069178 (program): Centered 21-gonal numbers.
  • A069180 (program): F(n) and n! are relatively prime where F(n) are the Fibonacci numbers.
  • A069181 (program): Decimal expansion of 1/1024.
  • A069183 (program): Expansion of 1/((1-x)(1-x^2)^2(1-x^3)(1-x^6)).
  • A069184 (program): Sum of divisors d of n such that d or n/d is odd.
  • A069190 (program): Centered 24-gonal numbers.
  • A069192 (program): Sum of the reversals of the divisors of n.
  • A069193 (program): a(n) = Sum_{d|n} d*phi(n)/phi(d).
  • A069194 (program): a(n) = Sum_{d|n} (n/d)*phi(n)/phi(d).
  • A069201 (program): a(n) = Sum_{k=1..n} mu(k)^2 * 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k.
  • A069202 (program): A Collatz-Fibonacci mixture: a(1) = 1, a(2) = 2, a(n+2) = a(n+1)/2+a(n)/2 if a(n+1) and a(n) have the same parity, a(n+2) = a(n+1)+a(n) otherwise.
  • A069203 (program): a(1)=0 a(2)=3 a(n+2)=(a(n+1)+a(n))/3 if (a(n+1)+a(n)==0 (mod 3)); a(n+2)=a(n+1)+a(n) otherwise.
  • A069205 (program): a(n) = Sum_{k=1..n} 2^bigomega(k).
  • A069208 (program): a(n) = Sum_{ d divides n } phi(n)/phi(d).
  • A069209 (program): Orders of non-Abelian Z-groups.
  • A069210 (program): a(1)=a(2)=1, a(n+2) = a(n+1)+1 if sign(sin(a(n+1)) = sign(sin(a(n)), a(n+2) = a(n)+1 otherwise.
  • A069212 (program): a(n) = Sum_{k=1..n} 3^omega(k).
  • A069213 (program): a(n) = n-th positive integer relatively prime to n.
  • A069214 (program): Let u(n,k) be the recursion defined by u(n,1)=1, u(n,2)=2, u(n,3)=n, and u(n,k+3) = (u(n,k+2) + u(n,k+1))/u(n,k) if u(n,k) divides u(n,k+2) + u(n,k+1), u(n,k+3) = u(n,k) otherwise. Then u(n,k) is periodic and a(n) = Max(u(n,k), k >= 1).
  • A069220 (program): Denominator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.
  • A069226 (program): a(n) = gcd(n, 2^n + 1).
  • A069227 (program): a(1)=1, a(2)=2; a(n+2) = (a(n+1) + a(n))/b(n) where b(n) = gcd(a(n+1) + a(n), 4).
  • A069228 (program): a(1)=1, a(2)=4, a(n+2)=(a(n+1)+a(n))/b(n), where b(n)=gcd(a(n+1)+a(n),4).
  • A069229 (program): a(n) = n*(2^n + 1).
  • A069237 (program): Composite n such that tau(n) divides phi(n), where tau(n) is the number of divisors of n and phi(n) the Euler totient function.
  • A069239 (program): Denominator of coefficient G_n defined by Sum_{ (m,m’) != (0,0)} 1/(m+m’*sqrt(-2))^(2*n) = (4*w)^(2*n)*G_n/(2*n)!, where 2w is one of the periods of the associated Weierstrass P-function.
  • A069241 (program): Number of Hamiltonian paths in the graph on n vertices {1,…,n}, with i adjacent to j iff |i-j| <= 2.
  • A069248 (program): Number of positive divisors of n themselves divisible by largest prime that divides n.
  • A069249 (program): n^2-phi(n)*sigma(n).
  • A069250 (program): Sum of the reversals of the proper divisors of n.
  • A069256 (program): Size of the Sylow 2-subgroup of the group GL_2(Z_n): maximal power of 2 that divides A000252(n).
  • A069260 (program): a(n) = core(1)*core(2)*…*core(n) where core(n) is the squarefree part of n (A007913).
  • A069262 (program): a(n) = 4*prime(n)^2.
  • A069264 (program): Inverse Moebius transform of bigomega(n).
  • A069265 (program): Numbers k such that Sum_{d|k} d/core(d) > k, where core(d) is the squarefree part of d.
  • A069267 (program): a(n) = (2^(n-1)/(2n)!)*Product_{k=1..n} q(k) where q(n) is the denominator of B(2n), the 2n-th Bernoulli number.
  • A069268 (program): Greatest common divisor of n! and n*(n+1)/2.
  • A069269 (program): Second level generalization of Catalan triangle (0th level is Pascal’s triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270).
  • A069270 (program): Third level generalization of Catalan triangle (0th level is Pascal’s triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).
  • A069271 (program): a(n) = binomial(4*n+1,n)*2/(3*n+2).
  • A069272 (program): 11-almost primes (generalization of semiprimes).
  • A069273 (program): 12-almost primes (generalization of semiprimes).
  • A069274 (program): 13-almost primes (generalization of semiprimes).
  • A069275 (program): 14-almost primes (generalization of semiprimes).
  • A069276 (program): 15-almost primes (generalization of semiprimes).
  • A069277 (program): 16-almost primes (generalization of semiprimes).
  • A069278 (program): 17-almost primes (generalization of semiprimes).
  • A069279 (program): Products of exactly 18 primes (generalization of semiprimes).
  • A069280 (program): 19-almost primes (generalization of semiprimes).
  • A069281 (program): 20-almost primes (generalization of semiprimes).
  • A069283 (program): a(n) = -1 + number of odd divisors of n.
  • A069288 (program): Number of odd divisors of n <= sqrt(n).
  • A069289 (program): Sum of odd divisors of n <= sqrt(n).
  • A069290 (program): Sum of square roots of square divisors of n.
  • A069294 (program): Number of n X 3 binary arrays with a path of adjacent 1’s from upper left corner to anywhere in right hand column.
  • A069306 (program): Number of 2 X n binary arrays with a path of adjacent 1’s from upper left corner to anywhere in right hand column.
  • A069321 (program): Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)*k*k! for n >= 1.
  • A069345 (program): n minus the number of its prime-factors: a(n) = n - A001222(n).
  • A069352 (program): Total number of prime factors of 3-smooth numbers.
  • A069353 (program): Numbers of form 2^i*3^j - 1 with i, j >= 0.
  • A069357 (program): Numbers of form 2^i*3^j + (i+j) with i, j >= 0.
  • A069359 (program): a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.
  • A069360 (program): Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.
  • A069361 (program): Number of 3 X n binary arrays with a path of adjacent 1’s from top row to bottom row.
  • A069403 (program): a(n) = 2*Fibonacci(2*n+1) - 1.
  • A069429 (program): Half the number of 3 X n binary arrays with no path of adjacent 1’s or adjacent 0’s from top row to bottom row.
  • A069440 (program): Half the number of n X 2 binary arrays with no path of adjacent 1’s or adjacent 0’s from top row to bottom row.
  • A069459 (program): a(n) = prime(n)^n - 1.
  • A069466 (program): Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.
  • A069470 (program): a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).
  • A069473 (program): First differences of (n+1)^6-n^6 (A022522).
  • A069474 (program): First differences of A069473.
  • A069475 (program): First differences of A069474, successive differences of (n+1)^6-n^6.
  • A069476 (program): First differences of A069475, successive differences of (n+1)^6-n^6.
  • A069477 (program): a(n) = 60*n^2 + 180*n + 150.
  • A069478 (program): First differences of A069477, successive differences of (n+1)^5 - n^5.
  • A069480 (program): Determinant of rank n matrix of 1..n^2 filled successively along antidiagonals.
  • A069482 (program): a(n) = prime(n+1)^2 - prime(n)^2.
  • A069483 (program): Largest prime factor of prime(n+1)^2 - prime(n)^2.
  • A069484 (program): a(n) = prime(n+1)^2 + prime(n)^2.
  • A069485 (program): Greatest prime factor of prime(n+1)^2 + prime(n)^2.
  • A069486 (program): a(n) = 2*prime(n)*prime(n+1).
  • A069495 (program): Squares which are the arithmetic mean of two consecutive primes.
  • A069496 (program): Smaller member of a twin prime pair with a square sum.
  • A069497 (program): Triangular numbers of the form 6*k.
  • A069498 (program): Triangular numbers of the form 10*k.
  • A069499 (program): Triangular numbers of the form 21*k.
  • A069505 (program): a(1) = 1; a(n) = smallest palindromic number of the form k*a(n-1) + 1 with k > 1.
  • A069506 (program): a(1) = 2; a(n) = smallest palindromic number of the form k*a(n-1) + 1 with k > 1.
  • A069507 (program): a(1) = 4; a(n) = smallest palindromic number of the form k*a(n-1) + 1 with k > 1.
  • A069508 (program): a(1) = 6; a(n) = smallest palindromic number of the form k*a(n-1) + 1 with k > 1.
  • A069510 (program): a(1) = 1; a(n) = smallest palindrome of the form k*a(n-1) + 1.
  • A069511 (program): Numbers in which starting from most significant digit the n-th digit is obtained by adding n to the (n-1)-st digit (the digit to the left of it) and then ignoring the carry. Alternately the n-th digit starting from the most significant digit is the n-th triangular number mod 10.
  • A069513 (program): Characteristic function of the prime powers p^k, k >= 1.
  • A069515 (program): Number of transpositions (interchanges of adjacent digits, sometimes called inversions) needed to change all n-digit base 3 numbers into nondecreasing order.
  • A069517 (program): a(n) = (-1)*sum( d divides n, moebius(d)*(-1)^d).
  • A069531 (program): Smallest positive k such that 10^k + 1 is divisible by n, or 0 if no such number exists.
  • A069532 (program): Smallest even number with digit sum n.
  • A069533 (program): sum(p,floor(n^2/p^2)) where the sum is over all the primes.
  • A069537 (program): Multiples of 2 whose digit sum is 2.
  • A069540 (program): Multiples of 5 with digit sum 5.
  • A069543 (program): Multiples of 8 with digit sum 8.
  • A069546 (program): a(n) = Sum_{d|n} sigma(n*d).
  • A069547 (program): n^2 mod n-th prime.
  • A069549 (program): Smallest composite k such that phi(k) > k*(1-1/n).
  • A069553 (program): Define S(k) to be the sequence of divisors and multiples of k, e.g. S(1) = 1,2,3,4,5… S(2) = 1,2,4,6,8,10,… S(10) = 1,2,5,10,20,30,40,50,…; a(n) = n-th term of the n-th sequence S(n).
  • A069562 (program): Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.
  • A069577 (program): Smallest prime p such that pi(n) <= pi(p)*2, where pi(n) is the number of primes <= n, A000720.
  • A069584 (program): a(n) = n - largest perfect power <= n.
  • A069622 (program): Let Power(n) be the sequence of integer roots or powers of n. Power(1) is 1,1,1,1,… Power(4) is 1,2,4,16,64,256,… Power(27) is 1,3,9,27,729,… Power (p^k) is 1,p,p^a,p^b,…p^k, p^2k,p^3k,…where p is a prime and a,b etc. are divisors of k. This is the sequence of the n-th term of Power(n).
  • A069623 (program): Number of perfect powers <= n.
  • A069624 (program): Number of different values taken by the integer part of n^(1/k) for all k > 1.
  • A069627 (program): Sum_{k=1..n} floor(n*(n-1)/(2*k)).
  • A069637 (program): Number of prime powers <= n with exponents > 1.
  • A069638 (program): “Sorted” sum of two previous terms, beginning with 0,1. “Sorted” means to sort the digits of the sum in ascending order.
  • A069639 (program): Smallest composite k such that phi(k)>k*(1-1/n^2).
  • A069649 (program): Let M_n be the n X n matrix with M_n(i,j)=i^3/(i+j); then a(n)=1/det(M_n).
  • A069651 (program): For n >= 1, let M_n be the n X n matrix with M_n(i,j) = i^2/(i+j); then a(n) = 1/det(M_n). Also, a(0) = 1 by convention.
  • A069658 (program): a(1) = 1; a(n) = smallest nontrivial n-digit perfect power.
  • A069659 (program): Largest n-digit perfect power.
  • A069660 (program): Order of the subgroup of the symmetric group S_n generated by the cycles (1,3) and (1,2,3,…,n).
  • A069685 (program): Denominators of coefficients in -log(1+x)log(1-x) power series.
  • A069703 (program): a(n) = n! - n^k where n^(k+1) > n! >= n^k.
  • A069705 (program): a(n) = 2^n mod 7.
  • A069715 (program): GCD of digits of n is 1.
  • A069720 (program): a(n) = 2^(n-1)*binomial(2n-1, n).
  • A069721 (program): Number of rooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).
  • A069722 (program): Number of rooted unicursal planar maps with n edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).
  • A069723 (program): a(n) = 2^(n-1)*binomial(2*n-3, n-1).
  • A069726 (program): Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.
  • A069731 (program): Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).
  • A069733 (program): Number of divisors d of n such that d or n/d is odd. Number of non-orientable coverings of the Klein bottle with n lists.
  • A069734 (program): Number of pairs (p,q), 0<=p<=q, such that p+q divides n.
  • A069735 (program): Number of regular orientable coverings of the Klein bottle with 2n lists.
  • A069736 (program): Total number of Eulerian circuits in labeled multigraphs with n edges.
  • A069739 (program): Size of the key space for isomorphism verification of circulant graphs of order n.
  • A069745 (program): Expansion of 1/((1-x^2)(1-x^3)(1-x^6)(1-x^7)(1-x^8)).
  • A069750 (program): a(1)=1; a(n+1) is the smallest integer such that 1/a(n+1) = 0.0…00a(n)xxxxx…
  • A069754 (program): Counts transitions between prime and composite to reach the number n.
  • A069755 (program): Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.
  • A069756 (program): Frobenius number of the numerical semigroup generated by consecutive squares.
  • A069759 (program): Frobenius number of the numerical semigroup generated by consecutive hex numbers.
  • A069760 (program): Frobenius number of the numerical semigroup generated by consecutive centered square numbers.
  • A069763 (program): Frobenius number of the numerical semigroup generated by consecutive cubes.
  • A069764 (program): Frobenius number of the numerical semigroup generated by consecutive octahedral numbers.
  • A069777 (program): Array of q-factorial numbers n!_q, read by ascending antidiagonals.
  • A069778 (program): q-factorial numbers 3!_q.
  • A069779 (program): q-factorial numbers 4!_q.
  • A069780 (program): a(n) = gcd(d(n^3), d(n)).
  • A069782 (program): Numbers k such that gcd(d(k^3), d(k)) = 2^w for some w.
  • A069783 (program): a(n) = gcd(d(n!^3), d(n!)) = A069780(n!), where d() is the number of divisors function.
  • A069785 (program): a(n)=A061680(n!).
  • A069811 (program): a(n) = Sum_{k=1..n} omega(k)^2.
  • A069812 (program): a(n) = Sum_{k=1..n} (bigomega(k)-omega(k)).
  • A069813 (program): Maximum number of triangles in polyiamond with perimeter n.
  • A069816 (program): a(n) = (sum of digits of n)^2 - (sum of digits^2 of n).
  • A069828 (program): Sum of positive integers k for k <= n and gcd(k,n) = gcd(k+1,n).
  • A069829 (program): a(n) = PS(n)(2n), where PS is described in A057032.
  • A069830 (program): Multiplicative inverse of prime(n) modulo prime(n+1).
  • A069834 (program): a(n) = n-th reduced triangular number: n*(n+1)/{2^k} where 2^k is the largest power of 2 that divides product n*(n+1).
  • A069835 (program): Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).
  • A069836 (program): Inverse permutation to A057033: a(n) is the m such that A057033(m) = n, or 0 if no such m exists.
  • A069856 (program): E.g.f.: exp(x)/(1+LambertW(x)).
  • A069858 (program): 1/n has period 4 in base 10.
  • A069859 (program): (Largest prime factor of n) modulo (smallest prime factor of n).
  • A069864 (program): Decimal expansion of 2/log(4/3).
  • A069865 (program): a(n) = Sum_{k = 0..n} C(n,k)^6.
  • A069876 (program): a(1) = 1, a(2) = 2^2 + 3^2; a(n) = (k-n+1)^n + (k-n)^n + ….(k-1)^n + k^n, where k = n(n+1)/2.
  • A069879 (program): Number of pairs {i,j} with i different from j; 1<=i<=n; 1<= j <=n such that i+j is a prime number.
  • A069882 (program): Numbers n such that n and 2n-1 are both palindromes.
  • A069891 (program): a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.
  • A069894 (program): Centered square numbers: a(n) = 4*n^2 + 4*n + 2.
  • A069895 (program): 2^a(n) divides (2n)^(2n): exponent of 2 in (2n)^(2n).
  • A069896 (program): GCD of consecutive values of Chowla’s function.
  • A069897 (program): Integer quotient of the largest and the smallest prime factors of n, with a(1) = 1.
  • A069899 (program): Integer quotient of largest and smallest prime factors of n is 1.
  • A069900 (program): Integer quotient of largest and smallest prime factors of n is greater than one.
  • A069901 (program): Smallest prime factor of n-th triangular number.
  • A069902 (program): Largest prime factor of n-th triangular number n(n+1)/2.
  • A069903 (program): Number of distinct prime factors of n-th triangular number.
  • A069904 (program): Number of prime factors of n-th triangular number (with multiplicity).
  • A069905 (program): Number of partitions of n into 3 positive parts.
  • A069908 (program): Numbers congruent to +-2, +-3, +-4 or +-5 mod 16.
  • A069909 (program): Numbers congruent to +-1, +-4, +-6, +-7 mod 16.
  • A069910 (program): Expansion of Product_{i in A069908} 1/(1 - x^i).
  • A069911 (program): Expansion of Product_{i in A069909} 1/(1 - x^i).
  • A069912 (program): a(n) = A067552(n)/9 where A067552(n) = SumOfDigits(n)^2 - SumOfDigits(n^2), with SumOfDigits = A007953.
  • A069913 (program): a(n) = Sum_{d|n} (d-1)*tau(n/d).
  • A069914 (program): a(n) = Sum_{d|n} (d-1)*sigma(n/d).
  • A069921 (program): Define C(n) by the recursion C(0) = 1 + I where I^2 = -1, C(n+1) = 1/(1+C(n)); then a(n) = (-1)^n/Im(C(n)) where Im(z) is the imaginary part of the complex number z.
  • A069924 (program): Number of k, 1<=k<=n, such that phi(k) divides k.
  • A069926 (program): Number of k, 1<=k<=n, such that k divides sigma(k).
  • A069928 (program): Number of k, 1<=k<=n, such that tau(k) divides sigma(k) where tau(x) is the number of divisors of x and sigma(x) the sum of divisors of x.
  • A069930 (program): Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.
  • A069931 (program): Number of k, 1<=k<=n, such that k divides sigma(n).
  • A069935 (program): Maximal power of 2 that divides the n-th partition number.
  • A069939 (program): 1/3!*((Sum of digits of n)^3 + 3*(Sum of digits of n)*(Sum of digits^2 of n) + 2*(Sum of digits^3 of n)).
  • A069940 (program): (1/2)*((Sum of digits of n)^2 + (Sum of digits^2 of n)).
  • A069943 (program): a(n) = numerator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).
  • A069944 (program): a(n) = denominator(b(n)), where b(1) = b(2) = 1, b(n) = (b(n-1) + b(n-2))/(n-1).
  • A069945 (program): Let M_k be the k X k matrix M_k(i,j)=1/binomial(i+n,j); then a(n)=1/det(M_(n+1)).
  • A069949 (program): a(n) = Sum_{d|n} phi(d+1).
  • A069955 (program): Let W(n) = Product_{k=1..n} (1 - 1/4k^2), the partial Wallis product (lim_{n->infinity} W(n) = 2/Pi); then a(n) = numerator(W(n)).
  • A069958 (program): (Sum of digits of n)^3 - (sum of digits^3 of n).
  • A069959 (program): Define C(n) by the recursion C(0) = 2*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 2*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of the complex number z.
  • A069960 (program): Define C(n) by the recursion C(0) = 3*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 3*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of the complex number z.
  • A069961 (program): Define C(n) by the recursion C(0) = 4*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 4*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.
  • A069962 (program): Define C(n) by the recursion C(0) = 5*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 5*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.
  • A069963 (program): Define C(n) by the recursion C(0) = 6*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 6*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.
  • A069971 (program): Table by antidiagonals of variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -k for the first time.
  • A069972 (program): Sum_{d|2*n,d+1|2*n} d.
  • A069973 (program): (Sum of digits of n)^3 - (sum of digits of n^3).
  • A069975 (program): a(n) = n*(16*n^2-1).
  • A069976 (program): Interior angle of a regular polygon of n sides, rounded to nearest integer.
  • A069977 (program): Numbers k such that k and k+2 are squarefree.
  • A069978 (program): (Sum of digits of n)^4 - (sum of digits of n^4).
  • A069980 (program): (Sum of digits of n)^6 - (sum of digits of n^6).
  • A069981 (program): Hermite’s problem: number of positive integral solutions to x + y + z = n subject to x <= y + z, y <= z + x and z <= x + y.
  • A069982 (program): Number of 4-gonal compositions of n into positive parts.
  • A069984 (program): 1123+21460n.
  • A069986 (program): Denominator of b(n) = binomial(2n,n)^3*(42n+5)/2^(12n+4).
  • A069987 (program): Squarefree numbers of form k^2 + 1.
  • A069989 (program): (-1)^(n+1)/2*sum(k=1,2n,C(2n+1,k)*B(k)*4^k) where C(n,k) are the binomial coefficients, B(k) the Bernoulli numbers.
  • A069993 (program): a(n) = 2^(2n+1)*Sum_{k=1..2*n} binomial(2n+1,k)*Bernoulli(k)/2^k.
  • A069996 (program): Number of spanning trees on the bipartite graph K_{3,n}.
  • A069998 (program): Decimal expansion of sqrt(Pi/2).
  • A070003 (program): Numbers divisible by the square of their largest prime factor.
  • A070004 (program): Numbers of the form 5*2^n or 5*3*2^n; a(n) = 5*A029744(n).
  • A070010 (program): GCD of consecutive values of sum-of-proper divisors.
  • A070012 (program): Floor of number of prime factors of n divided by the number of n’s distinct prime factors.
  • A070014 (program): Ceiling of number of prime factors of n divided by the number of n’s distinct prime factors.
  • A070021 (program): 1/n has period 1 in base 10 (but not terminating).
  • A070022 (program): 1/n has period 2 in base 10.
  • A070023 (program): 1/n has period 1 in base 10 (including terminating decimals).
  • A070031 (program): Expansion of (1+x*C)*C^3, where C = (1-sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
  • A070032 (program): Integer part of sigma(n)/phi(n).
  • A070038 (program): a(n) = sum of divisors of n that are at least sqrt(n).
  • A070039 (program): Sum of divisors of n that are smaller than sqrt(n).
  • A070043 (program): Numbers of the form 6*j*k+j+k for positive integers j and k.
  • A070044 (program): Numbers k such that ceiling(k^1.5) is prime.
  • A070045 (program): Primes of the form ceiling(k^1.5) for some integer k.
  • A070046 (program): Number of primes between prime(n) and 2*prime(n) exclusive.
  • A070047 (program): Number of partitions of n in which no part appears more than twice and no two parts differ by 1.
  • A070050 (program): Number of Bottleneck-Monge matrices with 2 rows. In the formula below, P = 2.
  • A070071 (program): a(n) = n*B(n), where B(n) are the Bell numbers, A000110.
  • A070072 (program): Number of distinct rectangles with integer sides <=n and squarefree area.
  • A070077 (program): Greatest common divisor of n-th squarefree number and n-th cubefree number.
  • A070083 (program): Perimeters of integer triangles, sorted by perimeter, sides lexicographically ordered.
  • A070087 (program): P(n) > P(n+1) where P(n) (A006530) is the largest prime factor of n.
  • A070089 (program): P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.
  • A070091 (program): Number of isosceles integer triangles with perimeter n and relatively prime side lengths.
  • A070098 (program): Number of integer triangles with perimeter n which are acute and isosceles.
  • A070106 (program): Number of integer triangles with perimeter n which are obtuse and isosceles.
  • A070162 (program): Numbers k such that k - phi(k) - 1 is a prime.
  • A070163 (program): Primes arising in A070162(n).
  • A070169 (program): Rounded total surface area of a regular tetrahedron with edge length n.
  • A070172 (program): Smallest k such that sigma(k) >= n.
  • A070176 (program): Let s(n) be smallest number >= n which is a sum of two squares (A001481); sequence gives s(n) - n.
  • A070178 (program): Coefficients of Lehmer’s polynomial.
  • A070189 (program): 12345679n.
  • A070194 (program): List the phi(n) numbers from 1 to n-1 which are relatively prime to n; sequence gives size of maximal gap.
  • A070196 (program): a(n)=n plus the sorted version of the base-10 digits of n.
  • A070198 (program): Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.
  • A070199 (program): Number of palindromes of length <= n.
  • A070207 (program): Expansion of (1-x-5*x^2)/(1-3*x-2*x^2-x^3).
  • A070212 (program): Number of 5 X 5 pandiagonal magic squares with sum n.
  • A070214 (program): Maximal number of occupied cells in all monotonic matrices of order n.
  • A070216 (program): Triangle T(n, k) = n^2 + k^2, 1 <= k <= n, read by rows.
  • A070219 (program): Smallest prime obtained as a concatenation of n and a number m greater than n.
  • A070221 (program): a(n)=LPF(n+1)-LPF(n), where LPF(n) denotes the largest prime factor of n.
  • A070229 (program): Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.
  • A070232 (program): a(1) = 4; a(n) = smallest composite number greater than the sum of all previous terms.
  • A070238 (program): Sign of core(n)-phi(n) where core(n) is the squarefree part of n and phi the Euler totient function.
  • A070246 (program): a(n) = lcm(n, R(n)) / gcd(n, R(n)), where R(n) (A004086) is the digit reversal of n.
  • A070251 (program): Unrelated-factorial numbers: product of numbers unrelated to n (numbers which have a common divisor with n but do not divide n).
  • A070252 (program): Number of n-digit palindromes.
  • A070258 (program): Smallest of 3 consecutive numbers each divisible by a square.
  • A070260 (program): Third diagonal of triangle defined in A051537.
  • A070261 (program): 4th diagonal of triangle defined in A051537.
  • A070262 (program): 5th diagonal of triangle defined in A051537.
  • A070263 (program): Triangle T(n,k), n>=0, 1 <= k <= 2^n, read by rows, giving minimal distance-sum of any set of k binary vectors of length n.
  • A070271 (program): n^reverse(n) (ignore leading 0’s).
  • A070279 (program): Sum of digits of n equals the sum of digits of 2n.
  • A070280 (program): a(1) = 1; a(n) = a(n-1) + product of the digits of n-1.
  • A070285 (program): a(n) = n^(n-2) * (n-1)^(n-1).
  • A070290 (program): a(n) = lcm(8,n)/gcd(8,n).
  • A070291 (program): a(n) = lcm(10,n)/gcd(10,n).
  • A070292 (program): a(n) = lcm(12,n)/gcd(12,n).
  • A070293 (program): a(n) = lcm(30,n)/gcd(30,n).
  • A070302 (program): Number of 3 X 3 X 3 magic cubes with sum 3n.
  • A070303 (program): Primes p such that the equation p^2*x^2==0 (mod p^2+x^2) has no solution.
  • A070306 (program): a(n) = 2*phi(n)/2^omega(n).
  • A070307 (program): Number of n X n matrices with nonnegative integer entries such that every row sum equals 3.
  • A070312 (program): a(1) = a(2) = 1; a(n) = a(n-1) + concatenation of a(n-2) and a(n-1).
  • A070313 (program): a(n) = 2^n - (2*n+1).
  • A070315 (program): Third diagonal of triangle in A046739.
  • A070318 (program): Max( sigma(k)-k : k=1,2,3,…,n ) where sigma(x)-x is the sum of proper divisors of x.
  • A070319 (program): Max( tau(k) : k=1,2,3,…,n ) where tau(n)=A000005(n) is the number of divisors of x.
  • A070320 (program): Max( phi(k) : k=1,2,3,…,n ).
  • A070321 (program): Greatest squarefree number <= n.
  • A070323 (program): Let M_n be the n X n matrix m(i,j) = min(prime(i), prime(j)); then a(n) = det(M_n).
  • A070324 (program): Max( sigma(k) : k=1,2,3,…,n ).
  • A070331 (program): Numbers in which suffixing or inserting a 0 anywhere yields only composite numbers.
  • A070333 (program): Expansion of (1 + x)*(1 - x + x^2)/((1 - x)^4*(1 + x + x^2)).
  • A070335 (program): a(n) = 2^n mod 23.
  • A070336 (program): a(n) = 2^n mod 25.
  • A070337 (program): a(n) = 2^n mod 27.
  • A070338 (program): a(n) = 2^n mod 33.
  • A070339 (program): a(n) = 2^n mod 35.
  • A070340 (program): a(n) = 2^n mod 39.
  • A070341 (program): a(n) = 3^n mod 11: Repeat (1, 3, 9, 5, 4), period 5.
  • A070342 (program): a(n) = 3^n mod 19.
  • A070343 (program): a(n) = 3^n mod 25.
  • A070344 (program): a(n) = 3^n mod 29.
  • A070345 (program): a(n) = 3^n mod 35.
  • A070346 (program): a(n) = 3^n mod 37.
  • A070347 (program): a(n) = 2^n mod 21.
  • A070348 (program): a(n) = 2^n mod 41.
  • A070349 (program): a(n) = 2^n mod 43.
  • A070350 (program): a(n) = 2^n mod 45.
  • A070351 (program): a(n) = 2^n mod 47.
  • A070352 (program): a(n) = 3^n mod 5; or period 4, repeat [1, 3, 4, 2].
  • A070353 (program): a(n) = 3^n mod 14.
  • A070354 (program): a(n) = 3^n mod 16.
  • A070355 (program): a(n) = 3^n mod 22.
  • A070356 (program): a(n) = 3^n mod 23.
  • A070357 (program): a(n) = 3^n mod 28.
  • A070358 (program): a(n) = 3^n mod 32.
  • A070359 (program): a(n) = 3^n mod 34.
  • A070360 (program): a(n) = 3^n mod 38.
  • A070361 (program): a(n) = 3^n mod 41.
  • A070362 (program): a(n) = 3^n mod 44.
  • A070363 (program): a(n) = 3^n mod 46.
  • A070364 (program): a(n) = 3^n mod 47.
  • A070365 (program): a(n) = 5^n mod 7.
  • A070366 (program): a(n) = 5^n mod 9.
  • A070367 (program): a(n) = 5^n mod 11.
  • A070368 (program): a(n) = 5^n mod 13.
  • A070369 (program): a(n) = 5^n mod 14.
  • A070370 (program): a(n) = 5^n mod 16.
  • A070371 (program): a(n) = 5^n mod 17.
  • A070372 (program): a(n) = 5^n mod 18.
  • A070373 (program): a(n) = 5^n mod 19.
  • A070374 (program): a(n) = 5^n mod 21.
  • A070375 (program): a(n) = 5^n mod 22.
  • A070376 (program): a(n) = 5^n mod 26.
  • A070377 (program): a(n) = 5^n mod 27.
  • A070378 (program): a(n) = 5^n mod 28.
  • A070379 (program): a(n) = 5^n mod 29.
  • A070380 (program): a(n) = 5^n mod 32.
  • A070381 (program): a(n) = 5^n mod 33.
  • A070382 (program): a(n) = 5^n mod 34.
  • A070383 (program): a(n) = 5^n mod 36.
  • A070384 (program): a(n) = 5^n mod 37.
  • A070385 (program): a(n) = 5^n mod 38.
  • A070386 (program): a(n) = 5^n mod 39.
  • A070387 (program): a(n) = 5^n mod 41.
  • A070388 (program): a(n) = 5^n mod 42.
  • A070389 (program): a(n) = 5^n mod 43.
  • A070390 (program): a(n) = 5^n mod 44.
  • A070391 (program): a(n) = 5^n mod 46.
  • A070392 (program): a(n) = 6^n mod 11.
  • A070393 (program): a(n) = 6^n mod 13.
  • A070394 (program): a(n) = 6^n mod 17.
  • A070395 (program): a(n) = 6^n mod 19.
  • A070396 (program): a(n) = 6^n mod 23.
  • A070397 (program): a(n) = 6^n mod 25.
  • A070398 (program): a(n) = 6^n mod 29.
  • A070399 (program): a(n) = 6^n mod 31.
  • A070400 (program): a(n) = 6^n mod 37.
  • A070401 (program): a(n) = 6^n mod 47.
  • A070402 (program): a(n) = 2^n mod 5.
  • A070403 (program): a(n) = 7^n mod 9.
  • A070404 (program): a(n) = 7^n mod 11.
  • A070405 (program): a(n) = 7^n mod 13.
  • A070406 (program): a(n) = 7^n mod 15.
  • A070407 (program): a(n) = 7^n mod 17.
  • A070408 (program): a(n) = 7^n mod 22.
  • A070409 (program): a(n) = 7^n mod 23.
  • A070410 (program): a(n) = 7^n mod 25.
  • A070411 (program): a(n) = 7^n mod 26.
  • A070412 (program): a(n) = 7^n mod 27.
  • A070413 (program): a(n) = 7^n mod 29.
  • A070414 (program): a(n) = 7^n mod 30.
  • A070415 (program): a(n) = 7^n mod 31.
  • A070416 (program): a(n) = 7^n mod 32.
  • A070417 (program): a(n) = 7^n mod 33.
  • A070419 (program): a(n) = 7^n mod 36.
  • A070420 (program): a(n) = 7^n mod 37.
  • A070421 (program): a(n) = 7^n mod 38.
  • A070422 (program): a(n) = 7^n mod 39.
  • A070423 (program): a(n) = 7^n mod 40.
  • A070424 (program): a(n) = 7^n mod 41.
  • A070425 (program): a(n) = 7^n mod 43.
  • A070426 (program): a(n) = 7^n mod 44.
  • A070427 (program): a(n) = 7^n mod 45.
  • A070429 (program): a(n) = 7^n mod 47.
  • A070430 (program): a(n) = n^2 mod 5.
  • A070431 (program): a(n) = n^2 mod 6.
  • A070432 (program): Period 4: repeat [0, 1, 4, 1]; a(n) = n^2 mod 8.
  • A070433 (program): a(n) = n^2 mod 9.
  • A070434 (program): a(n) = n^2 mod 11.
  • A070435 (program): a(n) = n^2 mod 12, or alternately n^4 mod 12.
  • A070436 (program): a(n) = n^2 mod 13.
  • A070437 (program): a(n) = n^2 mod 14.
  • A070438 (program): a(n) = n^2 mod 15.
  • A070439 (program): a(n) = n^2 mod 16.
  • A070440 (program): a(n) = n^2 mod 18.
  • A070441 (program): n^2 mod 19.
  • A070442 (program): a(n) = n^2 mod 20.
  • A070443 (program): a(n) = n^2 mod 21.
  • A070444 (program): a(n) = n^2 mod 22.
  • A070445 (program): a(n) = n^2 mod 23.
  • A070446 (program): a(n) = n^2 mod 24.
  • A070447 (program): a(n) = n^2 mod 25.
  • A070448 (program): a(n) = n^2 mod 26.
  • A070449 (program): a(n) = n^2 mod 27.
  • A070450 (program): a(n) = n^2 mod 28.
  • A070451 (program): a(n) = n^2 mod 29.
  • A070452 (program): a(n) = n^2 mod 30.
  • A070453 (program): a(n) = n^2 mod 31.
  • A070454 (program): a(n) = n^2 mod 32.
  • A070455 (program): a(n) = n^2 mod 33.
  • A070456 (program): a(n) = n^2 mod 34.
  • A070457 (program): a(n) = n^2 mod 35.
  • A070458 (program): a(n) = n^2 mod 36.
  • A070459 (program): a(n) = n^2 mod 37.
  • A070460 (program): a(n) = n^2 mod 38.
  • A070461 (program): a(n) = n^2 mod 39.
  • A070462 (program): a(n) = n^2 mod 40.
  • A070463 (program): a(n) = n^2 mod 41.
  • A070464 (program): a(n) = n^2 mod 42.
  • A070465 (program): a(n) = n^2 mod 43.
  • A070466 (program): a(n) = n^2 mod 44.
  • A070467 (program): a(n) = n^2 mod 45.
  • A070468 (program): a(n) = n^2 mod 46.
  • A070469 (program): a(n) = n^2 mod 47.
  • A070470 (program): a(n) = n^2 mod 48.
  • A070471 (program): a(n) = n^3 mod 5.
  • A070472 (program): a(n) = n^3 mod 7.
  • A070473 (program): a(n) = n^3 mod 11.
  • A070474 (program): a(n) = n^3 mod 12, n^5 mod 12.
  • A070475 (program): a(n) = n^3 mod 13.
  • A070476 (program): a(n) = n^3 mod 14.
  • A070477 (program): a(n) = n^3 mod 15.
  • A070478 (program): a(n) = n^3 mod 16.
  • A070479 (program): a(n) = n^3 mod 17.
  • A070480 (program): a(n) = n^3 mod 18.
  • A070481 (program): a(n) = n^3 mod 19.
  • A070482 (program): a(n) = n^3 mod 20.
  • A070483 (program): a(n) = n^3 mod 21.
  • A070484 (program): a(n) = n^3 mod 22.
  • A070485 (program): a(n) = n^3 mod 23.
  • A070486 (program): a(n) = n^3 mod 24 (or equivalently, n^5 mod 24).
  • A070487 (program): a(n) = n^3 mod 25.
  • A070488 (program): a(n) = n^3 mod 26.
  • A070489 (program): a(n) = n^3 mod 27.
  • A070490 (program): a(n) = n^3 mod 28.
  • A070491 (program): a(n) = n^3 mod 29.
  • A070492 (program): a(n) = n^3 mod 30.
  • A070493 (program): a(n) = n^3 mod 31.
  • A070494 (program): a(n) = n^3 mod 32.
  • A070495 (program): a(n) = n^3 mod 33.
  • A070496 (program): a(n) = n^3 mod 34.
  • A070497 (program): a(n) = n^3 mod 35.
  • A070498 (program): a(n) = n^3 mod 36.
  • A070499 (program): a(n) = n^3 mod 37.
  • A070500 (program): a(n) = n^3 mod 38.
  • A070501 (program): a(n) = n^3 mod 39.
  • A070502 (program): a(n) = n^3 mod 40.
  • A070503 (program): a(n) = n^3 mod 41.
  • A070504 (program): a(n) = n^3 mod 42.
  • A070505 (program): a(n) = n^3 mod 43.
  • A070506 (program): a(n) = n^3 mod 44.
  • A070507 (program): a(n) = n^3 mod 45.
  • A070508 (program): a(n) = n^3 mod 46.
  • A070509 (program): a(n) = n^3 mod 47.
  • A070510 (program): a(n) = n^3 mod 48.
  • A070511 (program): a(n) = n^4 mod 6.
  • A070512 (program): a(n) = n^4 mod 7.
  • A070513 (program): a(n) = n^4 mod 9.
  • A070514 (program): Final digit of n^4: n^4 mod 10.
  • A070515 (program): a(n) = n^4 mod 11.
  • A070517 (program): a(n) = n^4 mod 13.
  • A070521 (program): Value of prime(n)-th cyclotomic polynomial at n.
  • A070532 (program): a(n) = n^4 mod 14.
  • A070533 (program): n^4 mod 15.
  • A070534 (program): a(n) = n^4 mod 17.
  • A070535 (program): a(n) = n^4 mod 18.
  • A070537 (program): Numbers k such that the k-th cyclotomic polynomial has more terms than the largest prime factor of k.
  • A070538 (program): a(n) = n^4 mod 19.
  • A070539 (program): a(n) = n^4 mod 20.
  • A070540 (program): a(n) = n^4 mod 21.
  • A070541 (program): a(n) = n^4 mod 22.
  • A070543 (program): Triangular array read by rows: T(n,k) = number of k-dimensional isotropic subspaces of Spin(2n+1,C), n >= 1, 1 <= k <= n.
  • A070544 (program): Number of squarefree numbers s such that n < s < 2n.
  • A070545 (program): a(n) = Card( k, 0<k<=n such that k is relatively prime to sigma(k)).
  • A070546 (program): a(n) = Card( k, 0<k<=n such that k is relatively prime to tau(k)=A000005(k)).
  • A070548 (program): a(n) = Cardinality{ k in range 1 <= k <= n such that Moebius(k) = 1 }.
  • A070549 (program): a(n) = Card(k 0<k<=n such that mu(k)=-1).
  • A070550 (program): a(n) = a(n-1) + a(n-3) + a(n-4), starting with a(0..3) = 1, 2, 2, 3.
  • A070551 (program): n^4 mod 23.
  • A070552 (program): Semiprimes k such that k+1 is also a semiprime.
  • A070553 (program): Rectangular array read by rows: n-th row gives the 7 numbers k*10^n mod 7 for 0 <= k < 7.
  • A070554 (program): Number of positive integers, k, where k <= 2n+1 and gcd(k, 2n+1) = gcd(k+1, 2n+1) = 1.
  • A070555 (program): Sum of positive integers k, where k <= n and gcd(k,2n+1) = gcd(k+1,2n+1).
  • A070556 (program): a(n) = cototient(totient(n)).
  • A070563 (program): a(n) = 0 if 3 divides the Ramanujan number tau(n) (A000594(n)), otherwise 1.
  • A070564 (program): Partial sums of A070563.
  • A070565 (program): n - product of digits of n.
  • A070567 (program): a(n) = n^4 mod 24.
  • A070568 (program): n^4 mod 25.
  • A070569 (program): n^4 mod 26.
  • A070570 (program): n^4 mod 27.
  • A070571 (program): n^4 mod 28.
  • A070572 (program): n^4 mod 29.
  • A070573 (program): n^4 mod 30.
  • A070574 (program): n^4 mod 31.
  • A070575 (program): n^4 mod 32.
  • A070576 (program): n^4 mod 33.
  • A070577 (program): a(n) = n^4 mod 34.
  • A070578 (program): a(n) = n^4 mod 35.
  • A070579 (program): n^4 mod 36.
  • A070580 (program): a(n) = n^4 mod 37.
  • A070581 (program): n^4 mod 38.
  • A070582 (program): n^4 mod 39.
  • A070583 (program): n^4 mod 40.
  • A070584 (program): n^4 mod 41.
  • A070585 (program): n^4 mod 42.
  • A070586 (program): a(n) = n^4 mod 43.
  • A070587 (program): n^4 mod 44.
  • A070588 (program): a(n) = n^4 mod 45.
  • A070589 (program): n^4 mod 46.
  • A070590 (program): n^4 mod 47.
  • A070591 (program): n^4 mod 48.
  • A070593 (program): a(n) = n^5 mod 7.
  • A070595 (program): n^5 mod 9.
  • A070596 (program): n^5 mod 11.
  • A070598 (program): n^5 mod 13.
  • A070599 (program): n^5 mod 14.
  • A070600 (program): Largest number with n prime factors where all factors are less than or equal to n.
  • A070601 (program): n^5 mod 17.
  • A070602 (program): n^5 mod 18.
  • A070603 (program): n^5 mod 19.
  • A070604 (program): n^5 mod 20.
  • A070605 (program): n^5 mod 21.
  • A070606 (program): n^5 mod 22.
  • A070607 (program): a(n) = n^5 mod 23.
  • A070609 (program): a(n) = n^5 mod 25.
  • A070611 (program): n^5 mod 27.
  • A070612 (program): n^5 mod 28.
  • A070613 (program): n^5 mod 29.
  • A070614 (program): a(n) = n^5 mod 31.
  • A070616 (program): n^5 mod 33.
  • A070617 (program): n^5 mod 34.
  • A070618 (program): a(n) = n^5 mod 35.
  • A070619 (program): n^5 mod 36.
  • A070620 (program): a(n) = n^5 mod 37.
  • A070621 (program): a(n) = n^5 mod 38.
  • A070622 (program): a(n) = n^5 mod 39.
  • A070623 (program): n^5 mod 40.
  • A070624 (program): n^5 mod 41.
  • A070625 (program): n^5 mod 42.
  • A070626 (program): n^5 mod 43.
  • A070627 (program): n^5 mod 44.
  • A070628 (program): n^5 mod 45.
  • A070629 (program): n^5 mod 46.
  • A070630 (program): n^5 mod 47.
  • A070631 (program): n^5 mod 48.
  • A070634 (program): n^6 mod 11.
  • A070636 (program): n^6 mod 13.
  • A070637 (program): n^6 mod 14.
  • A070638 (program): a(n) = n^6 mod 15.
  • A070639 (program): a(n) = (1/phi(n))*Sum_{k=1..n} phi(n*k).
  • A070640 (program): n^6 mod 17.
  • A070641 (program): n^6 mod 18.
  • A070642 (program): n^6 mod 19.
  • A070644 (program): n^6 mod 21.
  • A070645 (program): n^6 mod 22.
  • A070646 (program): n^6 mod 23.
  • A070647 (program): Largest prime factor of sequence of numbers of the form p*q (p, q distinct primes).
  • A070648 (program): n^6 mod 25.
  • A070649 (program): n^6 mod 26.
  • A070650 (program): n^6 mod 27.
  • A070651 (program): n^6 mod 28.
  • A070652 (program): n^6 mod 29.
  • A070653 (program): a(n) = n^6 mod 30.
  • A070654 (program): n^6 mod 31.
  • A070656 (program): a(n) = n^6 mod 33.
  • A070657 (program): n^6 mod 34.
  • A070658 (program): n^6 mod 35.
  • A070659 (program): n^6 mod 36.
  • A070660 (program): n^6 mod 37.
  • A070661 (program): n^6 mod 38.
  • A070662 (program): n^6 mod 39.
  • A070663 (program): n^6 mod 40.
  • A070664 (program): n^6 mod 41.
  • A070665 (program): n^6 mod 42.
  • A070666 (program): n^6 mod 43.
  • A070667 (program): Smallest m in range 2..n-1 such that m^2 == 1 mod n, or 1 if no such number exists.
  • A070668 (program): Smallest m in range 2..n-1 such that m^3 == 1 mod n, or 1 if no such number exists.
  • A070669 (program): Smallest m in range 2..n-1 such that m^4 == 1 mod n, or 1 if no such number exists.
  • A070670 (program): Smallest m in range 2..n-1 such that m^5 == 1 mod n, or 1 if no such number exists.
  • A070671 (program): Smallest m in range 2..n-1 such that m^6 == 1 mod n, or 1 if no such number exists.
  • A070673 (program): Smallest m in range 2..n-1 such that m^8 == 1 mod n, or 1 if no such number exists.
  • A070674 (program): Smallest m in range 2..n-1 such that m^9 == 1 mod n, or 1 if no such number exists.
  • A070675 (program): Smallest m in range 2..n-1 such that m^10 == 1 mod n, or 1 if no such number exists.
  • A070676 (program): Smallest m in range 1..phi(n) such that 3^m == 1 mod n, or 0 if no such number exists.
  • A070677 (program): Smallest m in range 1..phi(n) such that 5^m == 1 mod n, or 0 if no such number exists.
  • A070678 (program): Smallest m in range 1..phi(n) such that 7^m == 1 mod n, or 0 if no such number exists.
  • A070679 (program): Smallest m in range 1..phi(n) such that 9^m == 1 mod n, or 0 if no such number exists.
  • A070680 (program): Smallest m in range 1..phi(n) such that 11^m == 1 mod n, or 0 if no such number exists.
  • A070681 (program): Smallest m in range 1..phi(2n+1) such that 6^m == 1 mod 2n+1, or 0 if no such number exists.
  • A070682 (program): Smallest m in range 1..phi(2n+1) such that 10^m == 1 mod 2n+1, or 0 if no such number exists.
  • A070683 (program): Smallest m in range 1..phi(2n+1) such that 12^m == 1 mod 2n+1, or 0 if no such number exists.
  • A070684 (program): n^6 mod 44.
  • A070685 (program): n^6 mod 45.
  • A070686 (program): n^6 mod 46.
  • A070687 (program): n^6 mod 47.
  • A070688 (program): a(n) = n^6 mod 48.
  • A070689 (program): Numbers k such that k+1 and k^2+1 are primes.
  • A070690 (program): a(n) = n^7 mod 5.
  • A070691 (program): (Sum of digits of n)^n.
  • A070692 (program): a(n) = n^7 mod 9.
  • A070693 (program): a(n) = n^7 mod 11.
  • A070695 (program): a(n) = n^7 mod 13.
  • A070696 (program): a(n) = n mod 14.
  • A070697 (program): n^7 mod 15.
  • A070699 (program): a(n) = n^7 mod 17.
  • A070700 (program): a(n) = n^7 mod 18.
  • A070701 (program): a(n) = n^7 mod 19.
  • A070702 (program): a(n) = n^7 mod 20.
  • A070703 (program): a(n) = n^7 mod 22.
  • A070704 (program): a(n) = n^7 mod 23.
  • A070706 (program): a(n) = n^7 mod 25.
  • A070707 (program): n^7 mod 26.
  • A070708 (program): n^7 mod 27.
  • A070709 (program): n^7 mod 28.
  • A070710 (program): n^7 mod 29.
  • A070711 (program): a(n) = n^7 mod 30.
  • A070712 (program): a(n) = n^7 mod 31.
  • A070714 (program): n^7 mod 33.
  • A070715 (program): n^7 mod 34.
  • A070716 (program): n^7 mod 35.
  • A070717 (program): a(n) = n^7 mod 36.
  • A070718 (program): n^7 mod 37.
  • A070719 (program): n^7 mod 38.
  • A070720 (program): n^7 mod 39.
  • A070721 (program): n^7 mod 40.
  • A070722 (program): a(n) = n^7 mod 41.
  • A070723 (program): n^7 mod 43.
  • A070724 (program): n^7 mod 44.
  • A070725 (program): n^7 mod 45.
  • A070726 (program): a(n) = n^7 mod 46.
  • A070727 (program): n^7 mod 47.
  • A070728 (program): n^7 mod 48.
  • A070732 (program): Size of largest conjugacy class in the group GL(2,Z_n).
  • A070733 (program): Size of largest conjugacy class in A_n, the alternating group on n symbols.
  • A070734 (program): Order of the subgroup of the symmetric group S_n generated by the cycles (1,2,3) and (1,2,3,…,n).
  • A070739 (program): Primes of form 2^x + 2^y + 1.
  • A070745 (program): z such that the Diophantine equation x^2 + y^3 = z^2 has solutions.
  • A070747 (program): a(n) = signum(sin(n)), where signum=A057427.
  • A070750 (program): 0 if n-th prime is even, 1 if n-th prime is == 1 mod 4, and -1 if n-th prime is == 3 mod 4.
  • A070751 (program): Numbers n such that sin(n) < 0.
  • A070752 (program): Numbers k such that sin(k) > 0.
  • A070755 (program): Squarefree tetrahedral numbers.
  • A070770 (program): b + c + d where b >= c >= d >= 0 ordered by b then c then d.
  • A070771 (program): b+c+d+e where b>=c>=d>=e>=0 ordered by b then c then d then e.
  • A070772 (program): b+c+d+e+f where b>=c>=d>=e>=f>=0 ordered by b then c then d then e then f.
  • A070775 (program): a(n) = Sum_{k=0..n} binomial(4*n,4*k).
  • A070776 (program): Numbers k such that number of terms in the k-th cyclotomic polynomial is equal to the largest prime factor of k.
  • A070777 (program): a(1) = 1; a(n) = (largest prime factor of n) - 1.
  • A070778 (program): Let M denote the 6 X 6 matrix = row by row /1,1,1,1,1,1/1,1,1,1,1,0/1,1,1,1,0,0/1,1,1,0,0,0/1,1,0,0,0,0/1,0,0,0,0,0/ and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = u(n).
  • A070779 (program): Expansion of e.g.f.: (exp(x/(1-x))*(2-x)-1+x)/(1-x)^3.
  • A070780 (program): Binomial((n+1)^2,n).
  • A070781 (program): a(n) = binomial((n+1)^2, n^2).
  • A070782 (program): a(n) = Sum_{k=0..n} binomial(5*n,5*k).
  • A070799 (program): Numbers of the form 6jk-j-k.
  • A070800 (program): Smallest prime greater than phi(n): a(n) = nextprime(phi(n)).
  • A070801 (program): Largest prime <= sigma(n): a(n) = prevprime(sigma(n)), where prevprime(n) = A007917(n), the largest prime less than or equal to n.
  • A070803 (program): Number of primes not exceeding sum of divisors of n.
  • A070804 (program): Number of primes not exceeding phi(n).
  • A070808 (program): Sum(((-1)^k*binomial(4*n,k)),k=n..2*n).
  • A070814 (program): Solutions to phi(gpf(x)) - gpf(phi(x)) = 14 = c are special multiples of 17, x = 17k, where greatest prime factors of factor k were observed from {2, 3, 5}, i.e., it is smaller than 17. See solutions to other even cases of c (=A070813): A007283 for 0, A070004 for 2, A070815 for 254, A070816 for 65534. Gpf = greatest prime factor.
  • A070815 (program): Solutions to phi(gpf(x)) - gpf(phi(x)) = 254 = c are special multiples of 257, x = 257k, where largest prime factors of factor k were observed from {2, 3, 5, 17}. See solutions to other even cases of c (=A070813): A007283 for 0, A070004 for 2, A070814 for 14, A070816 for 65534.
  • A070816 (program): Solutions to phi(gpf(x)) - gpf(phi(x)) = 65534 = c are special multiples of 65537, x=65537*k, where the largest prime factors of factor k were observed in {2, 3, 5, 17, 257}.
  • A070817 (program): a(n) = floor(n/2) - gpf(phi(n)), where gpf(n) is the largest prime factor of n.
  • A070819 (program): Values of commutator[phi,gpf] = commutator[A000010, A006530] at prime arguments; a(1)=0 by convention.
  • A070820 (program): Difference between n-th prime and the value of commutator[phi,gpf] = commutator[A000010, A006530] at the same prime argument.
  • A070821 (program): Integer part of n/(lpf(n)*gpf(n)), where lpf = A020639 is the least prime factor and gpf = A006530 the greatest prime factor.
  • A070824 (program): Number of divisors of n which are > 1 and < n (nontrivial divisors).
  • A070825 (program): One half of product of first n+1 Lucas numbers A000032.
  • A070826 (program): One half of product of first n primes A000040.
  • A070832 (program): a(n) = Sum_{k=0..n} binomial(8*n,8*k).
  • A070833 (program): a(n) = Sum_{k=0..n} binomial(10*n,10*k).
  • A070834 (program): Reverse(n)^n.
  • A070846 (program): Smallest prime == 1 (mod 2n).
  • A070847 (program): Smallest prime == 1 mod (3n).
  • A070848 (program): Smallest prime == 1 mod (4n).
  • A070849 (program): Smallest prime == 1 mod (5n).
  • A070850 (program): Smallest prime == 1 mod (6n).
  • A070851 (program): Smallest prime == 1 mod (7n).
  • A070852 (program): Smallest prime == 1 mod (8n).
  • A070853 (program): Smallest prime == 1 mod (9n).
  • A070857 (program): Expansion of (1+x*C)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A070859 (program): Expansion of (1+x*C^4)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A070864 (program): a(1) = a(2) = 1; a(n) = 2 + a(n - a(n-1)).
  • A070865 (program): Smallest prime such that the difference of successive terms is strictly increasing.
  • A070866 (program): Smallest prime such that the difference of successive terms is nondecreasing.
  • A070869 (program): a(1) = 16; a(n+1) = sum of a(n) and (a(n) written in base 2 and reversed).
  • A070870 (program): a(1) = 6; a(n+1) = (a(n)+1)/2 if a(n) odd, or 5*a(n)/2 if a(n) even.
  • A070871 (program): a(n) = A002487(n) * A002487(n+1) (Conway’s alimentary function).
  • A070875 (program): Binary expansion is 1x100…0 where x = 0 or 1.
  • A070876 (program): Binary expansion is 1xx100…0 where xx = 00 or 11.
  • A070877 (program): Expansion of Product_{k>=1} (1 - 2x^k).
  • A070878 (program): Stern’s diatomic array read by rows (version 2).
  • A070879 (program): Stern’s diatomic array read by rows (version 3 - same as version 2, A070878, but with final 0 in each row omitted).
  • A070883 (program): Bitwise XOR of n and n-th prime.
  • A070884 (program): 7 + x where x is congruent to {0, 4, 6, 10, 12, 16, 22, 24} mod 30.
  • A070885 (program): a(n) = (3/2)*a(n-1) if a(n-1) is even; (3/2)*(a(n-1)+1) if a(n-1) is odd.
  • A070886 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 90”.
  • A070893 (program): Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)*s(i)*t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,…,1,…,n-3,n-1}.
  • A070895 (program): Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n<k.
  • A070896 (program): Determinant of the Cayley addition table of Z_{n}.
  • A070906 (program): Every third Bell number A000110.
  • A070907 (program): Every fourth Bell number A000110.
  • A070909 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 28” and by “Rule 156”.
  • A070910 (program): Decimal expansion of BesselI(0,2).
  • A070912 (program): Binary expansion of BesselI(0,2).
  • A070914 (program): Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.
  • A070916 (program): a(1)=1, a(n) is the smallest integer => a(n-1) such that a(n)a(n-1)-1 is prime.
  • A070919 (program): a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.
  • A070920 (program): a(n) = Card{ (x,y,z,u) | lcm(x,y,z,u)=n }.
  • A070923 (program): a(n) is the smallest value >= 0 of the form x^3 - n^2.
  • A070929 (program): Smallest integer >= 0 of the form x^2 - n^3.
  • A070930 (program): Smallest integer >= 0 of the form x^3 - n^4.
  • A070933 (program): Expansion of Product_{k>=1} 1/(1 - 2*t^k).
  • A070935 (program): Largest proper divisor of n^2.
  • A070939 (program): Length of binary representation of n.
  • A070940 (program): Number of digits that must be counted from left to right to reach the last 1 in the binary representation of n.
  • A070941 (program): Length of binary representation of 2n+1.
  • A070945 (program): Number of permutations on n letters that have only cycles of length 4 or less.
  • A070946 (program): Number of permutations on n letters that have only cycles of length 5 or less.
  • A070947 (program): Number of permutations on n letters that have only cycles of length 6 or less.
  • A070951 (program): Number of 0’s in n-th row of triangle in A070950.
  • A070952 (program): Number of 1’s in n-th generation of 1-D CA using Rule 30, started with a single 1.
  • A070953 (program): Order of the group GU(n,2), the general unitary n X n matrices over the finite field GF(4).
  • A070956 (program): Number of pairs (x,y) such that n = gcd(x,y) + lcm(x,y).
  • A070959 (program): First minimum value > 0 of the form x^3-k^2 when k > n^3.
  • A070960 (program): a(1) = 1; a(n) = n!*(3/2) for n>=2.
  • A070964 (program): a(n) = Sum_{1<=k<=n, gcd(k,n)=1} Fibonacci(k).
  • A070966 (program): a(n) = Sum_{k|n, k<=sqrt(n)} phi(k); where the sum is over the positive divisors, k, of n, which are <= the square root of n; and phi(k) is the Euler totient function.
  • A070967 (program): a(n) = Sum_{k=0..n} binomial(6*n,6*k).
  • A070968 (program): Number of cycles in the complete bipartite graph K(n,n).
  • A070972 (program): Length of longest run of consecutive 1’s in binary expansion of 3^n (A004656).
  • A070973 (program): Smallest integer k such that n divides floor((3/2)^k).
  • A070974 (program): Number of steps to reach 1 in `3x+1’ (or Collatz) problem starting with n!.
  • A070975 (program): Number of steps to reach 1 in `3x+1’ (or Collatz) problem starting with prime(n).
  • A070976 (program): Number of steps to reach 1 in ‘3x+1’ (or Collatz) problem starting with 3^n.
  • A070981 (program): Smallest integer k such that n divides floor((4/3)^k).
  • A070990 (program): First differences of A002487.
  • A070992 (program): Partial sums of A002487.
  • A070997 (program): a(n) = 8*a(n-1) - a(n-2), a(0)=1, a(-1)=1.
  • A070998 (program): a(n) = 9*a(n-1) - a(n-2) for n > 0, a(0)=1, a(-1)=1.
  • A070999 (program): Numbers n such that the denominator of Sum_{k=1..n} 1/gcd(n,k) is not equal to n.
  • A071000 (program): Numbers m such that the denominator of Sum_{k=1..n} 1/gcd(m,k) equals m.
  • A071004 (program): Binary expansion of AGM(1,sqrt(2)) where AGM(x,y) denote the arithmetic-geometric mean of (x,y).
  • A071005 (program): Binary expansion of Pi/3 (A019670).
  • A071010 (program): Sigma(k)/4 when k is not a sum of 2 squares.
  • A071014 (program): Binomial transform of A002487.
  • A071015 (program): Inverse binomial transform of A002487.
  • A071018 (program): Inverse Moebius transform of A002487.
  • A071022 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 70” and by “Rule 198”.
  • A071023 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 78”.
  • A071024 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 92”.
  • A071026 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 188”.
  • A071027 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 230”.
  • A071028 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 50”.
  • A071029 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 22”.
  • A071030 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 54”.
  • A071033 (program): a(n) = n-th state of cellular automaton generated by “Rule 94” when started with a single ON cell.
  • A071035 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 126”.
  • A071036 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 150” when started with a single ON cell.
  • A071037 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 158”.
  • A071038 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 182”.
  • A071041 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 246”.
  • A071042 (program): Number of 0’s in n-th row of triangle in A070886.
  • A071043 (program): Number of 0’s in n-th row of triangle in A071029.
  • A071044 (program): Number of ON cells at generation n of 1-D CA defined by Rule 22, starting with a single ON cell.
  • A071045 (program): Number of 0’s in n-th row of triangle in A071030.
  • A071046 (program): Number of 0’s in n-th row of triangle in A071031, cellular automaton “rule 62”.
  • A071047 (program): Number of 1’s in n-th row of triangle in A071031, cellular automaton “rule 62”.
  • A071048 (program): Number of 0’s in n-th row of triangle in A070887.
  • A071049 (program): Number of 1’s in n-th generation of 1-D CA using Rule 110, started with a single 1.
  • A071050 (program): Number of 0’s in n-th row of triangle in A071035.
  • A071051 (program): Number of 1’s in n-th row of triangle in A071035.
  • A071052 (program): Number of 0’s in n-th row of triangle in A071036 (cellular automaton “Rule 150”).
  • A071053 (program): Number of ON cells at n-th generation of 1-D CA defined by Rule 150, starting with a single ON cell at generation 0.
  • A071054 (program): a(2n)=3n+1, a(2n+1)=2n+2.
  • A071055 (program): Number of 0’s in n-th row of triangle in A071038.
  • A071061 (program): Abjad values of the Arabic letters in the traditional order for abjad calculations.
  • A071068 (program): Number of ways to write n as a sum of two unordered squarefree numbers.
  • A071072 (program): Minimal “multiples of 4” set in base 10.
  • A071089 (program): Remainder when sum of first n primes is divided by n-th prime.
  • A071090 (program): Sum of middle divisors of n.
  • A071099 (program): a(n) = (n-1)*(n+3) - 2^n + 4.
  • A071100 (program): Expansion of (5 + 3*x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.
  • A071101 (program): Expansion of (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.
  • A071108 (program): CONTINUANT transform of {d(n)}, 1, 2, 2, 3, 2, 4, … (A000005).
  • A071111 (program): a(n) is the least integer x such that there exists an integer in the open interval (x/(i+1), x/i) for i= n-1, n-2 …, 3, 2, 1.
  • A071117 (program): Sum of first n digits of decimal expansion of e is prime.
  • A071118 (program): Size of the automorphism group of the group Z X Z_n.
  • A071121 (program): a(n) = a(n-1) + sum of decimal digits of n^3.
  • A071122 (program): a(n) = a(n-1) + sum of decimal digits of 2^n.
  • A071123 (program): a(n) = a(n-1) + sum of decimal digits of n!.
  • A071126 (program): Length of least repunit which is a multiple of the n-th prime, or 0 if no such multiple exists.
  • A071138 (program): CONTINUANT transform of {sigma(n)}, 1, 3, 4, 7, 6, 12, … (A000203).
  • A071139 (program): Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.
  • A071142 (program): Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 3 distinct prime factors and n is squarefree.
  • A071148 (program): Partial sums of sequence of odd primes (A065091); a(n) = sum of the first n odd primes.
  • A071152 (program): Łukasiewicz words for the rooted plane binary trees (interpretation d in Stanley’s exercise 19) with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0.
  • A071154 (program): Totally balanced decimal numbers: if we assign the weight w(d) = d-1 to each digit d (i.e., w(0) = -1, w(1) = 0, …, w(9) = 8) and then read the digits of the term from left to right, the partial sum of the weights is never negative and the total weighted sum is zero.
  • A071160 (program): Łukasiewicz words that are also valid asynchronic siteswap juggling patterns.
  • A071161 (program): Integers whose decimal expansion satisfies the condition that if we read each term from the left to right (the most significant to the least significant digit) then each nonzero digit gives a distance to the next nonzero digit to right (with a cyclic wrap-over from the least-significant to the most significant nonzero digit).
  • A071162 (program): Simple rewriting of binary expansion of n resulting A014486-codes for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf).
  • A071166 (program): a(n) = n - A006530(A000203(n)), difference between n and largest prime factor of the sum of its divisors.
  • A071168 (program): n-th prime reduced modulo phi(n).
  • A071170 (program): n-th prime reduced modulo sigma(n).
  • A071171 (program): L_2 norm of vertices of Permuto-Associahedron in R^n.
  • A071178 (program): Exponent of the largest prime factor of n.
  • A071179 (program): n - (sum of prime factors of n) is prime.
  • A071182 (program): SPF(n+1)-SPF(n), where SPF(n) denotes the smallest prime factor of n.
  • A071187 (program): Smallest prime factor of number of divisors of n.
  • A071188 (program): Largest prime factor of number of divisors of n.
  • A071189 (program): Smallest prime factor of sum of divisors of n.
  • A071190 (program): Greatest prime factor of sum of divisors of n, for n >= 2; a(1) = 1.
  • A071193 (program): Least m>n such that the number of prime factors of m and n differ at least by 1.
  • A071203 (program): Integer part of n divided by its largest digit (decimal notation).
  • A071204 (program): Numbers which are multiples of their largest digit (decimal notation).
  • A071205 (program): Largest digit of A071204(n).
  • A071207 (program): Triangular array T(n,k) read by rows, giving number of rooted trees on the vertex set {1..n+1} where k children of the root have a label smaller than the label of the root.
  • A071210 (program): Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the degree k of the root.
  • A071211 (program): Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the label k of the root.
  • A071213 (program): Number of labeled planar trees with n nodes such that the root is smaller than all its children.
  • A071214 (program): Number of labeled ordered trees with n nodes such that the root is smaller than all its children.
  • A071215 (program): Number of distinct prime factors of sum of 2 successive primes.
  • A071216 (program): a(n) is the largest prime factor of prime(n) + prime(n+1).
  • A071222 (program): Smallest k such that gcd(n,k) = gcd(n+1,k+1).
  • A071224 (program): LCM of n and n-th composite number.
  • A071227 (program): Number of solutions 1<=x<=n to gcd(A033950(n),x) = tau(A033950(n)).
  • A071228 (program): a(n) = n*(n-th composite number).
  • A071229 (program): a(n) = n*(14*n^2-21*n+13)/6.
  • A071230 (program): a(n) = n*(6*n^2 - 7*n + 3)/2.
  • A071231 (program): a(n) = (n^8 + n^4)/2.
  • A071232 (program): a(n) = (n^6 + n^3)/2.
  • A071233 (program): a(n) = 2*(n-1)*(n^2 + 1).
  • A071235 (program): a(n) = (n^12 + n^6)/2.
  • A071236 (program): a(n) = (n^10 + n^5)/2.
  • A071237 (program): a(n) = n*(n+1)*(n^2+1)/2.
  • A071238 (program): a(n) = n*(n+1)*(2*n^2+1)/6.
  • A071239 (program): a(n) = n*(n+1)*(n^2+2)/6.
  • A071240 (program): Arithmetic mean of k and R(k) where k is a number using all odd digits and R(k) is its digit reversal (A004086).
  • A071241 (program): Arithmetic mean of k and R(k) where k is a number using all even digits and R(k) is its digit reversal (A004086).
  • A071242 (program): Arithmetic mean of n and R(n) where n is a number such that the least significant digit and the most significant digits are of same parity and R(n) is its digit reversal (A004086).
  • A071243 (program): Record terms in A005179.
  • A071244 (program): n*(n-1)*(n^2+2)/6.
  • A071245 (program): a(n) = n*(n-1)*(2*n^2+1)/6.
  • A071246 (program): a(n) = n*(n - 1)*(2*n^2 + n + 2)/6.
  • A071248 (program): a(n) = Product_{k=1..n} lcm(n,k).
  • A071249 (program): Numbers k such that gcd(k, R(k)) > 1, where R(k) (A004086) is the digit reversal of k.
  • A071251 (program): Squarefree palindromes.
  • A071252 (program): a(n) = n*(n - 1)*(n^2 + 1)/2.
  • A071253 (program): a(n) = n^2*(n^2+1).
  • A071256 (program): Smallest multiple of n sandwiched between twin primes.
  • A071259 (program): Integer part of the arithmetic mean of the n-th prime p(n) and the n-th composite number C(n).
  • A071260 (program): Integer part of the geometric mean of the n-th prime prime(n) and the n-th composite number C(n).
  • A071263 (program): Smallest nontrivial composite number beginning with n.
  • A071264 (program): Expansion of (1+x*C^4)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071270 (program): a(n) = n^2*(2*n^2+1)/3.
  • A071273 (program): Concatenation of R(n) (A004086) and n, omitting leading 0’s.
  • A071274 (program): A071273 divided by 11.
  • A071279 (program): Kissing number of regular n-gon.
  • A071281 (program): Numerators of Peirce sequence of order 3.
  • A071282 (program): Denominators of Peirce sequence of order 3.
  • A071289 (program): a(n) = n*(n^2 + 1) if n is even, otherwise (n - 1/2)*(n^2 + 1).
  • A071295 (program): Product of numbers of 0’s and 1’s in binary representation of n.
  • A071302 (program): a(n) = (1/2) * (number of n X n 0..2 matrices M with MM’ mod 3 = I, where M’ is the transpose of M and I is the n X n identity matrix).
  • A071317 (program): a(n) = a(n-1) + sum of digits of n^2.
  • A071321 (program): Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n).
  • A071322 (program): Alternating sum of all prime factors of n; primes nonincreasing, starting with the largest prime factor: A006530(n).
  • A071323 (program): Alternating sum of all divisors of n; divisors nondecreasing, starting with 1.
  • A071324 (program): Alternating sum of all divisors of n; divisors nonincreasing, starting with n.
  • A071325 (program): Number of squares > 1 dividing n.
  • A071326 (program): Sum of squares > 1 dividing n.
  • A071327 (program): Sum of the squared primes dividing n.
  • A071328 (program): Smallest prime q such that q - prime(n) >= n.
  • A071329 (program): Largest prime q such that q - prime(n) <= n.
  • A071330 (program): Number of decompositions of n into sum of two prime powers.
  • A071336 (program): Number of vertices of Goldberg-Casper-Klug pseudo-icosahedra.
  • A071353 (program): First term of the continued fraction expansion of (3/2)^n.
  • A071354 (program): Floor(2^n/n) is odd.
  • A071355 (program): a(n) = 2*n^2 + 11*n + 12.
  • A071356 (program): Expansion of (1 - 2*x - sqrt(1 - 4*x - 4*x^2))/(4*x^2).
  • A071357 (program): Expansion of (1 - 4*x - (1-2*x)*sqrt(1-4*x-4*x^2))/(8*x^3).
  • A071364 (program): Smallest number with same sequence of exponents in canonical prime factorization as n.
  • A071374 (program): 0 iff n is of the form 4^a*(8k+7), otherwise 1.
  • A071377 (program): Number of positive integers <= n which are the sum of 3 squares.
  • A071378 (program): Largest proper divisor of n^3.
  • A071396 (program): Rounded total surface area of a regular octahedron with edge length n.
  • A071398 (program): Rounded total surface area of a regular icosahedron with edge length n.
  • A071399 (program): Rounded volume of a regular tetrahedron with edge length n.
  • A071400 (program): Rounded volume of a regular octahedron with edge length n.
  • A071403 (program): Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.
  • A071404 (program): Which nonsquarefree number is a square number? a(n)-th nonsquarefree number equals n^2, the n-th square.
  • A071407 (program): Least k such that k*prime(n) + 1 and k*prime(n) - 1 are twin primes.
  • A071408 (program): a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.
  • A071411 (program): “Sum of n first primes” minus “sum of first n nonprimes”.
  • A071412 (program): A002487 mod 3.
  • A071413 (program): a(n) = if n=0 then 0 else a(floor(n/2))+n*(-1)^(n mod 2).
  • A071415 (program): Maximal m such that all numbers in [0,m] are expressible as a*b + c with a + c <= n, b <= n and a,b,c positive integers.
  • A071416 (program): a(n) = gcd(n, binomial(2*n, n)).
  • A071418 (program): a(1)=0, a(n+1)=(a(n)+n)/2 if a(n)+n is even, a(n+1)=(3*(a(n)+n)+1)/2 otherwise.
  • A071419 (program): a(1)=1, a(n+1)=(a(n)+n)/2 if a(n)+n is even, a(n+1)=(3*(a(n)+n)+1)/2 otherwise.
  • A071421 (program): a(n) = a(n-1) + sum of decimal digits of n^n.
  • A071422 (program): a(n) = a(n-1) + sum of decimal digits of sigma(n), the sum of divisors of n.
  • A071423 (program): a(n) = a(n-1) + number of decimal digits of 2^n. Number of decimal digits of concatenation of first n powers of 2.
  • A071424 (program): a(n) = a(n-1) + number of decimal digits of n!. Number of decimal digits of concatenation of first n factorials.
  • A071425 (program): Total number of 1-s in binary representation of all factorials from 1 to n.
  • A071440 (program): Start with 1; add the digits of the previous term and the squares of the digits of the previous term.
  • A071520 (program): Number of 5-smooth numbers (A051037) <= n.
  • A071521 (program): Number of 3-smooth numbers <= n.
  • A071523 (program): Number of 11-smooth numbers <= n.
  • A071532 (program): (-1) * Sum[ k =1,n, (-1)^floor((3/2)^k) ].
  • A071534 (program): Determinant of n X n matrix of form : [1 2 1 0 0 0 0 0 0 0 / 2 1 2 1 0 0 0 0 0 0 / 1 2 1 2 1 0 0 0 0 0 / 0 1 2 1 2 1 0 0 0 0 / 0 0 1 2 1 2 1 0 0 0 / 0 0 0 1 2 1 2 1 0 0 / 0 0 0 0 1 2 1 2 1 0 / 0 0 0 0 0 1 2 1 2 1 / 0 0 0 0 0 0 1 2 1 2 / 0 0 0 0 0 0 0 1 2 1].
  • A071535 (program): (-1)^(n+1) * Determinant of n X n matrix of form [1^2 2^2 3^2 4^2 5^2 / 2^2 1^2 2^2 3^2 4^2 / 3^2 2^2 1^2 2^2 3^2 / 4^2 3^2 2^2 1^2 2^2 / 5^2 4^2 3^2 2^2 1^2].
  • A071538 (program): Number of twin prime pairs (p, p+2) with p <= n.
  • A071539 (program): Number of n-tuples of elements e_1,e_2,…,e_n in the symmetric group S_3 such that the subgroup generated by e_1,e_2,…,e_n is S_3.
  • A071542 (program): Number of steps to reach 0 starting with n and using the iterated process : x -> x - (number of 1’s in binary representation of x).
  • A071544 (program): Smallest k such that n+k divides (k+1)!-k!.
  • A071545 (program): Smallest k such that n+k divides (k+1)!+k!.
  • A071549 (program): a(n) = (7n)!/n!^7.
  • A071554 (program): Smallest x > 1 such that x^prime(n) == 1 mod(prime(i)) 2<=i<=n.
  • A071555 (program): Smallest x > 1 such that x^prime(n) == 1 mod(prime(i)) 3<=i<=n.
  • A071556 (program): Smallest x > 1 such that x^prime(n) == 1 mod(prime(i)) 4<=i<=n.
  • A071557 (program): Numbers k such that A065876(k) = k^2-k+1.
  • A071558 (program): Smallest k such that n*k + 1 and n*k - 1 are twin primes.
  • A071561 (program): Numbers with no middle divisors (cf. A071090).
  • A071562 (program): Numbers n such that the sum of the middle divisors of n (A071090) is not zero.
  • A071568 (program): Smallest k>n such that n^3+1 divides k*n^2+1.
  • A071569 (program): Det(M_n) where M_n is the n X n matrix m(i,j)=1 if floor(i/j) is even, 0 otherwise.
  • A071570 (program): a(n) = 2 * Sum_{d|n} 2^mu(d).
  • A071571 (program): Smallest number whose square has exactly 2n+1 divisors.
  • A071575 (program): Number of iterations of cototient(n) needed to reach 1 (cototient(n) = n-phi(n)).
  • A071578 (program): Number of iterations of Pi(n) needed to reach 1, where Pi(x) denotes the number of primes <= x.
  • A071582 (program): Powers of 4 written backwards.
  • A071583 (program): Powers of 5 written backwards.
  • A071584 (program): Powers of 7 written backwards.
  • A071585 (program): Numerator of the continued fraction expansion whose terms are the first-order differences of exponents in the binary representation of 4*n, with the exponents of 2 being listed in descending order.
  • A071586 (program): Powers of 8 written backwards.
  • A071587 (program): Powers of 9 written backwards.
  • A071588 (program): Powers of 6 written backwards.
  • A071589 (program): Numbers n such that Reversal(n) > n.
  • A071590 (program): Numbers k such that reversal(k) < k.
  • A071591 (program): Numbers m such that Reversal(m) is squarefree.
  • A071592 (program): Numbers m such that Reversal(m) is not squarefree.
  • A071602 (program): Sum of the reverses of the first n primes.
  • A071604 (program): a(n) is the number of 7-smooth numbers <= n.
  • A071617 (program): A063439[A000040(n)]=Phi[p]^Phi[p].
  • A071618 (program): a(n+1) - 3*a(n) + a(n-1) = (2/3)(1+w^(n+1)+w^(2n+2)), where w = exp(2 Pi I / 3).
  • A071619 (program): a(n) = ceiling( 2*n^2/3 ).
  • A071621 (program): Primes that can be written as “a * b + c * d”, where a, b, c and d are also primes.
  • A071622 (program): a(n) = (-1)*Sum_{k=1..n} (-1)^floor((4/3)^k).
  • A071623 (program): Least k such that n = A071532(k).
  • A071627 (program): Terms of Chernoff sequence A006939 divided by n!
  • A071635 (program): Number of decompositions of 4*n+2 into sum of two primes of form 4*k+1.
  • A071637 (program): Largest exponent k >=0 such that (n+1)^k divides n!.
  • A071640 (program): a(n) = Sum_{i=1..n} A040051(i).
  • A071642 (program): Numbers n such that x^n + x^(n-1) + x^(n-2) + … + x + 1 is irreducible over GF(2).
  • A071646 (program): Number of base 4 n-digit numbers with digit sum n.
  • A071648 (program): Sum of even decimal digits of n.
  • A071649 (program): Sum of odd decimal digits of n.
  • A071650 (program): Difference between sums of odd and even digits of n.
  • A071675 (program): Array read by antidiagonals of trinomial coefficients.
  • A071676 (program): Array read by antidiagonals of signed variant of trinomial coefficients with T(n,k)=T(n-1,k)+T(n-1,k-1)-T(n-1,k-2) starting with T(0,0)=1.
  • A071678 (program): GCD of n! and the reverse of n!.
  • A071679 (program): Least k such that the maximum number of elements among the continued fractions for k/1, k/2, k/3, k/4, …, k/k equals n.
  • A071683 (program): Nonprimes which are the average of two consecutive Fibonacci numbers.
  • A071684 (program): Number of plane trees with n edges and having an odd number of leaves.
  • A071688 (program): Number of plane trees with even number of leaves.
  • A071695 (program): Lesser members of twin prime pairs of form (4*k+1, 4*k+3), k > 0.
  • A071696 (program): Greater members of twin prime pairs of form (4*k+1,4*k+3), k>0.
  • A071697 (program): Product of twin primes of form (4*k+1,4*k+3), k>0.
  • A071698 (program): Lesser members of twin prime pairs of form (4*k+3, 4*k+5), k >= 0.
  • A071699 (program): Greater members of twin prime pairs of form (4*k+3, 4*k+5), k >= 0.
  • A071700 (program): Product of twin primes of form (4*k+3,4*(k+1)+1), k>=0.
  • A071701 (program): Number of twin prime pairs <= n of form (4*k+1,4*k+3), k>0.
  • A071715 (program): Expansion of (1+x*C^4)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071716 (program): Expansion of (1+x^2*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071717 (program): Expansion of (1 + x^2*C)*C^2, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071718 (program): Expansion of (1+x^2*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071719 (program): Expansion of (1+x^2*C)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071720 (program): Number of spanning trees in K_{n}-e, the complete graph on n nodes minus an edge (n > 1).
  • A071721 (program): Expansion of (1+x^2*C^2)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071722 (program): Expansion of (1+x^2*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071723 (program): Expansion of (1+x^2*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071724 (program): a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.
  • A071725 (program): Expansion of (1+x^2*C^4)*C, where C = (1 - sqrt(1-4*x))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071726 (program): Expansion of (1+x^3*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071727 (program): Expansion of (1+x^3*C)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071728 (program): Expansion of (1+x^3*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071730 (program): Smallest prime p > prime(n) such that p-prime(n) is a square.
  • A071731 (program): Expansion of (1+x^3*C^2)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071732 (program): Expansion of (1+x^3*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071733 (program): Expansion of (1+x^3*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071735 (program): Expansion of (1+x^3*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071736 (program): Expansion of (1+x^3*C^3)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071737 (program): Expansion of (1+x^3*C^3)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071738 (program): Expansion of (1+x^3*C^4)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071739 (program): Expansion of (1+x^3*C^4)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071740 (program): Expansion of (1+x^3*C^4)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071741 (program): Expansion of (1+x^3*C^4)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071742 (program): Expansion of (1+x^4*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071743 (program): Expansion of (1+x^4*C)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071744 (program): Expansion of (1+x^4*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071747 (program): Expansion of (1+x^4*C^2)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071752 (program): Expansion of (1+x^4*C^3)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071754 (program): Sum(k=0,n, pp(k)) where pp(k) is the parity of p(k) the k-th partition number = A040051(k).
  • A071755 (program): Expansion of (1+x^4*C^4)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071756 (program): Expansion of (1+x^4*C^4)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071757 (program): Expansion of (1+x^4*C^4)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
  • A071761 (program): Dealing cards in a game of solitaire.
  • A071762 (program): Leftmost 1 is converted to a 2, which then propagates one step at a time until it is rightmost; then it changes to a pair of 1’s and the process repeats.
  • A071765 (program): Number of n-tuples of elements e_1,e_2,…,e_n in the alternating group A_4 such that the subgroup generated by e_1,e_2,…,e_n is A_4.
  • A071766 (program): Denominator of the continued fraction expansion whose terms are the first-order differences of exponents in the binary representation of 4n, with the exponents of 2 being listed in descending order.
  • A071768 (program): Determinant of the n X n matrix whose element (i,j) equals |i-j| (Mod 3).
  • A071772 (program): Absolute values of the numerator of B(prime(n)-1) where B(k) are the Bernoulli numbers.
  • A071773 (program): a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.
  • A071778 (program): Number of ordered triples (a, b, c) with gcd(a, b, c) = 1 and 1 <= {a, b, c} <= n.
  • A071784 (program): Determinant of the n X n matrix whose element (i,j) equals the floor( Phi^(i-j) + 1).
  • A071789 (program): Decimal expansion of the first (of 10) decimal selvage numbers; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071790 (program): Decimal expansion of the second (of 10) decimal selvage numbers; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071791 (program): Decimal expansion of the third (of 10) decimal selvage numbers; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071792 (program): Decimal expansion of the fourth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071793 (program): Decimal expansion of the fifth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071797 (program): Restart counting after each new odd integer (a fractal sequence).
  • A071798 (program): Number of paths on the surface of the n-dimensional lattice [0..2]^n; i.e., the lattice paths that do not pass through the point (1,1,…,1).
  • A071799 (program): Number of lattice paths in the lattice [0..2n] X [0..2n] which do not pass through the point (n,n).
  • A071801 (program): a(n) = binomial(2n, n) - binomial(n, floor(n/2))^2.
  • A071814 (program): Number of 1’s in binary representation of n equals bigomega(n), the number of prime divisors of n (counted with multiplicity).
  • A071816 (program): Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.
  • A071820 (program): Kolakoski-(2,3) sequence: a(n) is length of n-th run.
  • A071821 (program): Numbers whose largest prime factor is of the form 4k+1.
  • A071822 (program): Numbers whose largest prime factor is of the form 4k+3.
  • A071823 (program): Number of numbers x <= n with largest prime factor of the form 4k+3.
  • A071824 (program): Number of x with largest prime factor of the form 4k+1 less than or equal to n.
  • A071838 (program): a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.
  • A071840 (program): Number of primes == 3 mod 8 <= n.
  • A071841 (program): Number of primes == 5 mod 8 <= n.
  • A071842 (program): Number of primes == 7 mod 8 <= n.
  • A071850 (program): Smallest k > n such that n divides prime(k) - prime(n).
  • A071854 (program): a(n) = (number of distinct prime factors in C(2n,n)) - pi(n).
  • A071858 (program): (Number of 1’s in binary expansion of n) mod 3.
  • A071860 (program): Number of k 1<=k<=n such that sigma(k) is odd.
  • A071868 (program): Number of k (1 <= k <= n) such that k^2+1 is prime.
  • A071869 (program): Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.
  • A071870 (program): Numbers k such that gpf(k) > gpf(k+1) > gpf(k+2) where gpf(k) denotes the largest prime factor of k.
  • A071873 (program): Decimal expansion of the sixth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071874 (program): Decimal expansion of the seventh (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071875 (program): Decimal expansion of the eighth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071876 (program): Decimal expansion of the ninth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071877 (program): Decimal expansion of the tenth (of 10) decimal selvage number; the n-th digit of a decimal selvage number, x, is equal to the tenths digit of n*x.
  • A071878 (program): G.f. D(x) satisfies: D(x) = (1 + x*D(x))*(1 + 2*x*D(x))*(1 + 3*x*D(x)).
  • A071879 (program): G.f. satisfies: A(x) = 1 + x*A(x) + x^3*A(x)^3.
  • A071883 (program): A002487(n)*A002487(n+2).
  • A071895 (program): CONTINUANT transform of Fibonacci number 1, 2, 3, 5, 8, …
  • A071896 (program): CONTINUANT transform of triangular numbers 1, 3, 6, 10, …
  • A071897 (program): CONTINUANT transform of Catalan numbers 1, 2, 5, 14, 42, …
  • A071898 (program): CONTINUANT transform of A002487: 1, 1, 2, 1, 3, 2, …
  • A071902 (program): Sum_{k=0..n^2} (k^2 - n^2)/n.
  • A071903 (program): Number of x less than or equal to n and divisible only by primes congruent to 3 mod 4 (i.e., in A004614).
  • A071904 (program): Odd composite numbers.
  • A071906 (program): Sum of digits of 2^n (mod 2).
  • A071907 (program): Kolakoski-(1,4) sequence: a(n) is length of n-th run.
  • A071908 (program): A002487(n)*A002487(n+1)*A002487(n+2).
  • A071909 (program): A002487(n)*A002487(n+1)*A002487(n+2)/2.
  • A071910 (program): a(n) = t(n)*t(n+1)*t(n+2), where t() are the triangular numbers.
  • A071911 (program): Numbers m such that Stern’s diatomic A002487(m) is divisible by 3.
  • A071913 (program): Smallest k such that A069624(k) = n.
  • A071917 (program): Number of pairs (x,y) where x is even, y is odd, 1<=x<=n, 1<=y<=n and x+y is prime.
  • A071919 (program): Number of monotone nondecreasing functions [n]->[m] for n >= 0, m >= 0, read by antidiagonals.
  • A071920 (program): Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=0 for all m>=0, read by antidiagonals.
  • A071921 (program): Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=1 by definition, read by antidiagonals.
  • A071922 (program): Unimodal analog of binomial coefficient, such that A071921(n,m) = a(n+m-1,n) for all (n,m) different from (0,0), arranged in a Pascal-like triangle.
  • A071925 (program): Digitally balanced numbers: binary numbers which have the same number of 0’s as 1’s; decimal representation: A031443.
  • A071928 (program): Kolakoski-(2,4) sequence: a(n) is length of n-th run.
  • A071930 (program): Number of words of length 2n in the two letters s and t that reduce to the identity 1 by using the relations ssTT=1, ststSS=1 and ststTT=1, where S and T are the inverses of s and t, respectively (i.e., sS=1 and tT=1). The generators s and t and the three stated relations generate the quaternion group Q4.
  • A071932 (program): a(n) = 4*Sum_{i=1..n} K(i,i+1) - n, where K(x,y) is the Kronecker symbol (x/y).
  • A071933 (program): a(n) = Sum_{i=1..n} K(i,i+1), where K(x,y) is the Kronecker symbol (x/y).
  • A071934 (program): a(n) = Sum_{i=1..n} K(i+1,i), where K(x,y) is the Kronecker symbol (x/y).
  • A071935 (program): K(n,n+1) where K(x,y) is the Kronecker symbol (x/y).
  • A071936 (program): K(n+1,n) where K(x,y) is the Kronecker symbol (x/y).
  • A071937 (program): Reverse(n)!.
  • A071942 (program): Kolakoski-(3,4) sequence: a(n) is length of n-th run.
  • A071943 (program): Triangle T(n,k) (n>=0, 0 <= k <= n) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R=(1,0), V=(0,1) and D=(1,2).
  • A071947 (program): Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.
  • A071948 (program): Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.
  • A071952 (program): Diagonal T(n,3) of triangle in A071951.
  • A071953 (program): Diagonal T(n,n-2) of triangle in A071951.
  • A071954 (program): a(n) = 4*a(n-1) - a(n-2) - 4, with a(0) = 2, a(1) = 4.
  • A071955 (program): a(n) = remainder when n is reduced mod reverse(n).
  • A071960 (program): Largest k >= 0 such that Product_{i=0..k} (n+i) divides n!.
  • A071963 (program): Largest prime factor of p(n), the n-th partition number A000041(n) (with a(0) = a(1) = 1 by convention).
  • A071968 (program): Denominators of coefficients of expansion of arctan(x)^2 = x^2-2/3*x^4+23/45*x^6-44/105*x^8+563/1575*x^10-3254/10395*x^12+ …
  • A071969 (program): a(n) = Sum_{k=0..floor(n/3)} (binomial(n+1, k)*binomial(2*n-3*k, n-3*k)/(n+1)).
  • A071975 (program): Denominator of rational number i/j such that Sagher map sends i/j to n.
  • A071978 (program): Right diagonal of A071977.
  • A071981 (program): Parity of the digits of e in base 10.
  • A071982 (program): Parity of the decimal digits of sqrt(2).
  • A071986 (program): Parity of the prime-counting function pi(n).
  • A071990 (program): Numerators of the partial sums of the reciprocals of the upper members of twin prime pairs.
  • A071991 (program): a(1) = a(2) = 1; a(n) = a(floor(n/3)) + a(n - floor(n/3)).
  • A071993 (program): a(n) = 3*n - 2* A003159(n).
  • A071994 (program): a(n) = Sum_{k=1..n} A003159(k).
  • A071995 (program): a(1) = 1, a(2) = 0, a(n) = a(floor(n/3)) + a(n - floor(n/3)).
  • A071996 (program): a(1) = 0, a(2) = 1, a(n) = a(floor(n/3)) + a(n - floor(n/3)).
  • A071998 (program): Write n in binary, interpret that as a decimal number, convert back to binary.
  • A071999 (program): Determinant of n X n matrix whose element A(i,j) is 1 if i=j, i if n=i+j and 0 otherwise.
  • A072000 (program): Number of semiprimes (A001358) <= n.
  • A072001 (program): Variant of the factorial base representation of n.
  • A072003 (program): 10’s complement of final digit of n-th prime.
  • A072004 (program): Remainder when sum of squares of first n primes is divided by n-th prime.
  • A072010 (program): In prime factorization of n replace all primes of form k*4+1 with k*4+3 and primes of form k*4+3 with k*4+1.
  • A072012 (program): a(n) = A072010(A072010(n)).
  • A072013 (program): Numbers k such that A072012(k) = k.
  • A072015 (program): Maxima when the mapping of A072010 is applied to n repeatedly.
  • A072020 (program): Sum of an infinite series: a(n) = Sum_{ k = 0..infinity} ((1/27) * (3^n)^3 * GAMMA(n+1/3*k+1/3) * GAMMA(n+1/3*k+2/3) * GAMMA(n+1/3*k+1) / (GAMMA(4/3+1/3*k) * GAMMA(5/3+1/3*k) * GAMMA(2+1/3*k) * exp(1) * k!).
  • A072024 (program): Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).
  • A072025 (program): a(n) = n^4 + 2*n^3 + 4*n^2 + 3*n + 1 = ((n+1)^5+n^5) / (2*n+1).
  • A072031 (program): Row sums of A072030.
  • A072032 (program): a(n) = gcd(2^n, reverse(2^n)) = gcd(2^n, A004086(2^n)) = A055483(2^n).
  • A072034 (program): a(n) = Sum_{k=0..n} binomial(n,k)*k^n.
  • A072035 (program): Sum(binomial(n,k)*n^k*k^n,k=1..n).
  • A072042 (program): a(n+2) = a(n+1)*a(n)*(1+1/n), a(1)=a(2)=1.
  • A072046 (program): Greatest common divisor of product of divisors of n and product of non-divisors < n.
  • A072047 (program): Number of prime factors of the squarefree numbers: omega(A005117(n)).
  • A072048 (program): Number of divisors of the squarefree numbers: tau(A005117(n)).
  • A072055 (program): a(n) = 2*prime(n)+1.
  • A072056 (program): Number of divisors of 2*prime(n)+1.
  • A072057 (program): Sum of divisors of 2*prime(n)+1.
  • A072058 (program): Squarefree kernel of 2*prime(n)+1.
  • A072061 (program): [t], 1+[t], [2t], 2+[2t], [3t], 3+[3t], …, where t=tau = (1+sqrt(5))/2 and []=floor.
  • A072062 (program): Inverse permutation to A072061.
  • A072063 (program): Smallest prime of form prime(n)+k*n, k>0.
  • A072064 (program): Least k>0 such that prime(n)+k*n is prime.
  • A072065 (program): Define a “piece” to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.
  • A072068 (program): Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.
  • A072078 (program): Number of 3-smooth divisors of n.
  • A072079 (program): Sum of 3-smooth divisors of n.
  • A072081 (program): Numbers divisible by the square of the sum of their digits in base 10.
  • A072082 (program): Numbers divisible by the cube of the sum of their digits in base 10.
  • A072084 (program): In prime factorization of n replace all primes with their numbers of 1’s in binary representation.
  • A072085 (program): a(n) = A072084(A072084(n)).
  • A072086 (program): Number of steps to reach 1, starting with n and applying the A072084-map repeatedly.
  • A072097 (program): Decimal expansion of 180/Pi.
  • A072100 (program): Column 2 of the array m(i,1)=m(1,j)=1 m(i,j)=m(i-1,j-1)+m(i-1,j+1) (a(n)=m(n,2)).
  • A072103 (program): Sorted perfect powers a^b for a, b > 1 with duplication.
  • A072107 (program): a(n) = Sum_{k=1..n} A014963(k).
  • A072110 (program): a(n) = 4*a(n-1) - a(n-2) - 2, with a(0)=1, a(1)=2.
  • A072114 (program): Number of 3-almost primes (A014612) <= n.
  • A072122 (program): Numbers with 12 odd integers in their Collatz (or 3x+1) trajectory.
  • A072123 (program): Remainder when Fibonacci(n) is divided by prime(n).
  • A072126 (program): Parity of the decimal digits of log(2).
  • A072130 (program): a(n+1) -3*a(n) + a(n-1) = (2/3)*(1+w^(n+1)+w^(2*n+2)); a(1) = 0, a(2) = 1; where w is the cubic root of unity.
  • A072134 (program): Second generation sequence in which each number is skipped that can be written as sum of distinct previous entries. To make the first generation we start with all natural numbers: this gives the powers of 2 (A000079). For the second generation we start with the natural numbers from which are removed the numbers of the first generation.
  • A072154 (program): Coordination sequence for the planar net 4.6.12.
  • A072165 (program): Values of Moebius function of the products of two (not necessarily distinct) primes (semiprimes or 2-almost primes, A001358).
  • A072166 (program): Triangle in which first row is {1}; to get n-th row take first n numbers greater than last number in previous row which are congruent to 1 (mod n).
  • A072169 (program): Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.
  • A072172 (program): a(n) = (2*n+1)*5^(2*n+1).
  • A072173 (program): a(n) = (2*n+1)*239^(2*n+1).
  • A072175 (program): a(1)=1, a(2)=2, a(n) = a(n-1) + 1 - 2*sign(a(n-2)) for n>2.
  • A072176 (program): Unimodal analog of Fibonacci numbers: a(n+1) = Sum_{k=0..floor(n/2)} A071922(n-k,k).
  • A072190 (program): Indices of primes with primitive root 2.
  • A072191 (program): a(n) = a(n-1)^2 + 2.
  • A072192 (program): Indices of Sophie Germain primes: p and 2p+1 are primes.
  • A072194 (program): Replace all prime factors p of n with n-p.
  • A072195 (program): Replace all prime factors p of n with n/p.
  • A072196 (program): Multiples of 3 which on one operation of the Collatz function T (N -> 3N+1/2^r) yield the number 5.
  • A072197 (program): a(n) = 4*a(n-1) + 1 with a(0) = 3.
  • A072198 (program): E12 range of preferred resistor values in electronic engineering.
  • A072201 (program): a(n) = 4*a(n-1) + 1, a(1) = 15.
  • A072202 (program): Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity.
  • A072203 (program): (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).
  • A072205 (program): a(n) = (p^2 - p + 2)/2 for p = prime(n); number of squares modulo p^2.
  • A072206 (program): Third terms of triple Peano sequence A071988.
  • A072210 (program): a(1)=a(2)=1; a(n)=reverse(reverse(a(n-1))+reverse(a(n-2))) for n > 2.
  • A072211 (program): a(n) = p-1 if n=p, p if n=p^e and e<>1, 1 otherwise; p a prime.
  • A072219 (program): Any number n can be written uniquely in the form n = 2^k_1 - 2^k_2 + 2^k_3 - … + 2^k_{2r+1} where the signs alternate, there are an odd number of terms, and k_1 > k_2 > k_3 > … > k_{2r+1} >= 0; sequence gives number of terms 2r+1.
  • A072221 (program): a(n) = 6*a(n-1) - a(n-2) + 2, with a(0)=1, a(1)=4.
  • A072223 (program): Decimal expansion of unimodal analog of golden section with respect to A072176: a=lim A072176(n)/A072176(n+1).
  • A072229 (program): Witt index of the standard bilinear form <1,1,1,…,1> over the 2-adic rationals.
  • A072230 (program): a(n) = n! (mod n^2), that is, n factorial modulo n^2.
  • A072251 (program): (3*a(n)+1)/2^(2*n + 1) = 23-6*n.
  • A072253 (program): Numbers n for which one step of the Collatz iteration (3n+1)/2^r gives rise to values 59,53,47,41,35,29,23,17,11 and 5 for r=1,3,5,..,19.
  • A072256 (program): a(n) = 10*a(n-1) - a(n-2) for n > 1, a(0) = a(1) = 1.
  • A072257 (program): a(n) = ((6*n-17)*4^n - 1)/3.
  • A072258 (program): a(n) = ((6*n+1)*4^n - 1)/3.
  • A072259 (program): a(n) = ((6*n+37)*4^n - 1)/3.
  • A072260 (program): a(n) = ((6*n+19)*4^n - 1)/3.
  • A072261 (program): a(n) = 4*a(n-1) + 1, a(1)=7.
  • A072262 (program): a(n) = 4*a(n-1) + 1, a(1)=11.
  • A072263 (program): a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=2, a(1)=3.
  • A072264 (program): a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=1.
  • A072265 (program): Variant of Lucas numbers: a(n) = a(n-1) + 4*a(n-2) starting with a(0)=2 and a(1)=1.
  • A072266 (program): Number of words of length 2n generated by the two letters s and t that reduce to the identity 1 using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.
  • A072272 (program): Number of active cells in n-th stage of growth of two-dimensional cellular automaton defined by “Rule 614”, based on the 5-celled von Neumann neighborhood.
  • A072273 (program): Index of powers of 2 that equal the number of noncongruent roots to the congruence x^2 == k (mod n) for (k,n)=1 and assuming solvability.
  • A072277 (program): Smallest integer > 1 which is both n-gonal and centered n-gonal.
  • A072284 (program): Numbers k begins a new chain of squarefree integers. I.e., k is squarefree but k-1 is not.
  • A072290 (program): Number of digits in the decimal expansion of the Champernowne constant that must be scanned to encounter all n-digit strings.
  • A072292 (program): Number of proper powers b^d <= n (b > 1, d > 1).
  • A072301 (program): Number of positive integers not exceeding n that are relatively prime to sigma(n).
  • A072302 (program): Number of positive integers not exceeding n that are relatively prime to phi(n).
  • A072303 (program): Numbers n for which n is congruent to n^2 mod phi(n).
  • A072327 (program): Numbers k such that k^2 is a term of A072510.
  • A072328 (program): a(n+1) = 2*a(n-2) + a(n-1), with a(0) = 3, a(1) = 0, and a(2) = 2.
  • A072329 (program): a(n+1)=3*a(n-2)+2*a(n-1) a(n)=x^n+y^n+z^n.
  • A072331 (program): a(n) = 2^(n-1)*sum(k=0..n),((n+k)!/(n-k)!)/k!).
  • A072334 (program): Decimal expansion of e^2.
  • A072335 (program): Expansion of 1/((1-x^2)*(1-4*x+x^2)).
  • A072339 (program): Any number n can be written (in two ways, one with m even and one with m odd) in the form n = 2^k_1 - 2^k_2 + 2^k_3 - … + 2^k_m where the signs alternate and k_1 > k_2 > k_3 > … >k_m >= 0; sequence gives minimal value of m.
  • A072341 (program): a(n) = the least natural number k such that k*sigma(n) + 1 is prime.
  • A072342 (program): a(n) = the least natural number k such that k*reverse(n) + 1 is prime.
  • A072343 (program): a(n) = 2^(n-1)*sum(k=0,n,(n+k)!/k!^2).
  • A072344 (program): a(n) = the least natural number k such that k*phi(n) + 1 is prime.
  • A072345 (program): Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives numerator of C_n.
  • A072346 (program): Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives denominator of C_n.
  • A072347 (program): If n = pqr…st in binary, a(n) = value of continuant [p,q,r,…,s,t].
  • A072351 (program): Smallest n-digit Fibonacci number.
  • A072352 (program): a(n) is the largest n-digit Fibonacci number.
  • A072353 (program): a(n) is the index of the largest Fibonacci number containing n digits.
  • A072354 (program): a(n)-th Fibonacci number is the smallest Fibonacci number containing n digits.
  • A072357 (program): Cubefree nonsquares whose factorization into a product of primes contains exactly one square.
  • A072358 (program): Number of cubefree numbers <= n which are not squarefree.
  • A072365 (program): Decimal expansion of (1/3)^(1/3).
  • A072371 (program): a(0) = 0, a(1) = 1, a(n+1) = 2*a(n) + (2*n-1)^2*a(n-1).
  • A072372 (program): a(0) = 1, a(1) = 1, a(n) = 2*a(n-1) + (2*n-1)^2*a(n-2) for n > 1.
  • A072373 (program): Complexity of doubled cycle (regarding case n = 2 as a graph).
  • A072374 (program): a(1) = 1; a(n) = 1 + Sum_{i=1..n} Product_{j=i..2*i-1} (n-j).
  • A072375 (program): Number of cubefree numbers <= n which are nonsquares having exactly one square in their factorization.
  • A072376 (program): a(n) = a(floor(n/2)) + a(floor(n/4)) + a(floor(n/8)) + … starting with a(0)=0 and a(1)=1.
  • A072378 (program): Numbers n such that 12*n divides F(12*n), where F(m) is the m-th Fibonacci number.
  • A072379 (program): Sum_{k<=n} (sigma(k)^2), where sigma(k) denotes the sum of the divisors of k A000203.
  • A072380 (program): Third differences of partition numbers A000041.
  • A072388 (program): a(0) = 1; for n > 0, a(n) = floor((prime(n+1) + prime(n) + a(n-1))/3).
  • A072390 (program): Sum of two powers of 13.
  • A072391 (program): D2(n,n) = Sum_{1<=k<=n} (d_n(k^2)), where d_a(k^2)=card{d: d|k and 1<=d<=a} for real a.
  • A072400 (program): (Factors of 4 removed from n) modulo 8.
  • A072401 (program): 1 iff n is of the form 4^m*(8k+7).
  • A072403 (program): Numerator of the Reingold-Tarjan sequence, denominator=A072404.
  • A072405 (program): Triangle T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise, read by rows.
  • A072406 (program): Number of values of k for which C(n,k)-C(n-2,k-1) is odd.
  • A072418 (program): Parity of floor(3^n/2^n).
  • A072436 (program): Remove prime factors of form 4*k+3.
  • A072437 (program): Numbers with no prime factors of form 4*k+3.
  • A072438 (program): Remove prime factors of form 4*k+1.
  • A072442 (program): Least k such that Sum( Cos(1/Sqrt(i)) i=1..k) > n.
  • A072448 (program): Squares of the terms of the decimal expansion of Pi.
  • A072451 (program): Number of odd terms in the reduced residue system of 2*n-1.
  • A072452 (program): a(n) = reversal(a(n-1)+n) for n>0, a(0) = 0.
  • A072453 (program): Shadow transform of A000522.
  • A072457 (program): Shadow transform of tetrahedral numbers A000292.
  • A072458 (program): Shadow transform of Catalan numbers A000108.
  • A072464 (program): Code word lengths for non-redundant MML code for positive integers.
  • A072465 (program): A Fibonacci-like model in which each pair of rabbits dies after the birth of their 4th litter: a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5).
  • A072466 (program): Numbers with 11 odd integers in their Collatz (or 3x+1) trajectory.
  • A072473 (program): a(n) = prime(2*n) - prime(n).
  • A072474 (program): Sum of next n squares.
  • A072475 (program): Sum of next n composite numbers.
  • A072476 (program): Difference between the sum of first n prime numbers and the sum of first n composite numbers.
  • A072477 (program): (2*n)!*binomial(2*n,n)/8.
  • A072478 (program): Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1) = n*Pi^(n/2)*r^(n-1)/(n/2)! = S_n*Pi^floor(n/2)*r^(n-1); sequence gives numerator of S_n.
  • A072479 (program): Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1) = n*Pi^(n/2)*r^(n-1)/(n/2)! = S_n*Pi^floor(n/2)*r^(n-1); sequence gives denominator of S_n.
  • A072480 (program): Shadow transform of factorials A000142.
  • A072481 (program): a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d).
  • A072486 (program): a(1) = 1, a(n) = a(n-1) times smallest prime factor of n.
  • A072487 (program): a(1) = 1, a(n) = a(n-1) times largest nontrivial divisor if n is composite.
  • A072488 (program): a(1) = 1, a(n) = a(n-1) times largest divisor of n <= n^(1/2).
  • A072489 (program): a(1) = 1, a(n) = a(n-1) times smallest divisor of n >= n^(1/2).
  • A072490 (program): Number of squarefree numbers (excluding 1) less than n.
  • A072491 (program): Define f(1) = 0. For n>=2, let f(n) = n - p where p is the largest prime <= n. a(n) = number of iterations of f to reach 0, starting from n.
  • A072492 (program): Values of n for which A072491(n)=3.
  • A072493 (program): a(1) = 1 and a(n) = ceiling((Sum_{k=1..n-1} a(k))/3) for n >= 2.
  • A072497 (program): Numbers n such that n^2 is member of A072498.
  • A072498 (program): n is not equal to the product of the k smallest divisors of n for any k.
  • A072499 (program): Product of divisors of n which are <= n^(1/2).
  • A072500 (program): Product of divisors of n which are >= n^(1/2).
  • A072501 (program): Ratio of the product of divisors of n which are > n^(1/2) to product of divisors of n which are < n^(1/2).
  • A072502 (program): Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways.
  • A072504 (program): a(n) = LCM of divisors of n which are <= sqrt(n).
  • A072505 (program): a(n) = n / (LCM of divisors of n which are <= sqrt(n)).
  • A072507 (program): Smallest start of n consecutive integers with n divisors, or 0 if no such number exists.
  • A072510 (program): Numbers n with property that n = product of first k divisors of n for some k.
  • A072512 (program): Product of all n - d, where 1 < d < n and d is a divisor of n.
  • A072513 (program): Product of all n - d, where d < n and d is a divisor of n.
  • A072514 (program): Sum of n mod k for k in {1…n} with gcd(k,n) > 1.
  • A072515 (program): Let u(1) = u(2) = v(1) = v(2) = 1, u(n+2) = u(n+1)+v(n), v(n+2) = abs(u(n+1)-v(n)), then a(n) = u(n).
  • A072516 (program): Sum of remainders when n is divided by lesser squares.
  • A072524 (program): Sum of the remainders when the n-th triangular number is divided by all smaller triangular numbers > 1.
  • A072527 (program): Number of values of k such that n divided by k leaves a remainder 3.
  • A072528 (program): Table T(n,k) read by rows, giving number of occurrences of the remainder k when n is divided by i=1,2,3,…,n.
  • A072529 (program): Product of the next n multiples of n.
  • A072541 (program): List of pairs of numbers (k, k+4), where k-1 and k+3 are primes.
  • A072547 (program): Main diagonal of the array in which first column and row are filled alternatively with 1’s or 0’s and then T(i,j) = T(i-1,j) + T(i,j-1).
  • A072548 (program): a(n) = sigma(n) mod PrimePi(n).
  • A072553 (program): Sigma of n-th composite number equals a(n)-th composite number if it is also a composite or equals zero if sigma[c] is prime.
  • A072557 (program): Let w(n) be defined by the following recurrence: w(1)=w(2)=w(3)=1, w(n)=(w(n-1)*w(n-2)+(w(n-1)+w(n-2))/3) / w(n-3); sequence gives values of n such that w(n) is an integer.
  • A072560 (program): Denominators of w(n) where w(1)=w(2)=w(3)=1, w(n)=(w(n-1)*w(n-2)+(w(n-1)+w(n-2))/3) / w(n-3).
  • A072561 (program): Denominators of w(n) equals 3 where w(1)=w(2)=w(3)=1, w(n)=(w(n-1)*w(n-2)+(w(n-1)+w(n-2))/3) / w(n-3).
  • A072563 (program): 9*w(n) where : w(1)=w(2)=w(3)=1 w(n)=(w(n-1)*w(n-2)+(w(n-1)+w(n-2))/3) / w(n-3).
  • A072565 (program): a(n) = prime(n+1)*prime(n+2)+1 mod prime(n), where prime(n) is the n-th prime.
  • A072566 (program): Numbers n such that n and sigma(n) end with the same digit in base 10.
  • A072568 (program): Even interprimes.
  • A072569 (program): Odd interprimes.
  • A072572 (program): Odd interprimes divisible by 3.
  • A072586 (program): Number of numbers <= n having prime factors with odd exponents only.
  • A072587 (program): Numbers having at least one prime factor with an even exponent.
  • A072588 (program): Numbers having at least one prime factor with an odd and one with an even exponent.
  • A072590 (program): Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals.
  • A072592 (program): Even numbers with at least one prime factor of form 4*k+1.
  • A072597 (program): Expansion of 1/(exp(-x) - x) as exponential generating function.
  • A072600 (program): Numbers which in base 2 have fewer 0’s than 1’s.
  • A072601 (program): Numbers which in base 2 have at least as many 1’s as 0’s.
  • A072602 (program): Numbers such that in base 2 the number of 0’s is >= the number of 1’s.
  • A072603 (program): Numbers which in base 2 have more 0’s than 1’s.
  • A072608 (program): Parity of remainder Mod[p(n),n]=A004648(n).
  • A072609 (program): Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range […1111..]. AC-range corresponds to 0, while DC-range labeled by 1.
  • A072613 (program): Number of numbers of the form p*q (p, q distinct primes) less than or equal to n.
  • A072614 (program): a(n) = Sum_{k = 1..n} (-1)^(n reduced mod k).
  • A072624 (program): Mod[Prime[n^2],n^2]
  • A072627 (program): Number of divisors d of n such that d-1 is prime.
  • A072628 (program): Number of divisors d of n such that d-1 is not prime.
  • A072629 (program): Parity of n*floor(log n).
  • A072630 (program): Values of n where A072629 switches from 01010.. into 0000.. or back.
  • A072632 (program): Solutions to A072631[n]=0.
  • A072633 (program): Smallest positive integer m where 1^n+2^n+3^n+…+m^n is greater than or equal to (m+1)^n.
  • A072638 (program): Number of unary-binary rooted trees of height at most n.
  • A072639 (program): a(0) = 0, a(n) = Sum_{i=0..n-1} 2^((2^i)-1).
  • A072641 (program): Binary widths of the terms of A072638.
  • A072643 (program): Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2.
  • A072648 (program): a(n) = [log_{Phi}(n*sqrt(5))], where log_{Phi} is logarithm in the base Phi ( = (sqrt(5)+1)/2) and [] stands for the floor function.
  • A072649 (program): n occurs Fibonacci(n) times (cf. A000045).
  • A072650 (program): Starting from the right (the least significant end) rewrite 0 to 0 and x1 to 1 in the binary expansion of n.
  • A072654 (program): Maximum position in A072645 where the value n occurs.
  • A072655 (program): Binary widths of the terms of A072654.
  • A072661 (program): Composition of the A059905 and A048679, i.e., a(n) = A059905(A048679(n)).
  • A072662 (program): Composition of the A059906 and A048679, i.e., a(n) = A059906(A048679(n)).
  • A072666 (program): Numbers n such that prime(n) + prime(n+1) - 1 is prime.
  • A072667 (program): Consider m such that prime(m) + prime(m+1) = prime(k) + 1 for some k; sequence gives values of prime(m).
  • A072668 (program): Numbers one less than composite numbers.
  • A072669 (program): Primes of the form prime(x) + prime(x+1) - 1.
  • A072670 (program): Number of ways to write n as i*j + i + j, 0 < i <= j.
  • A072672 (program): Prime(n)*prime(2*n)+prime(n)+prime(2*n).
  • A072674 (program): 3^n+2*2^n-3.
  • A072675 (program): Last digit of F(n) is 1 where F(n) is the n-th Fibonacci number.
  • A072677 (program): a(n) = prime(prime(n)+1) where prime(k) is the k-th prime.
  • A072678 (program): Generalized Bell numbers B_{4,2}.
  • A072680 (program): Difference between (least prime >= n) and (largest prime <= n).
  • A072681 (program): a(n) = (n - A007917(n)) * (A007918(n) - n).
  • A072682 (program): Numbers congruent to {3, 36, 54, 57} mod 60.
  • A072683 (program): Numbers k such that the last digit of F(k) is 3 where F(k) is the k-th Fibonacci number.
  • A072689 (program): Difference between (least square >= n) and (largest square <= n).
  • A072690 (program): a(n) = (n - A048760(n)) * (A048761(n) - n).
  • A072691 (program): Decimal expansion of Pi^2/12.
  • A072695 (program): a(n) = lcm(d(n^2),d(n)), where d(n) = A000005, the number of divisors of n.
  • A072696 (program): a(n) = lcm(d(n^3), d(n)), where d(n) = A000005, the number of divisors of n.
  • A072697 (program): Squarefree numbers such that the sum of the prime factors is a multiple of the number of prime factors.
  • A072698 (program): Sum of prime factors of A072697(n).
  • A072699 (program): Number of prime factors of A072697(n).
  • A072702 (program): Last digit of F(n) is 4 where F(n) is the n-th Fibonacci number.
  • A072703 (program): Indices of Fibonacci numbers whose last digit is 5.
  • A072708 (program): Last digit of F(n) is 6 where F(n) is the n-th Fibonacci number.
  • A072709 (program): Last digit of F(n) is 7 where F(n) is the n-th Fibonacci number.
  • A072710 (program): Last digit of F(n) is 8 where F(n) is the n-th Fibonacci number.
  • A072711 (program): Last digit of F(n) is 9 where F(n) is the n-th Fibonacci number.
  • A072713 (program): a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n)*a(n-5) = a(n-1)*a(n-2)*a(n-3)*a(n-4)+1.
  • A072726 (program): Numerator of the rationals >= 1 whose continued fractions consist of only even terms, in ascending order by the sum of the continued fraction terms and descending by lowest order continued fraction terms to highest.
  • A072727 (program): Denominator of the rationals >= 1 whose continued fractions consist of only even terms, in ascending order by the sum of the continued fraction terms and descending by lowest order continued fraction terms to highest.
  • A072731 (program): Difference of numbers of composite and prime numbers <= n.
  • A072747 (program): Counting factor 2 in the first n squarefree numbers.
  • A072748 (program): Counting factor 3 in the first n squarefree numbers.
  • A072749 (program): Count of factors of 5 in the first n squarefree numbers.
  • A072750 (program): Counting factor 7 in the first n squarefree numbers.
  • A072756 (program): a(n)=least positive integer not a(k)+Floor(a(k)/2) for k<n.
  • A072757 (program): Complement of A072756.
  • A072762 (program): n coded as binary word of length=n with k-th bit set iff k is prime (1<=k<=n), decimal value.
  • A072768 (program): The RASTxx transformation of the sequence A072643. The sizes of the parenthesizations produced by ‘cons’ combination A072764 & its transpose A072766.
  • A072770 (program): Triangle A072768 computed modulo 2.
  • A072774 (program): Powers of squarefree numbers.
  • A072775 (program): Squarefree kernels of powers of squarefree numbers.
  • A072776 (program): Exponents of powers of squarefree numbers.
  • A072779 (program): a(n) = sigma_2(n) + phi(n) * sigma(n).
  • A072780 (program): a(n) = sigma_2(n) + phi(n) * sigma(n) - 2*n^2, which is A072779(n) - 2*n^2.
  • A072783 (program): Differences between A072740 and A072736.
  • A072784 (program): Differences between A072741 and A072737.
  • A072785 (program): Differences between A072781 and A072738.
  • A072786 (program): Differences between A072782 and A072739.
  • A072795 (program): A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: . thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().
  • A072800 (program): Composition of A030101 and A014486. Binary encodings of parenthesizations, Dyck paths and other Catalan structures reversed.
  • A072805 (program): Primes of form 4k+3 written in base 3.
  • A072806 (program): Primes of the form 6k+5 written in base 5.
  • A072810 (program): a(0)=1, a(n) = a(n-1) - sum_{k=2..n} mu(k)a(n-k), where mu(k) is the Moebius function of k.
  • A072815 (program): Sum of proper divisors of 6n + 1.
  • A072818 (program): Possibly the only integers of the form sqrt(m^2*(m^2-1)*2/3) [only checked for the first 5 terms].
  • A072819 (program): Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time.
  • A072822 (program): The terms of A073215 (sums of two powers of 23) divided by 2.
  • A072823 (program): Numbers that are not the sum of two powers of 2.
  • A072830 (program): Absolute value of 2*b(n)-9*n, where b(n) = accumulative sum of the greatest digit of n minus the least digit of n (A037904).
  • A072831 (program): Number of bits in n!.
  • A072833 (program): Numbers that are congruent to 0, 5, 8, 9 mod 12.
  • A072834 (program): Exponents occurring in expansion of F_8(q^2).
  • A072835 (program): Exponents occurring in expansion of F_9(q^2).
  • A072844 (program): Number of words of length 2n+1 generated by the two letters s and t that reduce to the identity 1 by using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.
  • A072845 (program): {1, 3, 7, 9} -> Mod[ {1*{1, 3, 7, 9}, 3*{1, 3, 7, 9}, 7*{1, 3, 7, 9}, 9*{1, 3, 7, 9}}, 10}
  • A072860 (program): Highest power of 3 dividing the period length of 1/prime(n) = A002371(n).
  • A072861 (program): a(n) = sigma(n)^2.
  • A072862 (program): The smallest k>1 such that k divides sigma(k*n) is equal to 6.
  • A072863 (program): a(n) = 2^(n-3)*(n^2+3*n+8).
  • A072864 (program): Numbers n such that the smallest k dividing sigma(k*n^2) is equal to 3.
  • A072869 (program): a(n) = sigma(sigma(n)-n), where sigma = A000203, sum of the divisors of n.
  • A072872 (program): a(n) is the smallest positive number k such that n divides 2^k - k.
  • A072876 (program): a(1) = a(2) = a(3) = a(4) = 1 and a(n) = (a(n-1)*a(n-3) + a(n-2)^3)/a(n-4) for n >= 5.
  • A072877 (program): a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)*a(n-3) + a(n-2)^4)/a(n-4).
  • A072878 (program): a(n) = 4*a(n-1)*a(n-2)*a(n-3) - a(n-4) with a(1) = a(2) = a(3) = a(4) = 1.
  • A072881 (program): a(1)=a(2)=a(3)=1; for n>3, a(n)=(a(n-1)*a(n-2)+a(n-1)+a(n-2))/a(n-3).
  • A072882 (program): A nonlinear recurrence of order 3: a(1)=a(2)=a(3)=1; a(n)=(a(n-1)+a(n-2))^2/a(n-3).
  • A072886 (program): The s-aints, numbers generated like the Aronson series from a generating sentence, “S ain’t the second, third, fourth, fifth . . . letter of this sentence.”.
  • A072894 (program): Let c(k) be defined as follows: c(1)=1, c(2)=n, c(k+2) = c(k+1)/2 + c(k)/2 if c(k+1) and c(k) have the same parity; c(k+2) = c(k+1)/2 + c(k)/2 + 1/2 otherwise; a(n) = limit_{ k -> infinity} c(k).
  • A072898 (program): Numerator of c(n) where c(0)=1, c(n+1) = n/c(n) + 1.
  • A072899 (program): Denominator of c(n) where c(0)=1 c(n+1) = n/c(n) + 1.
  • A072900 (program): Discriminant of quadratic field Q(sqrt(prime(n))) where prime(n) is the n-th prime.
  • A072901 (program): Composite numbers n such that the discriminant of the quadratic field Q(sqrt(n)) equals 4n.
  • A072902 (program): Nonprime numbers m such that the discriminant of the quadratic field Q(sqrt(m)) equals m.
  • A072905 (program): a(n) is the least k > n such that k*n is a square.
  • A072906 (program): Least k >=1 such that floor(n/k) is squarefree.
  • A072909 (program): Least k>0 such that n+k is squarefree.
  • A072912 (program): Number of Fibonacci numbers F(k) <= 10^n which end in 0.
  • A072913 (program): Numerators of (1/4!)*(H(n,1)^4+6*H(n,1)^2*H(n,2)+8*H(n,1)*H(n,3)+3*H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.
  • A072914 (program): Denominators of 1/4!*(H(n,1)^4+6*H(n,1)^2*H(n,2)+8*H(n,1)*H(n,3)+3*H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.
  • A072917 (program): a(n) = p(n) - phi(n), where p(n) is the least prime greater than phi(n).
  • A072918 (program): a(n) = p(n) - sigma(n), where p(n) is the least prime greater than sigma(n).
  • A072920 (program): a(n) = Sum_{k=1..n} A034693(k).
  • A072921 (program): a(1)=1; a(n) = a(n-1) + [sum of all decimal digits present so far in the sequence].
  • A072926 (program): a(n) = Sum_{k=1..n} A051699(k).
  • A072929 (program): a(n) = Sum_{d dividing n} binomial(2d,d).
  • A072931 (program): Number of ways to write n as a sum of 2 semiprimes.
  • A072939 (program): Define a sequence c depending on n as follows: c(1)=1 and c(2)=n; c(k+2) = (c(k+1) + c(k))/2 if c(k+1) and c(k) have the same parity; otherwise c(k+2) = abs(c(k+1) - 2*c(k)); sequence gives values of n such that lim_{k->oo} c(k) = infinity.
  • A072944 (program): a(1)=2, a(n+1) = 2*a(n) - phi(a(n)) where phi is the Euler totient function A000010.
  • A072946 (program): Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+…+q^(n-1)) and q is a root of unity.
  • A072951 (program): a(n) = Sum_{k=1..n} binomial(k, n mod k).
  • A072953 (program): a(n)=sum(k=1,n,C(n,n reduced (mod k))).
  • A072960 (program): Numbers using only the curved digits 0, 3, 6, 8 and 9.
  • A072961 (program): Numbers using only the digits 2 and 5, or numbers that are both curved and linear.
  • A072966 (program): Numbers which are not the sum of two semiprimes.
  • A072975 (program): a(n) = 2^n*binomial(3*n,n)*(3*n+1).
  • A072976 (program): a(n) = 2^n*binomial(3*n,n)*(3*n+2).
  • A072978 (program): Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.
  • A072979 (program): a(n) = Sum_{k=1..n-1} gcd(k,n)*a(k), a(1) = 1.
  • A072985 (program): Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n >= 2, nu(n) = b*nu(n-1) + lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,3), where (n)_q = (1+q+…+q^(n-1)) and q is a root of unity.
  • A072987 (program): FIBMOD numbers: a(1) = a(2) = 1, a(n) = a(n-1) mod (n-1) + a(n-2) mod (n-2).
  • A072988 (program): Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(3,1), where (n)_q=(1+q+…+q^(n-1)) and q is a root of unity.
  • A072994 (program): Number of solutions to x^n==1 (mod n), 1<=x<=n.
  • A072996 (program): Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,1), where (n)_q=(1+q+…+q^(n-1)) and q is a root of unity.
  • A072998 (program): To get a(n), write n in balanced ternary notation (using only digits -1, 0, 1, -1), then change -1’s to 0’s, 0’s to 1’s, and 1’s to 2’s.
  • A073000 (program): Decimal expansion of arctangent of 1/2.
  • A073010 (program): Decimal expansion of Pi/sqrt(27).
  • A073013 (program): (1/2)*sum(k=1,n,(n+k)!/(k!)^2).
  • A073014 (program): (n!/2)*sum(k=1,n,(n+k)!/(k!)^3).
  • A073015 (program): a(n) is such that 2 = sqrt(1+sqrt(1+sqrt(1+….sqrt(a(n))….))) where there are n sqrt’s.
  • A073016 (program): Decimal expansion of Sum_{n>=1} 1/binomial(2n,n).
  • A073023 (program): Number of solutions to the equation n^x==1 (modx) 1<=x<=n.
  • A073028 (program): a(n) = max{ C(n,0), C(n-1,1), C(n-2,2), …, C(n-n,n) }.
  • A073030 (program): Sum_{k=1..n, gcd(n,k) = 1} 10^(k-1).
  • A073031 (program): Number of ways of making change for n cents using coins of sizes 1, 2, 5, 10 cents, when order matters.
  • A073040 (program): Numbers n such that sum of proper divisors of n is a square.
  • A073044 (program): Triangle read by rows: T(n,k) (n >= 1, n-1 >= k >= 0) = number of n-sequences of 0’s and 1’s with no pair of adjacent 0’s and exactly k pairs of adjacent 1’s.
  • A073046 (program): Write 2*n = p+q (p,q prime), p*q minimal; then a(n) = p*q.
  • A073047 (program): Least k such that x(k)=0 where x(1)=n and x(k)=k*floor(x(k-1)/k).
  • A073052 (program): Decimal expansion of cos(Pi/7).
  • A073055 (program): a(n) = product of first n digits in the decimal expansion of Pi, ignoring decimal point.
  • A073059 (program): a(n) = (1/2)*(A073504(n+1) - A073504(n)).
  • A073060 (program): Multiplication table for 1 and odd primes, read by antidiagonals.
  • A073061 (program): a(1)=1, a(2k)=2*a(k), a(2k+1)=3*a(k).
  • A073065 (program): a(n) = prime(n) * prime(prime(n)).
  • A073071 (program): Least k such that k! > prime(1)*prime(2)*…*prime(n) where prime(n) is the n-th prime.
  • A073072 (program): Minimum value of abs(n^2-x^3) x>0.
  • A073078 (program): Least k such that n divides C(2k,k).
  • A073080 (program): 3 appears three times, 2*3=6 appears six times, 2*6=12 appears twelve times etc.
  • A073085 (program): Numbers n such that 210*n+1 is prime.
  • A073086 (program): Floor[concatenation of eight consecutive numbers from n to n+7 divided by 8].
  • A073088 (program): Sum of first n terms of the simple continued fraction of Sum_{k>=0} 1/2^(2^k) (cf. A007400).
  • A073089 (program): a(n) = (1/2)*(4n - 3 - Sum_{k=1..n} A007400(k)).
  • A073092 (program): Number of numbers of the form x^2 + y^2 (0 <= x <= y) less than or equal to n.
  • A073093 (program): Number of prime power divisors of n.
  • A073094 (program): Final digit of C(2k,k) when not equal to zero.
  • A073102 (program): Primes of the form 210n + 1.
  • A073103 (program): Number of solutions to x^4 == 1 (mod n).
  • A073106 (program): Least k such that there are n primes among the numbers 2*n*x + 1 for 1 <= x <= k.
  • A073107 (program): Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp((1+y)*x)/(1-x) (with n >= 0 and 0 <= k <= n).
  • A073108 (program): Least k such that there are n primes among the numbers n^2 + x^2 for 1 <= x <= k.
  • A073117 (program): a(n+1) = a(n) + a(n) mod n; a(1) = 1.
  • A073121 (program): a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)) with a(1)=1 and (r,s)=(2,2).
  • A073122 (program): Minimal reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n. See A072339.
  • A073123 (program): a(n) is the largest number such that pi(a(n)) = prime(n).
  • A073124 (program): a(n) = prime(1+prime(n)) - prime(prime(n)).
  • A073131 (program): a(n) = prime(prime(n+1)) - prime(prime(n)).
  • A073133 (program): Table by antidiagonals of T(n,k) = n*T(n,k-1) + T(n,k-2) starting with T(n,1) = 1.
  • A073134 (program): Table by antidiagonals of T(n,k)=n*T(n,k-1)-T(n,k-2) starting with T(n,1)=1.
  • A073136 (program): a(n) = prime(n) + prime(prime(n)).
  • A073137 (program): a(n) is the least number whose binary representation has the same number of 0’s and 1’s as n.
  • A073138 (program): Largest number having in its binary representation the same number of 0’s and 1’s as n.
  • A073139 (program): Difference between the largest and smallest number having in binary representation the same number of 0’s and 1’s as n.
  • A073140 (program): Sum of the largest and smallest number having in binary representation the same number of 0’s and 1’s as n.
  • A073145 (program): a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.
  • A073146 (program): Triangle of numbers {a(n,k), n >= 0, 0 <= k <= n} defined by a(0,0)=1, a(n,0)=A000670(n), a(n,n)=A000629(n), a(n,k) = a(n,k-1) + a(n-1,k-1); a(n+1,0) = Sum_{k=0..n} a(n,k).
  • A073155 (program): Leftmost column sequence of triangle A073153.
  • A073156 (program): Main diagonal sequence of triangle A073153.
  • A073157 (program): Number of Schroeder n-paths containing no FFs.
  • A073168 (program): a(n) = A007821(n) - A022449(n).
  • A073169 (program): a(n)=A002808(n)-n, difference between n-th composite and n.
  • A073170 (program): a(1) = a(2) = 0; for n>2, a(n) = prime(n-1)-n+1.
  • A073171 (program): (n^2)-th composite number.
  • A073177 (program): (n-th digit of Pi) times (n-th digit of e).
  • A073180 (program): Number of divisors of n which are not greater than the squarefree kernel of n.
  • A073181 (program): Sum of divisors of n which are not greater than the squarefree kernel of n.
  • A073182 (program): Number of divisors of n which are not greater than the cubefree kernel of n.
  • A073183 (program): Sum of divisors of n that are not greater than the cubefree kernel of n.
  • A073184 (program): Number of cubefree divisors of n.
  • A073185 (program): Sum of cubefree divisors of n.
  • A073188 (program): n appears 1+[n/3] times.
  • A073189 (program): Integers 0..k three times, then 0..k+1 three times, etc.
  • A073211 (program): Sum of two powers of 11.
  • A073212 (program): n-th digit of Pi + n-th digit of e.
  • A073213 (program): Sum of two powers of 17.
  • A073214 (program): Sum of two powers of 19.
  • A073215 (program): Sum of two powers of 23.
  • A073216 (program): The terms of A055235 (sums of two powers of 3) divided by 2.
  • A073217 (program): The terms of A055237 (sums of two powers of 5) divided by 2.
  • A073218 (program): The terms of A055258 (sums of two powers of 7) divided by 2.
  • A073219 (program): The terms of A073211 (sums of two powers of 11) divided by 2.
  • A073220 (program): Terms of A072390 (sums of two powers of 13) divided by 2.
  • A073221 (program): The terms of A073213 (sums of two powers of 17) divided by 2.
  • A073222 (program): A073214/2.
  • A073223 (program): a(n) = abs((n-th digit of Pi) - (n-th digit of e)).
  • A073225 (program): a(n) = ceiling(n^n/n!).
  • A073245 (program): Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.
  • A073246 (program): Prime digits in the decimal expansion of e.
  • A073252 (program): Coefficients of replicable function number “48g”.
  • A073254 (program): Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.
  • A073255 (program): Sum of divisors of n-th composite number.
  • A073256 (program): a(n) = phi(n-th composite number).
  • A073258 (program): Numbers n such that gcd(c(n),n) = gcd(A002808(n),n) = A064814(n)=1 where c(n) is the n-th composite number.
  • A073260 (program): Length of FixedPointList leading to value of [10^n]-th composite number.
  • A073264 (program): Prime digits in the decimal expansion of Pi.
  • A073267 (program): Number of compositions (ordered partitions) of n into exactly two powers of 2.
  • A073268 (program): Number of plane binary trees whose right (or respectively: left) subtree is a unique “complete” tree of (2^m)-1 nodes with all the leaf-nodes at the same depth m and the left (or respectively: right) subtree is any plane binary tree of size n - 2^m + 1.
  • A073271 (program): a(n) = floor( prime(n)*prime(n+2) / prime(n+1) ).
  • A073272 (program): A000040(n+1) - A073271(n).
  • A073273 (program): a(n) = floor(sqrt(prime(n)*prime(n+2))).
  • A073274 (program): A000040(n+1) - A073273(n).
  • A073278 (program): A triangle constructed from the coefficients of the n-th derivative of the normal probability distribution function.
  • A073300 (program): If n=pqr…st in ternary, a(n)=value of the continuant [p,q,r,…,s,t].
  • A073302 (program): Indices of prime digits (2, 3, 5, 7) in the decimal expansion of e.
  • A073303 (program): Indices of prime digits in the decimal expansion of Pi.
  • A073304 (program): Remaining days in non-leap year at end of n-th month.
  • A073305 (program): Remaining days in leap year at end of n-th month.
  • A073306 (program): a(n) = Product_{2i<n} binomial(2*n-2*i-1, 2*i).
  • A073311 (program): Number of squarefree numbers in the reduced residue system of n.
  • A073312 (program): Number of nonsquarefree numbers in the reduced residue system of n.
  • A073313 (program): Binomial transform of generalized Lucas numbers S(n) = S(n-1) + S(n-2) + S(n-3), S(0)=3, S(1)=1, S(2)=3.
  • A073314 (program): Binomial transform, alternating in sign, of Lucas generalized numbers S(n): S(n) = S(n-1) + S(n-2) + S(n-3), S(0)=3, S(1)=1, S(2)=3.
  • A073333 (program): Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).
  • A073334 (program): The so-called “rhythmic infinity system” of Danish composer Per Nørgård [Noergaard].
  • A073337 (program): Primes of the form 4*k^2 - 10*k + 7 with k positive.
  • A073338 (program): Positive values of n for which 4n^2-10n+7 is prime.
  • A073342 (program): Average digit (rounded to the nearest integer) in the decimal expansion of n-th prime.
  • A073351 (program): n^2(n+1)(2n+1)^2(7n+1)/36.
  • A073352 (program): Positive integers making n^2*(n-1)*(2*n-1)^2*(7*n-1)/36 a square.
  • A073353 (program): Sum of n and its squarefree kernel.
  • A073354 (program): Binomial coefficient ( n, squarefree kernel(n) ).
  • A073355 (program): Sum of squarefree kernels of numbers <= n.
  • A073356 (program): Greatest common divisor of squarefree kernel of n and sum of squarefree kernels of numbers <= n.
  • A073357 (program): Binomial transform of tribonacci numbers.
  • A073358 (program): Binomial transform, alternating in sign, of the tribonacci numbers.
  • A073359 (program): Nested floor product of n and fractions (2k+2)/(2k+1) for all k>=0, divided by 2.
  • A073360 (program): Nested floor product of n and fractions (k+1)/k for all k>0 (mod 3), divided by 3.
  • A073361 (program): Nested floor product of n and fractions (k+1)/k for all k>0 (mod 4), divided by 4.
  • A073362 (program): Nested floor product of n and fractions (k+1)/k for all k>0 (mod 5), divided by 5.
  • A073363 (program): Nested floor product of n and fractions (k+1)/k for all k>0 (mod 6), divided by 6.
  • A073366 (program): Remainder when n-th prime is divided by number of composites not exceeding n.
  • A073367 (program): Remainder when n-th composite is divided by number of composites not exceeding n.
  • A073368 (program): Remainder when n is divided by number of composites not exceeding n.
  • A073370 (program): Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.
  • A073371 (program): Convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n >= 0, with itself.
  • A073372 (program): Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073373 (program): Third convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073374 (program): Fourth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073375 (program): Fifth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073376 (program): Sixth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073377 (program): Seventh convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073378 (program): Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073379 (program): Ninth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
  • A073380 (program): Third convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073381 (program): Fourth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073382 (program): Fifth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073383 (program): Sixth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073384 (program): Seventh convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073385 (program): Eighth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073386 (program): Ninth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
  • A073387 (program): Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.
  • A073388 (program): Convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073389 (program): Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073390 (program): Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073391 (program): Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073392 (program): Fifth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073393 (program): Sixth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073394 (program): Seventh convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
  • A073395 (program): Product of sums of prime factors of n: with and without repetition.
  • A073397 (program): Eighth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.
  • A073398 (program): Ninth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.
  • A073409 (program): Largest prime factor of the denominator of the Bernoulli number B(2*n) (A002445).
  • A073411 (program): cototient(x) - 1 where x are the odd semiprimes (i.e., x are odd terms in A001358).
  • A073414 (program): Numerator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).
  • A073415 (program): Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).
  • A073423 (program): Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.
  • A073424 (program): Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 …, m = 0,1,2,3, … n.
  • A073425 (program): a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.
  • A073432 (program): Numbers n such that prime(n) + composite(n) is even.
  • A073433 (program): Value of [p(n)+c(n)]/2 when it is integer; p(n) is n-th prime, c(n) is n-th composite.
  • A073437 (program): Smallest x such that remainder Mod[A065855(x), A000720(x)]=n.
  • A073438 (program): Remainder of division G[n]/Pi[n], where G[n] is the number of composites not exceeding n.
  • A073445 (program): Second differences of A002808, the sequence of composites.
  • A073446 (program): Product L(n)*S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) * A001644(n).
  • A073447 (program): Decimal expansion of csc(1).
  • A073448 (program): Decimal expansion of sec(1).
  • A073449 (program): Decimal expansion of cot(1).
  • A073453 (program): Number of distinct remainders arising when n is divided by all primes up to n.
  • A073458 (program): a(n) = floor(prime(n)/composite(n)).
  • A073464 (program): a(n) = phi(n) mod PrimePi(n).
  • A073469 (program): Expansion of x/B(x) where B(x) is the g.f. for A002487.
  • A073470 (program): Trisection of A007294.
  • A073471 (program): Trisection of A007294.
  • A073472 (program): Trisection of A007294.
  • A073474 (program): Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.
  • A073480 (program): Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp(x*y)*log(1+x)/(1-x).
  • A073481 (program): Least prime factor of the n-th squarefree number.
  • A073482 (program): Largest prime factor of the n-th squarefree number.
  • A073484 (program): Number of gaps in factors of the n-th squarefree number.
  • A073485 (program): Product of any number of consecutive primes; squarefree numbers with no gaps in their prime factorization.
  • A073490 (program): Number of prime gaps in factorization of n.
  • A073491 (program): Numbers having no prime gaps in their factorization.
  • A073492 (program): Numbers having at least one prime gap in their factorization.
  • A073493 (program): Numbers having exactly one prime gap in their factorization.
  • A073496 (program): Expansion of (3 + 2*x + 3*x^2)/(1 + x + 3*x^2 - x^3).
  • A073497 (program): a(n) = n^2 - prime(n).
  • A073498 (program): Binomial transform of A073145.
  • A073504 (program): A possible basis for finite fractal sequences: let u(1) = 1, u(2) = n, u(k) = floor(u(k-1)/2) + floor(u(k-2)/2); then a(n) = lim_{k->infinity} u(k).
  • A073521 (program): The set of 16 consecutive primes with the property that they form a 4 X 4 magic square with the smallest magic constant (258).
  • A073522 (program): A set of 25 consecutive primes that form a 5 X 5 magic square with the (non-minimal) magic constant 1703.
  • A073523 (program): The set of 36 consecutive primes that form a 6 X 6 pandiagonal magic square with the smallest magic constant (930).
  • A073525 (program): Result of applying the transformation on generating functions A -> 1/((1-x)*(1-x*A)) to the g.f. for A024718.
  • A073530 (program): a(0) = 1; for n>0 a(n)=sum(binomial(n,k)*binomial(n+k,k+1)*binomial(n+k,k),k=0..n)/n.
  • A073531 (program): Number of n-digit positive integers with all digits distinct.
  • A073533 (program): Let x(1)=1, x(n+1) = (4/3)*x(n) - floor((4/3)*x(n)); then a(n)=x(n)*3^n.
  • A073536 (program): Breaking indices for A058842 (i.e., n such that A058842(n) is not equal to 3*A058842 (n-1) ).
  • A073548 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 2.
  • A073549 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 6.
  • A073550 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 1.
  • A073551 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 3.
  • A073553 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 5.
  • A073554 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 7.
  • A073555 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 8.
  • A073556 (program): Number of Fibonacci numbers F(k), k <= 10^n, which end in 9.
  • A073570 (program): G.f.: Sum_{n >= 1} x^n/(1-x^n)^5.
  • A073573 (program): Numbers n such that n^3 + 4 is prime.
  • A073575 (program): Sum of factorial numbers dividing n.
  • A073576 (program): Number of partitions of n into squarefree parts.
  • A073577 (program): a(n) = 4*n^2 + 4*n - 1.
  • A073578 (program): a(n) = Sum_{k=1..n} mu(2*k).
  • A073579 (program): Signed primes: if prime(n) even, a(n) = 0; if prime(n) == 1 (mod 4), a(n) = prime(n); if prime(n) == -1 (mod 4), a(n) = -prime(n).
  • A073582 (program): Numbers n such that S(n) = n/2, where S(n) is the Kempner function A002034.
  • A073583 (program): Decimal expansion of 23/19.
  • A073587 (program): a(n)=a(n-1)*2^n+1 where a(0)=1.
  • A073588 (program): a(n) = a(n-1)*2^n-1 with a(1)=1.
  • A073590 (program): Expansion of e.g.f. exp(x) * log(1+x)/(1-x).
  • A073591 (program): A000522(n)+1.
  • A073592 (program): Euler transform of negative integers.
  • A073596 (program): Expansion of exp(x)*log(1-x)/(x-1).
  • A073598 (program): Numbers n such that n^3 + 5 is prime.
  • A073603 (program): Smallest multiple of n-th prime which is == 1 mod (n+1)-st prime.
  • A073604 (program): Smallest multiple of (n+1)-st prime which is == 1 mod n-th prime.
  • A073609 (program): a(0) = 2; a(n) for n > 0 is the smallest prime greater than a(n-1) that differs from a(n-1) by a square.
  • A073610 (program): Number of primes of the form n-p where p is a prime.
  • A073612 (program): Group the positive integers as (1, 2), (3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15, 16, 17), … the n-th group containing prime(n) elements. Except the first, all groups contain an odd number of elements and hence have a middle term. Sequence gives the middle terms starting from group 2.
  • A073613 (program): Triangular numbers which are the sum of two squares.
  • A073617 (program): Consider Pascal’s triangle A007318; a(n) = product of terms at +45 degrees slope with the horizontal.
  • A073618 (program): Consider Pascal’s triangle A007318; a(n) = LCM of terms at +45 degree slope with the horizontal.
  • A073627 (program): a(1)=a(2)=1; for n > 2, a(n) is the smallest integer such that a(n) > a(n-1) and a(n)+a(n-1) is prime.
  • A073632 (program): Numbers k such that floor((3/2)^k) = A002379(k) is odd.
  • A073634 (program): Numbers k such that floor((3/2)^k) = A002379(k) is even.
  • A073636 (program): Period 3: repeat [1, 8, 9] ; Digital root of A000578(n) = n^3 for n >= 1.
  • A073637 (program): Digital root (cf. A010888) of prime(n)^3.
  • A073642 (program): Replace 2^k in the binary representation of n with k (i.e., if n = 2^b + 2^c + 2^d + … then a(n) = b + c + d + …).
  • A073645 (program): a(1)=2 and, for all n>=1, a(n) is the length of the n-th run of increasing consecutive integers with each run after the first starting with 1.
  • A073649 (program): Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (4,2).
  • A073650 (program): Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (2,6).
  • A073663 (program): Total number of branches of length k (k>=1) in all ordered trees with n+k edges (it is independent of k).
  • A073668 (program): Decimal expansion of Sum_{k=1..inf} 1/(10^k-1).
  • A073675 (program): Rearrangement of natural numbers such that a(n) is the smallest proper divisor of n not included earlier but if no such divisor exists then a(n) is the smallest proper multiple of n not included earlier, subject always to the condition that a(n) is not equal to n.
  • A073681 (program): Smallest of three consecutive primes whose sum is a prime.
  • A073699 (program): Floor((Product of composite numbers up to n)/(product of primes up to n)).
  • A073701 (program): a(n) = n^2*a(n-1)+(-1)^n.
  • A073702 (program): a(n) = A073145(n)^2.
  • A073705 (program): a(n) = Sum_{ d divides n } (n/d)^(2d).
  • A073706 (program): a(n) = Sum_{ d divides n } (n/d)^(3d).
  • A073707 (program): Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.
  • A073708 (program): Generating function A(x) satisfies A(x) = (1+x)^2*A(x^2)^2, with A(0)=1.
  • A073709 (program): First differences of A073708.
  • A073710 (program): Convolution of A073709, which is also the first differences of the unique terms of A073709.
  • A073717 (program): a(n) = T(2n+1), where T(n) are the tribonacci numbers A000073.
  • A073718 (program): Powers of 2 with composite exponents.
  • A073719 (program): a(n) = floor(prime(2^n)/composite(2^n)).
  • A073720 (program): Let b(1) = 1, b(k+1) = b(k) - k*trunc(k/b(k)+1), where trunc(x) = floor(x) if x>= 0, trunc(x) = ceiling(x) otherwise. Sequence a(n) gives the successive absolute values taken by b(k).
  • A073721 (program): Remainder of division Sigma[n]/PrimePi[n] equals zero.
  • A073724 (program): a(n) = (4^(n+1) + 6n + 5)/9.
  • A073725 (program): a(n)-th composite number = phi(n-th composite number); a(1)=a(2)=0.
  • A073728 (program): a(n) = Sum_{k=0..n} S(k), where S(n) are the tribonacci generalized numbers A001644.
  • A073729 (program): Concatenation of initial and final digits of n in decimal representation.
  • A073730 (program): Concatenation of largest and smallest digits of n in decimal representation.
  • A073731 (program): Least k such that A073729(k) = n.
  • A073736 (program): Sum of primes whose index is congruent to n (mod 3); equals the partial sums of A073737 (in which sums of three successive terms form the primes).
  • A073737 (program): Sums of three successive terms form the odd primes.
  • A073738 (program): Sum of every other prime <= n-th prime down to 2 or 1; equals the partial sums of A036467 (in which sums of two consecutive terms form the primes).
  • A073739 (program): Least positive integers whose convolution forms a sequence whose odd-indexed terms are twice the odd primes (see: A073740).
  • A073742 (program): Decimal expansion of sinh(1).
  • A073743 (program): Decimal expansion of cosh(1).
  • A073744 (program): Decimal expansion of tanh(1).
  • A073745 (program): Decimal expansion of csch(1).
  • A073746 (program): Decimal expansion of sech(1).
  • A073747 (program): Decimal expansion of coth(1).
  • A073748 (program): a(n) = S(n)*S(n-1), where S(n) are the generalized tribonacci numbers A001644.
  • A073750 (program): Factors of 2 in the denominators of the fractional coefficients of the square-root of the prime power series: sum_{n=0..inf} p_n x^n, where p_n is the n-th prime and p_0 is defined to be 1.
  • A073752 (program): Greatest common divisor of n/spf(n) and n/gpf(n) where spf(n) is the smallest and gpf(n) is the greatest prime factor of n (see A020639, A006530).
  • A073753 (program): a(n) = A073752(A073752(n)), where A073752(n) = gcd(n/spf(n), n/gpf(n)), with spf(n) as the smallest and gpf(n) as the greatest prime factor of n (see A020639, A006530).
  • A073757 (program): a(n) = d(n) + phi(n) - 1.
  • A073758 (program): Smallest number that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.
  • A073759 (program): Largest number that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.
  • A073760 (program): Integers m such that A073758(m) = 4.
  • A073762 (program): a(n) = 24*n - 12.
  • A073763 (program): Least number of unrelated set belonging to these numbers is odd.
  • A073765 (program): Special binomial coefficient: C(prime(n), composite(n)).
  • A073766 (program): a(n) = binomial(composite(n+1), composite(n)) = binomial(A002808(n+1), A002808(n)).
  • A073767 (program): Bateman polynomial values n!*Z_n(-1).
  • A073773 (program): Number of plane binary trees of size n+2 and height n.
  • A073775 (program): Polynomial (1/3)*n^3 + (9/2)*n^2 + (85/6)*n - 2.
  • A073776 (program): a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.
  • A073778 (program): a(n) = Sum_{k=0..n} T(k)*T(n-k), where T is A000073; convolution of A000073 with itself.
  • A073779 (program): Number of 0’s in base-3 representation of n-th prime.
  • A073780 (program): Number of 1’s in base 3 representation of n-th prime.
  • A073781 (program): Number of 2’s in base-3 representation of n-th prime.
  • A073783 (program): First differences of composite numbers.
  • A073784 (program): Number of primes between successive composite numbers.
  • A073791 (program): Replace 4^k with (-4)^k in base 4 expansion of n.
  • A073792 (program): Replace 5^k with (-5)^k in base 5 expansion of n.
  • A073793 (program): Replace 6^k with (-6)^k in base 6 expansion of n.
  • A073794 (program): Replace 7^k with (-7)^k in base 7 expansion of n.
  • A073795 (program): Replace 8^k with (-8)^k in base 8 expansion of n.
  • A073796 (program): Replace 9^k with (-9)^k in base 9 expansion of n.
  • A073797 (program): a(n) = 2^n mod pi(n).
  • A073800 (program): Remainder of division 2^n/c[n], where c[n]=A002808, the n-th composite.
  • A073802 (program): Number of common divisors of n and sigma(n).
  • A073803 (program): Number of divisors of n is smaller than that of sigma(n).
  • A073804 (program): Number of divisors of n is greater than that of sigma[n].
  • A073805 (program): Numbers k such that 1 + k*R(k) is prime, where R(k) is the reverse of k.
  • A073806 (program): Number of divisors of sum of square of divisors.
  • A073807 (program): Number of divisors of sum of cube of divisors.
  • A073808 (program): Number of common divisors of sigma_1(n) and sigma_2(n).
  • A073811 (program): Number of common divisors of n and phi(n).
  • A073812 (program): Number of common divisors of sigma(n) and phi(n).
  • A073813 (program): Difference between n and largest unrelated number belonging to n, when n runs over composite numbers. For primes and for 4, unrelated set is empty.
  • A073814 (program): a(n) is the smallest number k such that A073813(k) = prime(n).
  • A073817 (program): Tetranacci numbers with different initial conditions: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) starting with a(0)=4, a(1)=1, a(2)=3, a(3)=7.
  • A073818 (program): a(n) = max(prime(i)*(n+1-i) | 1 <= i <= n).
  • A073821 (program): Decimal expansion of number with continued fraction expansion 0, 2, 4, 6, … (the even numbers).
  • A073829 (program): 4*((n-1)!+1)+n.
  • A073832 (program): k between A001359(n) and A001359(n+1) such that A073830(k) is maximal.
  • A073833 (program): Numerators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1).
  • A073834 (program): Denominators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1).
  • A073835 (program): Replace 10^k with (-10)^k in decimal expansion of n.
  • A073836 (program): Let C(n) = product of composite numbers between the n-th prime and (n+1)-th prime; a(n) = floor(C(n+1)/C(n)).
  • A073837 (program): Sum of primes p satisfying n <= p <= 2n.
  • A073838 (program): Product of primes p satisfying n <= p <= 2n.
  • A073839 (program): Sum of the composite numbers between n and 2n (both inclusive).
  • A073842 (program): a(1) = 1; for n>1, a(n) = the smallest positive integer root of n not included earlier, if such a root exists, otherwise the smallest power of n not included earlier, subject to a(n)<>n.
  • A073845 (program): a(1)=a(2)=1, a(n+2)=a(n+1)+a(n)+(-2)^n.
  • A073847 (program): a(0) = 1, c(0) = 1, a(n) for n > 0 is the smallest prime a(n-1) + c(n), where c(n) is composite and larger than c(n-1).
  • A073849 (program): Cumulative sum of initial digits of (n base 3).
  • A073850 (program): Cumulative sum of initial digits of (n base 4).
  • A073851 (program): Cumulative sum of initial digits of (n base 5).
  • A073855 (program): Number of steps to reach 0 starting with n and applying the process x ->bigomega(x), where bigomega = A001222.
  • A073869 (program): a(n) = Sum_{i=0..n} A002251(i)/(n+1).
  • A073881 (program): a(n) = smallest number m (obviously prime) such that pi(m) = 2*pi(n).
  • A073882 (program): Number of primes between n and n^2.
  • A073890 (program): Numerator of n/floor(sqrt(n)).
  • A073897 (program): a(1) = 1, a(n) = smallest odd or even number not occurring earlier according as n is prime or composite.
  • A073898 (program): a(1) = 1; for n>1, a(n) = smallest even or odd number not occurring earlier accordingly as n is prime or composite.
  • A073929 (program): a(1) = 1, a(n) = smallest number not included earlier such that the n-th partial sum (n>1) is divisible by n+1.
  • A073933 (program): Number of terms in n-th row of triangle in A073932.
  • A073934 (program): Sum of terms in n-th row of triangle in A073932.
  • A073937 (program): a(n) = a(n-1)-a(n-2)+a(n-3)+a(n-4), a(0)=4, a(1)=1, a(2)=-1, a(3)=1.
  • A073941 (program): a(n) = ceiling((Sum_{k=1..n-1} a(k)) / 2) for n >= 2 starting with a(1) = 1.
  • A073995 (program): Number of strings of length n over GF(4) with trace 0 and subtrace 0.
  • A073996 (program): Number of strings of length n over GF(4) with trace 0 and subtrace 1.
  • A073997 (program): Number of strings of length n over GF(4) with trace 1 and subtrace 0.
  • A073998 (program): Number of strings of length n over GF(4) with trace 1 and subtrace 1.
  • A073999 (program): Number of strings of length n over GF(4) with trace 1 and subtrace x where x = RootOf(z^2+z+1).
  • A074039 (program): If (n, n+2) is the k-th twin prime pair then k else 0.
  • A074040 (program): Product of first n twin prime pair products.
  • A074041 (program): Product of first n single (i.e., non-twin) primes.
  • A074042 (program): Numerator of Sum{1/A077800(k)|1<=k<=n}, denominator=A074043.
  • A074043 (program): Denominator of Sum{1/A077800(k)|1<=k<=n}, numerator=A074042.
  • A074046 (program): a(n)=a(n-1)*a(n-2)*a(n-3)*(1/a(n-1)+1/a(n-2)+1/a(n-3)) starting with a(1)=a(2)=1 and a(3)=2.
  • A074047 (program): a(n)=a(n-1)*a(n-2)*a(n-3)*(1/a(n-1)+1/a(n-2)+1/a(n-3)) starting with a(1)=a(2)=a(3)=1.
  • A074048 (program): Pentanacci numbers with initial conditions a(0)=5, a(1)=1, a(2)=3, a(3)=7, a(4)=15.
  • A074049 (program): Tree generated by the Wythoff sequences: a permutation of the positive integers.
  • A074051 (program): For each n there are uniquely determined numbers a(n) and b(n) and a polynomial p_n(x) such that for all integers m we have Sum_{i=1..m}i^n(i+1)! = a(n)*Sum_{i=1..m} (i+1)! + p_n(m)*(m+2)! + b(n). The sequence b(n) is A074052.
  • A074056 (program): a(n)=a(n-1)^3/a(n-2)^2+a(n-1)*a(n-2) with a(1)=a(2)=1.
  • A074057 (program): a(n) = 2*phi(n-2)-(n-1).
  • A074058 (program): Reflected tetranacci numbers A073817.
  • A074061 (program): Positive integers n such that 24*n^2-23 is a square.
  • A074065 (program): Numerators a(n) of fractions slowly converging to sqrt(3): let a(1) = 0, b(n) = n - a(n); if (a(n) + 1) / b(n) < sqrt(3), then a(n+1) = a(n) + 1, else a(n+1)= a(n).
  • A074066 (program): Zigzag modulo 3.
  • A074067 (program): Zigzag modulo 5.
  • A074068 (program): Zigzag modulo 7.
  • A074081 (program): Sum of determinants of 3rd order principal minors of powers of inverse of tetramatrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).
  • A074082 (program): Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(1,1).
  • A074084 (program): Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(2,1).
  • A074087 (program): Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(2,3).
  • A074092 (program): Number of plane binary trees of size n+3 and contracted height n.
  • A074093 (program): Number of values of k such that n = k - largest divisor of k (<k).
  • A074101 (program): Squares using no prime digit.
  • A074107 (program): a(n) = Product of (prime + 1) for first n primes - primorial (n).
  • A074109 (program): Smallest n-digit squarefree number.
  • A074110 (program): Largest n-digit squarefree number.
  • A074114 (program): Largest n-digit number of the form p^a*q^b… with the maximum value of a+b+…. where p, q etc. are primes.
  • A074116 (program): Largest n-digit power of 2.
  • A074117 (program): Smallest n-digit power of 3.
  • A074118 (program): Largest power of 3 <= 10^n.
  • A074132 (program): Row sums of triangle A074135.
  • A074133 (program): Average of the n-th group, if positive integers are rearranged in groups of k=1,2,3,… numbers whose sum is a multiple of k.
  • A074134 (program): First column of triangle A074135.
  • A074136 (program): Main diagonal of triangle A074135.
  • A074142 (program): Coefficients a(n) of a series connected with the odd primes.
  • A074143 (program): a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).
  • A074147 (program): (2n-1) odd numbers followed by 2n even numbers.
  • A074148 (program): a(n) = n + floor(n^2/2).
  • A074149 (program): Sum of terms in each group in A074147.
  • A074155 (program): Group the natural numbers so that the product of members of a group is a multiple of the sum: (1),(2,3,4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),…. This is the sequence of the ratio of product /sum.
  • A074166 (program): Product of first n palindromes.
  • A074167 (program): Product of prime divisors of composite numbers between consecutive primes.
  • A074171 (program): a(1) = 1. For n >= 2, a(n) is either a(n-1)+n or a(n-1)-n; we only use the minus sign if a(n-1) is prime. E.g. since a(2)=3 is prime, a(3)=a(2)-3=0.
  • A074179 (program): a(1) = 1, a(n) = smallest multiple of n divisible by the sum of all previous terms.
  • A074180 (program): a(1) = 1, a(n) = smallest multiple of n divisible by the product of all previous terms.
  • A074181 (program): Smallest power of n >= n!.
  • A074182 (program): Largest power of n <= n!.
  • A074184 (program): Index of the smallest power of n >= n!.
  • A074185 (program): a(1) = 1, for n > 1 a(n) is the smallest number such that the product of all previous terms is > n^n.
  • A074186 (program): a(1) = 1; a(n) is the largest number such that the product of all previous terms is < n^n.
  • A074188 (program): Smallest power (>=2) >= n!.
  • A074189 (program): a(1) = 1, a(2) = 2; for n > 2, a(n) = {a(n-1) +a(n+1)}/n or a(n+1) = n*a(n)-a(n-1).
  • A074201 (program): Let b(1) = 1, b(2) = 2, b(n+2) = (b(n+1)+1)/(b(n)+1); then a(n) = 1 if b(n) >= 1 and a(n) = 0 otherwise (also a(n) = floor(b(n)) for n > 2).
  • A074206 (program): Kalmár’s [Kalmar’s] problem: number of ordered factorizations of n.
  • A074209 (program): a(n) = Sum_{i=n+1..2n} i^n.
  • A074215 (program): Numbers m such that m and F(m) are relatively prime, where F(m) denotes the m-th Fibonacci number.
  • A074216 (program): Squares satisfying sigma(n)==0 (mod 3).
  • A074225 (program): a(n) = n * Sum_{d|n} d*2^(d-1).
  • A074226 (program): Numbers n such that Kronecker(3,n) = 1.
  • A074229 (program): Numbers n such that Kronecker(6,n)==mu(gcd(6,n)).
  • A074231 (program): Numbers n such that Kronecker(8,n) = mu(gcd(8,n)).
  • A074232 (program): Positive numbers that are not 3 or 6 (mod 9).
  • A074239 (program): Related to cumulative number of non-twin primes.
  • A074258 (program): Gaps between primes p such that 2p-1 is also prime.
  • A074259 (program): Gaps between primes p such that 2p+1 is also prime, i.e., Sophie-Germain primes A005384.
  • A074261 (program): Positions in the Kolakoski sequence (A000002) when the number of 2’s is greater than the number of 1’s.
  • A074262 (program): Positions of ‘11’ in Kolakoski sequence A000002.
  • A074263 (program): Positions of ‘22’ in Kolakoski sequence A000002.
  • A074264 (program): Values of Kolakoski sequence A000002 at positions n = 0 mod 3.
  • A074265 (program): Values of Kolakoski sequence A000002 at positions n = 0 mod 5.
  • A074268 (program): Primes of the form p^2 - p + 1 where p is prime.
  • A074272 (program): Partial alternating sums of the Kolakoski sequence A000002.
  • A074273 (program): Positions in the Kolakoski sequence (A000002) where there are an even number of 1’s and the current term is 1.
  • A074274 (program): Gaps between even number of 1’s in the Kolakoski sequence A000002.
  • A074278 (program): Positions in the Kolakoski sequence A000002 when there are a multiple of 3 1’s and the current term is 1.
  • A074279 (program): n appears n^2 times.
  • A074284 (program): Sum of the aliquot divisors of n-th triangular number.
  • A074285 (program): Sum of the divisors of n-th triangular number.
  • A074286 (program): Partial sum of the Kolakoski sequence (A000002) minus n.
  • A074287 (program): Even numbers n such that the partial sum of the Kolakoski sequence (A000002) at n is less than 3n/2.
  • A074288 (program): n-th term of the Kolakoski sequence (A000002) multiplied by the n-th partial sum.
  • A074290 (program): Difference between Kolakoski(n)=A000002(n) and 1 (n odd) or 2 (n even).
  • A074291 (program): Positions in the Kolakoski sequence where A000002(n) is 1 if n is odd, or 2 if n is even.
  • A074292 (program): Dominant digit in successive groups of 3 from the Kolakoski sequence (A000002).
  • A074294 (program): Integers 1 to 2*k followed by integers 1 to 2*k + 2 and so on.
  • A074299 (program): Lengths of subsequences such that the first ‘average’ value (a[n]*1.5) is not achieved from the starting position in the Kolakoski sequence (A000002).
  • A074305 (program): a(3m) = 2m, a(3m+1) = 4m+3, a(3m+2) = 4m+1.
  • A074306 (program): Inverse of permutation in A074305.
  • A074307 (program): Square of permutation in A074305.
  • A074308 (program): Inverse of permutation in A074307.
  • A074309 (program): Sum of next n terms of the form i^i.
  • A074313 (program): a(n) = the maximal length of a sequence of primes {s_1 = prime(n), s_2 = f(s1), s_3 = f(s_2), ….} formed by repeated application of f(m) = Floor(m/2) on prime(n).
  • A074314 (program): Deficient triangular numbers.
  • A074315 (program): Abundant triangular numbers.
  • A074320 (program): a(n) = sum of smallest and largest prime factors of n, for n>1; a(1)=2.
  • A074322 (program): 0 if pronounced name of n-th letter of English alphabet begin with a vowel sound, otherwise 1. Different from A054638.
  • A074323 (program): Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,2), where (n)_q=(1+q+…+q^(n-1)) and q is a root of unity.
  • A074324 (program): a(2n+1) = 3^n, a(2n) = 4*3^(n-1) except for a(0) = 1.
  • A074330 (program): a(n) = Sum_{k=1..n} 2^b(k) where b(k) denotes the number of 1’s in the binary representation of k.
  • A074331 (program): a(n) = Fibonacci(n+1) - (1 + (-1)^n)/2.
  • A074334 (program): a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.
  • A074335 (program): In music, with 0 = C natural, 1 = C#, etc.: The unfolding of a semitonal interval cycle, alternating the ascending and descending aspects of the cycle from a common point or axis of symmetry. Any regularly occurring alignment may be used, with predictable even or odd results.
  • A074337 (program): 18 primes in arithmetic progression.
  • A074352 (program): Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(1,2).
  • A074353 (program): Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(1,2).
  • A074355 (program): Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).
  • A074358 (program): Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = b*nu(n-1) + lambda*(1 + q + q^2 + … + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).
  • A074359 (program): Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).
  • A074361 (program): Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+…+q^(n-2))*nu(n-2) with (b,lambda)=(3,1).
  • A074367 (program): (p^2-5)/4 for odd primes p.
  • A074369 (program): Number of divisors of Sum_{i=1..n} prime(i).
  • A074370 (program): Sum of the divisors of Sum_{i=1..n} prime(i).
  • A074372 (program): 1 + the sum of the distinct primes dividing n.
  • A074373 (program): Square of the sum of the prime factors of n (with repetition).
  • A074374 (program): s(s+1)/2 where s is the sum of the prime factors of n (with repetition).
  • A074375 (program): s(s+3)/2 where s is the sum of the prime factors of n (with repetition).
  • A074376 (program): s(3s-1)/2 where s is the sum of the prime factors of n (with repetition).
  • A074377 (program): Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, …
  • A074378 (program): Even triangular numbers halved.
  • A074381 (program): (p-1)/2 mod 2, where p is the n-th prime for which p+2 is also prime; i.e., a(n)=0 if p==1 (mod 4), a(n)=1 if p==3 (mod 4).
  • A074389 (program): a(n) = gcd(n, sigma(n), phi(n)).
  • A074392 (program): a(n) = Lucas(n+1) + (3*(-1)^n - 1)/2.
  • A074394 (program): a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.
  • A074396 (program): a(n) = 10 - (p mod 10), where p is the n-th prime congruent to 1 (mod 4) for which p+2 is also prime.
  • A074398 (program): Number of primes between n and phi(n), with neither n nor phi(n) included in the count in case they are primes.
  • A074399 (program): a(n) is the largest prime divisor of n(n+1).
  • A074400 (program): Sum of the even divisors of 2n.
  • A074451 (program): Non-cubefree noncubes.
  • A074461 (program): Average digit (rounded down) in the decimal expansion of the n-th prime number.
  • A074462 (program): Average digit (rounded up) in the decimal expansion of prime(n).
  • A074466 (program): a(n) = gcd(n^3, sigma(n^3), phi(n^3)).
  • A074472 (program): Length of iteration sequence of Collatz-function (A006370) when initial value is 3^n (A000244) and final cycle is followed once.
  • A074475 (program): a(n) = Sum_{j=0..floor(n/2)} T(2*j + q), where T(n) are generalized tribonacci numbers (A001644) and q = n - 2*floor(n/2).
  • A074494 (program): Number of 2-input gates used to synthesize parity function in disjunctive normal form (DNF) with n inputs.
  • A074495 (program): a(n) = the first prime > sigma(n).
  • A074499 (program): Sum of three perfect powers.
  • A074500 (program): Difference between n*sqrt(n)+1 and prime(n), rounded to nearest integer.
  • A074501 (program): a(n) = 1^n + 2^n + 5^n.
  • A074502 (program): 1^n + 2^n + 6^n.
  • A074503 (program): a(n) = 1^n + 2^n + 7^n.
  • A074504 (program): a(n) = 1^n + 2^n + 8^n.
  • A074505 (program): a(n) = 1^n + 2^n + 9^n.
  • A074506 (program): a(n) = 1^n + 3^n + 4^n.
  • A074507 (program): a(n) = 1^n + 3^n + 5^n.
  • A074508 (program): 1^n + 3^n + 6^n.
  • A074509 (program): a(n) = 1^n + 3^n + 7^n.
  • A074510 (program): a(n) = 1^n + 3^n + 8^n.
  • A074511 (program): a(n) = 1^n + 4^n + 5^n.
  • A074512 (program): a(n) = 1^n + 4^n + 6^n.
  • A074513 (program): a(n) = 1^n + 4^n + 7^n.
  • A074514 (program): 1^n + 4^n + 8^n.
  • A074515 (program): a(n) = 1^n + 4^n + 9^n.
  • A074516 (program): a(n) = 1^n + 5^n + 6^n.
  • A074517 (program): a(n) = 1^n + 5^n + 7^n.
  • A074518 (program): a(n) = 1^n + 5^n + 8^n.
  • A074519 (program): a(n) = 1^n + 5^n + 9^n.
  • A074520 (program): 1^n + 6^n + 7^n.
  • A074521 (program): a(n) = 1^n + 6^n + 8^n.
  • A074522 (program): a(n) = 1^n + 6^n + 9^n.
  • A074523 (program): a(n) = 1^n + 7^n + 8^n.
  • A074524 (program): a(n) = 1^n + 7^n + 9^n.
  • A074525 (program): a(n) = 1^n + 8^n + 9^n.
  • A074526 (program): a(n) = 2^n + 3^n + 4^n.
  • A074527 (program): a(n) = 2^n + 3^n + 5^n.
  • A074528 (program): a(n) = 2^n + 3^n + 6^n.
  • A074529 (program): a(n) = 2^n + 3^n + 7^n.
  • A074530 (program): a(n) = 2^n + 3^n + 8^n.
  • A074531 (program): a(n) = 2^n + 3^n + 9^n.
  • A074532 (program): a(n) = 2^n + 4^n + 5^n.
  • A074533 (program): a(n) = 2^n + 4^n + 6^n.
  • A074534 (program): a(n) = 2^n + 4^n + 7^n.
  • A074535 (program): a(n) = 2^n + 4^n + 8^n.
  • A074536 (program): a(n) = 2^n + 4^n + 9^n.
  • A074537 (program): a(n) = 2^n + 5^n + 6^n.
  • A074538 (program): a(n) = 2^n + 5^n + 7^n.
  • A074539 (program): a(n) = 2^n + 5^n + 8^n.
  • A074540 (program): a(n) = 2^n + 5^n + 9^n.
  • A074541 (program): a(n) = 2^n + 6^n + 7^n.
  • A074542 (program): a(n) = 2^n + 6^n + 8^n.
  • A074543 (program): a(n) = 2^n + 6^n + 9^n.
  • A074544 (program): a(n) = 2^n + 7^n + 8^n.
  • A074545 (program): a(n) = 2^n + 7^n + 9^n.
  • A074546 (program): a(n) = 2^n + 8^n + 9^n.
  • A074547 (program): a(n) = 3^n + 4^n + 5^n.
  • A074548 (program): a(n) = 3^n + 4^n + 6^n.
  • A074549 (program): a(n) = 3^n + 4^n + 7^n.
  • A074550 (program): a(n) = 3^n + 4^n + 8^n.
  • A074551 (program): a(n) = 3^n + 4^n + 9^n.
  • A074552 (program): a(n) = 3^n + 5^n + 7^n.
  • A074553 (program): a(n) = 3^n + 5^n + 8^n.
  • A074554 (program): a(n) = 3^n + 5^n + 9^n.
  • A074555 (program): a(n) = 3^n + 6^n + 7^n.
  • A074556 (program): a(n) = 3^n + 6^n + 8^n.
  • A074557 (program): 3^n + 6^n + 9^n.
  • A074558 (program): a(n) = 3^n + 7^n + 8^n.
  • A074559 (program): a(n) = 3^n + 7^n + 9^n.
  • A074560 (program): a(n) = 3^n + 8^n + 9^n.
  • A074561 (program): a(n) = 4^n + 5^n + 6^n.
  • A074562 (program): a(n) = 4^n + 5^n + 7^n.
  • A074563 (program): a(n) = 4^n + 5^n + 8^n.
  • A074564 (program): a(n) = 4^n + 5^n + 9^n.
  • A074565 (program): a(n) = 4^n + 6^n + 7^n.
  • A074566 (program): a(n) = 4^n + 6^n + 8^n.
  • A074567 (program): a(n) = 4^n + 6^n + 9^n.
  • A074568 (program): a(n) = 4^n + 7^n + 8^n.
  • A074569 (program): a(n) = 4^n + 7^n + 9^n.
  • A074570 (program): a(n) = 4^n + 8^n + 9^n.
  • A074571 (program): a(n) = 5^n + 6^n + 7^n.
  • A074572 (program): a(n) = 5^n + 6^n + 8^n.
  • A074573 (program): a(n) = 5^n + 6^n + 9^n.
  • A074574 (program): a(n) = 5^n + 7^n + 8^n.
  • A074575 (program): a(n) = 5^n + 7^n + 9^n.
  • A074576 (program): a(n) = 5^n + 8^n + 9^n.
  • A074577 (program): a(n) = 6^n + 7^n + 8^n.
  • A074578 (program): a(n) = 6^n + 7^n + 9^n.
  • A074579 (program): a(n) = 6^n + 8^n + 9^n.
  • A074580 (program): a(n) = 7^n + 8^n + 9^n.
  • A074581 (program): a(n)=T(3n+1), where T(n) are tribonacci numbers A000073.
  • A074582 (program): a(n) = S(3n), where S(n) is the generalized tribonacci sequence A001644.
  • A074584 (program): Esanacci (hexanacci or “6-anacci”) numbers.
  • A074585 (program): a(n)= Sum_{j=0..floor(n/2)} A073145(2*j + q), where q = 2*(n/2 - floor(n/2)).
  • A074587 (program): Sum of the coefficients of the n-th Moebius polynomial, M(n,x), where M(n,-1) = mu(n), the Moebius function of n.
  • A074589 (program): Replace each number n in Pascal’s triangle with the n-th prime.
  • A074591 (program): If n is a prime power then 0 else n.
  • A074592 (program): Smallest prime factors of numbers that are not prime powers.
  • A074593 (program): Largest prime factors of numbers that are not prime powers.
  • A074594 (program): Number of distinct prime factors of numbers that are not prime powers.
  • A074595 (program): Number of prime factors of numbers that are not prime powers (with multiplicity).
  • A074597 (program): Numerator of 3 * H(n,3,2), a generalized harmonic number. See A075135.
  • A074598 (program): Numerator of 4 * H(n,4,1), a generalized harmonic number. See A075136.
  • A074599 (program): Numerator of 2 * H(n,2,1), a generalized harmonic number. See A075135. Also 2 * A025550.
  • A074600 (program): a(n) = 2^n + 5^n.
  • A074601 (program): a(n) = 2^n + 6^n.
  • A074602 (program): a(n) = 2^n + 7^n.
  • A074603 (program): a(n) = 2^n + 8^n.
  • A074604 (program): a(n) = 2^n + 9^n.
  • A074605 (program): a(n) = 3^n + 4^n.
  • A074606 (program): a(n) = 3^n + 5^n.
  • A074607 (program): a(n) = 3^n + 6^n.
  • A074608 (program): a(n) = 3^n + 7^n.
  • A074609 (program): a(n) = 3^n + 8^n.
  • A074610 (program): a(n) = 3^n + 9^n.
  • A074611 (program): 4^n + 5^n.
  • A074612 (program): a(n) = 4^n + 6^n.
  • A074613 (program): a(n) = 4^n + 7^n.
  • A074614 (program): a(n) = 4^n + 9^n.
  • A074615 (program): a(n) = 5^n + 6^n.
  • A074616 (program): a(n) = 5^n + 7^n.
  • A074617 (program): a(n) = 5^n + 8^n.
  • A074618 (program): a(n) = 5^n + 9^n.
  • A074619 (program): a(n) = 6^n + 7^n.
  • A074620 (program): a(n) = 6^n + 8^n.
  • A074621 (program): a(n) = 6^n + 9^n.
  • A074622 (program): a(n) = 7^n + 8^n.
  • A074623 (program): a(n) = 7^n + 9^n.
  • A074624 (program): a(n) = 8^n + 9^n.
  • A074627 (program): Numbers n such that sigma(n) is divisible by 6.
  • A074628 (program): Numbers k such that sigma(k) == 2 mod 6.
  • A074630 (program): Numbers k such that sigma(k) == 4 mod 6.
  • A074635 (program): a(0)=1, a(n) = Sum_{k=0..n} (binomial(n,k)^2 * binomial(n+k,k+1)^2)/n^2.
  • A074637 (program): Numerator of 4 * H(n,4,3), a generalized harmonic number.
  • A074638 (program): Denominator of 1/3 + 1/7 + 1/11 + … + 1/(4n-1).
  • A074644 (program): a(n) = A074639(n^2) - A074639(n) mod n.
  • A074649 (program): a(0) = 1; for n >= 1, a(n) = sum(binomial(n,k)^3*binomial(n+k,k+1)^2,k = 0..n)/n^2.
  • A074662 (program): a(n) = F(n+1)+cos(n*Pi/2).
  • A074664 (program): Number of algebraically independent elements of degree n in the algebra of symmetric polynomials in noncommuting variables.
  • A074677 (program): a(n) = Sum_{i = 0..floor(n/2)} (-1)^(i + floor(n/2)) F(2*i + e), where F = A000045 (Fibonacci numbers) and e = (1-(-1)^n)/2.
  • A074678 (program): a(n) = Sum_{j=0..floor(n/2)} (-1)^(j+floor(n/2))*S(2j+q), where S(n) are generalized tribonacci numbers (A001644) and q = (1-(-1)^n)/2.
  • A074695 (program): Greatest common divisor of n and floor(n^(1/2))^2.
  • A074700 (program): a(n) = tau(F(2^n)) where tau(x) is the number of divisors of x (A000005(x)) and F(k) the k-th Fibonacci number (A000045(k)).
  • A074701 (program): Numbers k such that k = Sum_{d|phi(k)} mu(phi(d))*phi(k)/d.
  • A074702 (program): a(n) = ((n+1)^2*(n-1)*a(n-1)+(-1)^(n+1))/n.
  • A074703 (program): a(n) = n^2*a(n-1)+1, a(1)=0.
  • A074704 (program): a(n) = floor(n^(3/2)) - n*floor(n^(1/2)).
  • A074705 (program): a(1)=4; for n>1: number of primes between squares of (n-1)-th and (n+1)-th primes.
  • A074706 (program): 1/(1+Sum_{n>0} (-x)^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.
  • A074707 (program): exp(Sum_{n>0} x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.
  • A074712 (program): Number of (interiors of) cells touched by a diagonal in a regular m X n grid (enumerated antidiagonally).
  • A074715 (program): Number of prime factors of F(2^n) where F(m) is the m-th Fibonacci number.
  • A074719 (program): ip(n): the number of primes not exceeding reverse(n).
  • A074722 (program): a(n) = Sum_{d divides n} phi(n/d)*(-1)^bigomega(d).
  • A074723 (program): Largest power of 2 dividing F(3n) where F(k) is the k-th Fibonacci number.
  • A074724 (program): Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.
  • A074728 (program): a(n) = Sum_{k=1..n} k^(n-k)*binomial(n,k-1).
  • A074732 (program): a(n+3) = floor( ( a(n) + 2*a(n+1) + 3*a(n+2) )/4 ), with a(0), a(1), a(2) equal to 0, 1, 2.
  • A074740 (program): a(n) = n!*2^(n-1)/Product_{k=1..n} tau(k) where tau = A000005.
  • A074741 (program): Sum of squares of gaps between consecutive primes.
  • A074742 (program): a(n) = (n^3 + 6n^2 - n + 12)/6.
  • A074745 (program): a(n) = sum_{k=1..n} prime(k)*prime(k+1).
  • A074752 (program): Number of combinatorially inequivalent cyclic subgroups of S_n of order 6. Number of partitions of n of order 6.
  • A074753 (program): Number of integers k such that sigma(k) < n.
  • A074754 (program): Number of integers k such that sigma(k) divides n.
  • A074757 (program): Numbers n such that tau(n) = (tau(n+1)+tau(n-1))/2.
  • A074763 (program): a(n) = (1/n) * Sum_{d divides n} (-1)^(n+d)*phi(n/d)*2^d.
  • A074764 (program): Numbers of smaller squares into which a square may be dissected.
  • A074766 (program): a(n) = prime(2n) - 2*prime(n) - n.
  • A074775 (program): Numbers n such that tau(n) < tau(n+1) where tau(x)=A000005(x).
  • A074784 (program): a(n) = a(n-1) + square of the sum of digits of n.
  • A074787 (program): Sum of squares of the number of unitary divisors of k from 1 to n.
  • A074789 (program): Partial sums of usigma(n)^2: square of the sum of unitary divisors of n.
  • A074790 (program): a(n) = (2*n+1)!*Sum_{k=0..n} (-1)^k/(2*k+1)!.
  • A074792 (program): Least k > 1 such that k^n == 1 (mod n).
  • A074793 (program): Sum of prime powers less than or equal to n.
  • A074794 (program): Number of numbers k <= n such that tau(k) == 1 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.
  • A074795 (program): Number of numbers k <= n such that tau(k) == 0 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.
  • A074796 (program): Number of numbers k <= n such that tau(k) == 2 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.
  • A074798 (program): Floor of S*n^2, where S equals sum of reciprocal terms of this same sequence.
  • A074800 (program): a(n) = denominator( (4*n+1)*(Product_{i=1..n} (2*i-1)/Product_{i=1..n} (2*i))^5 ).
  • A074801 (program): a(n) is the sum of the n-th row of the triangle formed by replacing each m in Pascal’s triangle with sigma(m).
  • A074802 (program): Number of numbers k <= n such that tau(k) = tau(k+1) where tau(x) = A000005(x) is the number of divisors of x.
  • A074803 (program): Kolakoski-(4,2) sequence: a(n) is length of n-th run.
  • A074804 (program): Kolakoski-(3,2) sequence: a(n) is length of n-th run.
  • A074805 (program): n mod 19 + 1 (“Golden Number”).
  • A074806 (program): Least k such that F(k) reduced (mod prime(n)) = prime(n)-1 where F(k) is the k-th Fibonacci number.
  • A074816 (program): a(n) = 3^A001221(n) = 3^omega(n).
  • A074818 (program): Number of integers in {1, 2, …, prime(n)} that are coprime to n.
  • A074819 (program): Numbers k such that mu(k)+mu(k+1) = 0.
  • A074822 (program): Primes p such that p + 4 is prime and p == 9 (mod 10).
  • A074823 (program): a(n) = 2^omega(n)*mu(n)^2.
  • A074827 (program): Numbers n such that tau(n) > tau(n+1) where tau(x) = A000005(x).
  • A074828 (program): a(1) = 1; for n>1, a(n) = smallest composite multiple of n if n is a prime else the smallest prime divisor of n if n is composite.
  • A074832 (program): Primes whose binary reversal is also prime.
  • A074837 (program): Numbers k such that the penultimate 3 divisors of k sum to k.
  • A074839 (program): a(0) = 1, a(n+1) = a(n) + next prime larger than a(n).
  • A074840 (program): Numerators a(n) of fractions slowly converging to sqrt(2): let a(1) = 0, b(n) = n - a(n); if (a(n) + 1) / b(n) < sqrt(2), then a(n+1) = a(n) + 1, else a(n+1)= a(n).
  • A074842 (program): Triplets: base 10 representation is the juxtaposition of three identical strings.
  • A074843 (program): Quadruplets: base 10 representation is the juxtaposition of four identical strings.
  • A074845 (program): Numbers n such that S(n) = largest difference between consecutive divisors of n (ordered by size), where S(n) is the Kempner function (A002034).
  • A074850 (program): Partial products of successive digits in the decimal expansion of Pi.
  • A074851 (program): Numbers k such that k and k+1 both have exactly 2 distinct prime factors.
  • A074854 (program): a(n) = Sum_{d|n} (2^(n-d)).
  • A074858 (program): a(n) = a(n-1) + a(n-2) + R(a(n-3)) where a(0) = a(1) = a(2) = 1 and R(n) (A004086) means the reverse of n.
  • A074860 (program): a(n) = a(n-1) + R(a(n-2)) + R(a(n-3)) where a(0)=a(1)=a(2)=1 and R(k) = A004086(k) is the reverse of k.
  • A074861 (program): Iccanobirt sequence: a(n) = R(a(n-1)) + R(a(n-2)) + R(a(n-3)) where a(1)=a(2)=a(3)=1 and R(n) (A004086) is the reverse of n.
  • A074867 (program): a(n) = M(a(n-1)) + M(a(n-2)) where a(1)=a(2)=1 and M(k) is the product of the digits of k in base 10.
  • A074872 (program): Inverse BinomialMean transform of the Fibonacci sequence A000045 (with the initial 0 omitted).
  • A074877 (program): Number of function calls required to compute ack(3,n), where ack denotes the Ackermann function.
  • A074878 (program): Row sums of triangle in A074829.
  • A074879 (program): 10 - Mod(Prime(n),10) when Prime(n) + 22 = Prime(n+1).
  • A074882 (program): Number of integers in {1, 2, …, sigma(n)} that are coprime to n.
  • A074890 (program): Decimal form of binary integers produced by a modified version of Wolfram’s Rule 30 one-dimensional cellular automaton.
  • A074909 (program): Running sum of Pascal’s triangle (A007318), or beheaded Pascal’s triangle read by beheaded rows.
  • A074919 (program): Number of integers in {1, 2, …, phi(n)} that are coprime to n.
  • A074922 (program): Number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 2 simple intersections.
  • A074927 (program): a(n) such that p(n)*p(n+1)+a(n) is a minimal square.
  • A074928 (program): a(n)>0 such that p(n)*p(n+1)+a(n) is a minimal prime.
  • A074929 (program): a(n)>0 such that p(n)*p(n+1)-a(n) is a maximal square.
  • A074930 (program): Number of integers in {1, 2, …, n!} that are coprime to n.
  • A074932 (program): Row sums of unsigned triangle A075513.
  • A074935 (program): Denominator of a(n), where for n > 2, a(n)=-1/a(n-1)+1/a(n-2), a(1)=1, a(2)=2.
  • A074937 (program): Let c(1) = c(2) = 1, c(n+2) = 1/(c(n+1)+c(n)); then a(n) = (1+sign(c(n)-sqrt(1/2))/2.
  • A074938 (program): Odd numbers such that base 3 representation contains no 2.
  • A074939 (program): Even numbers such that base 3 representation contains no 2.
  • A074940 (program): Numbers having at least one 2 in their ternary representation.
  • A074941 (program): a(n) = sigma(n) mod 3.
  • A074942 (program): a(n) = phi(n) mod 3.
  • A074943 (program): tau(n) (mod 3).
  • A074945 (program): Number of k with 1<=k<=n such that gcd(n,k) = floor(n/k).
  • A074946 (program): Positive integers n for which the sum of the prime-factorization exponents of n (bigomega(n) = A001222(n)) divides n.
  • A074953 (program): Numbers equidistant from consecutive twin prime pairs.
  • A074964 (program): Numbers k such that Max ( sigma(x*y) : 1 <= x <= k, 1 <= y <= k ) = sigma(k^2).
  • A074972 (program): a(n) == - prime(n) (modulo 20).
  • A074984 (program): m^p-n, for smallest m^p>=n.
  • A074985 (program): Squares of semiprimes (A001358).
  • A074988 (program): Numbers n such that the k-th binary digit of n equals mu(k)^2 for k=1 up to A029837(n+1).
  • A074989 (program): Distance from n to nearest cube.
  • A074990 (program): Number of primes in the interval (n,3n].
  • A074991 (program): Concatenation of n, n+1, n+2 divided by 3.
  • A074992 (program): a(n) = (10^(2*n)+10^n+1)/3.
  • A074993 (program): a(n) = floor((concatenation of n, n+1)/2).
  • A074994 (program): Floor of concatenation of n, n+1, n+2, n+3 divided by 4.
  • A074995 (program): Floor of concatenation of n, n+1, n+2, n+3, n+4 divided by 5.
  • A074996 (program): Floor of concatenation of n, n+1, n+2, n+3, n+4, n+5 divided by 6.
  • A074999 (program): Floor[concatenation of nine consecutive numbers from n to n+8 divided by 9].
  • A075003 (program): Floor[ concatenation of n+1 and n divided by 2 ].
  • A075004 (program): Floor[ concatenation of n+2, n+1 and n divided by 3 ].
  • A075005 (program): Floor[ concatenation of n+3, n+2, n+1 and n divided by 4 ].
  • A075008 (program): Floor[ concatenation of 7 numbers from n+6 to n in that order divided by 7].
  • A075010 (program): a(n) = floor( concatenation of 9 numbers from n+8 to n in that order divided by 9 ).
  • A075011 (program): Floor[ concatenation of numbers from n to 1 divided by concatenation of numbers from 1 to n].
  • A075020 (program): a(1) = 1; for n>1, a(n) = the smallest prime divisor of the number C(n) formed from the reverse concatenation of 1,2,3,… up to n.
  • A075045 (program): Coefficients A_n for the s=3 tennis ball problem.
  • A075054 (program): Smallest k such that (n+1)(n+2)…(n+k) is divisible by n!.
  • A075055 (program): Smallest integer of the form product (n+1)(n+2)…(n+k)/n!.
  • A075059 (program): a(n) = 1 + lcm(1, 2, …, n) = 1 + A003418(n).
  • A075061 (program): Triangle in A075059 read by rows.
  • A075062 (program): Row sums of triangle in A075059.
  • A075063 (program): Smallest prime == 1 mod first n composite numbers.
  • A075064 (program): Smallest composite number == 1 mod first n prime numbers.
  • A075065 (program): a(1) = 1 and then alternately even and odd composite numbers matching the parity of the index.
  • A075066 (program): Alternately odd and even composite numbers complementing the parity of the index.
  • A075069 (program): Product of prime(n) consecutive numbers starting from prime(n).
  • A075070 (program): a(n) = n-th compositorial number / (product of those primes which divide the n-th compositorial number).
  • A075071 (program): n! divided by product of factorials of all proper divisors of n, as n runs through the values for which the result is an integer.
  • A075084 (program): Number of composite numbers between n and 2n.
  • A075089 (program): Smallest prime == 1 mod n-th composite number.
  • A075091 (program): Sum of Lucas numbers and reflected Lucas numbers (comment to A061084).
  • A075092 (program): Sum of generalized tribonacci numbers (A001644) and reflected generalized tribonacci numbers (A073145).
  • A075101 (program): Numerator of 2^n/n.
  • A075104 (program): Greatest common divisor of n and integer part of log_2(n).
  • A075105 (program): Numerator of n/floor(log_2(n)); denominator is A075106(n).
  • A075106 (program): Denominator of n/floor(log_2(n)); numerator is A075105(n).
  • A075110 (program): Concatenation of n-th prime and n in decimal notation.
  • A075111 (program): a(n)=Sum((-1)^(i+Floor(n/2))T(2i+e),(i=0,..,Floor(n/2))), where T(n) are tribonacci numbers (A000073) and e=(1/2)(1-(-1)^n).
  • A075113 (program): a(n) = A000217(n) - A048702(n+1).
  • A075115 (program): Binomial transform of A073145: a(n)=Sum(binomial(n,k)*A073145(k),(k=0,..,n)).
  • A075116 (program): Binomial transform of A073817: a(n)=Sum(Binomial(n,k)*A073817(k),(k=0,..,n)).
  • A075117 (program): Table by antidiagonals of generalized Lucas numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=2 and T(n,1)=1.
  • A075118 (program): Variant on Lucas numbers: a(n) = a(n-1) + 3*a(n-2) with a(0)=2 and a(1)=1.
  • A075119 (program): Denominator of n/floor(sqrt(n)); numerator is A073890(n).
  • A075123 (program): a(n) is the least positive integer > a(n-1) and a(n) is not 2*a(i)+a(j) for 1<=i<j<n.
  • A075128 (program): Binomial transform of generalized tetranacci numbers A073817: a(n)=Sum((-1)^k Binomial(n,k)*A073817(k),(k=0,..,n)).
  • A075129 (program): Binomial transform of reflected tetranacci numbers A074058: a(n)=Sum((-1)^k Binomial(n,k)*A074058(k),(k=0,..,n)).
  • A075133 (program): Indices of double-safe primes: p=prime(n) is double-safe: q=(p-1)/2 & r=(q-1)/2 are both prime (and q is safe).
  • A075135 (program): Numerator of the generalized harmonic number H(n,3,1) described below.
  • A075136 (program): Numerator of the generalized harmonic number H(n,4,1).
  • A075137 (program): Numerator of the generalized harmonic number H(n,5,1).
  • A075138 (program): Denominator of the generalized harmonic number H(n,5,1).
  • A075139 (program): Numerator of the generalized harmonic number H(n,5,2).
  • A075140 (program): Denominator of the generalized harmonic number H(n,5,2).
  • A075141 (program): Numerator of the generalized harmonic number H(n,5,3).
  • A075142 (program): Denominator of the generalized harmonic number H(n,5,3).
  • A075143 (program): Numerator of the generalized harmonic number H(n,5,4).
  • A075144 (program): Denominator of the generalized harmonic number H(n,5,4).
  • A075145 (program): Prime basis of A061373.
  • A075148 (program): Table E(n,k) (listed antidiagonalwise as E(0,0), E(1,0), E(0,1), E(2,0), E(1,1), E(0,2), …) where E(n,k) is F(n+k) for all even n and L(n+k) for all odd n. F(n) and L(n) are the n-th Fibonacci (A000045) and Lucas (A000032) numbers respectively.
  • A075149 (program): Sum_{i=0..2*A053645(n)} (C(2*A053645(n),i) mod 2)*A000045(n-i) [where C(r,c) is the binomial coefficient (A007318) and A000045(n) is the n-th Fibonacci number).].
  • A075150 (program): a(n)=L(n)*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (see comment to A061084).
  • A075151 (program): a(n)=L(n)^2*C(n), L(n)=Lucas numbers (A000032), C(n)=reflected Lucas numbers (comment to A061084).
  • A075155 (program): Cubes of Lucas numbers.
  • A075156 (program): Binomial transform of pentanacci numbers A074048: a(n) = Sum_{k=0..n} binomial(n,k)*A074048(k).
  • A075157 (program): Run lengths in the binary expansion of n gives the vector of exponents in prime factorization of a(n)+1, with the least significant run corresponding to the exponent of the least prime, 2; with one subtracted from each run length, except for the most significant run of 1’s.
  • A075158 (program): Prime factorization of n+1 encoded with the run lengths of binary expansion.
  • A075159 (program): Run lengths in the binary expansion of n-1 gives the vector of exponents in prime factorization of a(n), with the least significant run corresponding to the exponent of the least prime, 2.
  • A075160 (program): Prime factorization of n encoded with the run lengths of binary expansion + 1.
  • A075172 (program): Number of edges in each rooted plane tree produced with the binary run length unranking algorithm presented in A075171.
  • A075177 (program): Indices of additive primes - primes with prime sum-of-digits, see A046704.
  • A075180 (program): Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.
  • A075183 (program): One half of third column of triangle A075181.
  • A075184 (program): One half of fourth column of triangle A075181.
  • A075185 (program): One-fourth of fifth column of triangle A075181.
  • A075190 (program): Numbers k such that k^2 is an interprime = average of two successive primes.
  • A075193 (program): Expansion of (1-2*x)/(1+x-x^2).
  • A075194 (program): Binomial transform of pentanacci numbers A074048: a(n)=Sum((-1)^k*Binomial(n,k)*A074048(k),(k=0,..,n)).
  • A075225 (program): Expansion of 2-AGM(1,1-8x) (where AGM denotes the arithmetic-geometric mean).
  • A075253 (program): Trajectory of 77 under the Reverse and Add! operation carried out in base 2.
  • A075254 (program): a(n) = n + (sum of prime factors of n taken with repetition).
  • A075255 (program): a(n) = n - (sum of primes factors of n (with repetition)).
  • A075263 (program): Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.
  • A075269 (program): Product of Lucas numbers and inverted Lucas numbers: a(n)=A000032(n)*A075193(n).
  • A075270 (program): Sum of Lucas numbers and inverted Lucas numbers: a(n) = A000032(n)*A075193(n).
  • A075271 (program): a(0) = 1 and, for n >= 1, (BM)a(n) = 2*a(n-1), where BM is the BinomialMean transform.
  • A075272 (program): BinomialMean (BM) transform of A075271, which see for the definition of (BM).
  • A075298 (program): Inverted (definition in A075193) generalized tribonacci numbers A001644.
  • A075300 (program): Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0,
  • A075301 (program): Inverse permutation to A075300.
  • A075302 (program): Transpose of array A075300.
  • A075303 (program): Inverse permutation to A075302.
  • A075311 (program): a(1) = 1; for n > 1, a(n) is the smallest number m > a(n-1) such that the number of 1’s in the binary expansion of m is not already in the sequence.
  • A075312 (program): Products of Wythoff pairs: [n*r]*[n*r^2], where [] is the floor function and r is the golden ratio, (1+sqrt(5))/2.
  • A075317 (program): Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1,3),(5,9),(7,13),(11,19),(15,25),(17,29),(21,35),(23,39),(27,45),… This is the sequence of the first member of pairs.
  • A075318 (program): Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1,3),(5,9),(7,13),(11,19),(15,25),(17,29),(21,35),(23,39),(27,45),… This is the sequence of the second member of pairs.
  • A075319 (program): Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1,3),(5,9),(7,13),(11,19),(15,25),(17,29),(21,35),(23,39),(27,45),… This is the sequence of the sum of the members of pairs.
  • A075320 (program): Pair the odd numbers such that the k-th pair is (r, r+2k) where r is the smallest odd number not included earlier: (1, 3), (5, 9), (7, 13), (11, 19), (15, 25), (17, 29), (21, 35), (23, 39), (27, 45), … This is the sequence of the product of the members of pairs.
  • A075325 (program): Pair the natural numbers such that the m-th pair is (r, s) where r, s and s-r are the smallest numbers which have not occurred earlier and also are not equal to the difference of any earlier pair: (1, 3), (4, 9), (6, 13), (8, 18), (11, 23), (14, 29), (16, 33), (19, 39), (21, 43), (24, 49), (26, 53), (28, 58), … Sequence gives first term of each pair.
  • A075326 (program): Anti-Fibonacci numbers: start with a(0) = 0, and extend by the rule that the next term is the sum of the two smallest numbers that are not in the sequence nor were used to form an earlier sum.
  • A075327 (program): Sum of n-th pair in A075325.
  • A075328 (program): Difference between n-th pair in A075325.
  • A075342 (program): a(1) = 1, a(n+1) is the smallest number such that there are n primes between a(n) and a(n+1) exclusive.
  • A075349 (program): a(1) = 1; first differences follow the pattern 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,…, i.e., the next n differences are n.
  • A075350 (program): 1-1, 2*3-(2+3), 4*5*6-(4+5+6), 7*8*9*10-(7+8+9+10), …
  • A075351 (program): a(n) = floor(2*binomial(n+1,2)!/(binomial(n,2)!*n*(n^2+1))).
  • A075353 (program): Initial term of n-th group in A075352.
  • A075354 (program): Final term of n-th group in A075352.
  • A075356 (program): Sum of terms in n-th group in A075352.
  • A075357 (program): a(n) = smallest k such that (n+1)(n+2)…(n+k) >= n!.
  • A075358 (program): a(n) = smallest (n+1)(n+2)…(n+k) that is >= n!.
  • A075359 (program): Sum of the digits of the next n numbers.
  • A075362 (program): Triangle read by rows with the n-th row containing the first n multiples of n.
  • A075363 (program): Triangle read by rows, in which n-th row gives n smallest powers of n.
  • A075364 (program): a(n) = floor( geometric mean of n-th row of A075363).
  • A075365 (program): Smallest k such that (n+1)(n+2)…(n+k) is divisible by the product of all the primes up to n.
  • A075369 (program): Square associated with twin primes (p,p+2): p(p+2) + 1. Square of the average of twin primes.
  • A075374 (program): a(n+2) = n*a(n+1) - a(n), with a(1)=1, a(2)=2.
  • A075393 (program): Positive integers k that are not divisible by A002034(k).
  • A075402 (program): Smallest number such that a(n) + T(n) is a prime, where T(n) is the n-th triangular number.
  • A075408 (program): Perfect powers pp such that pp+1 is prime.
  • A075411 (program): Squares of A002276.
  • A075412 (program): Squares of A002277.
  • A075413 (program): Squares of A002278.
  • A075414 (program): Squares of A002279: a(n) = (5*(10^n - 1)/9)^2.
  • A075415 (program): Squares of A002280 or numbers (666…6)^2.
  • A075416 (program): Squares of A002281.
  • A075417 (program): Squares of A002282: a(n) = (8*(10^n - 1)/9)^2.
  • A075418 (program): Sum of generalized tribonacci numbers A001644 and inverted tribonacci numbers A075298.
  • A075419 (program): Convolution of A073145 with A056594.
  • A075423 (program): rad(n) - 1, where rad(n) is the squarefree kernel of n (A007947).
  • A075424 (program): A075423(A075423(n)).
  • A075425 (program): Number of steps to reach 1 starting with n and iterating the map n ->rad(n)-1, where rad(n) is the squarefree kernel of n (A007947).
  • A075427 (program): a(0) = 1; a(n) = a(n-1)+1 if n is even, otherwise a(n) = 2*a(n-1).
  • A075431 (program): Primes of the form n+mu(n), where mu is the Moebius function (A008683).
  • A075435 (program): T(n,k) = right- or upward-moving paths connecting opposite corners of an n X n chessboard, visiting the diagonal at k points between start and finish.
  • A075436 (program): Right- or upward-moving paths connecting opposite corners of an n X n chessboard, visiting the diagonal in 0 up to (n-2) intermediate points between start and finish. Equivalently, subdivide the chessboard into 1 up to (n-1) blocks along the diagonal in all possible ways and sum the path-count over all sub-blocks.
  • A075438 (program): Triangle read by rows giving successive iterations of the Rule 60 elementary cellular automaton starting with a single ON cell where row n is of length 2n+1.
  • A075439 (program): Triangle read by rows giving successive iterations of the Rule 102 elementary cellular automaton starting with a single ON cell where row n is of length 2n+1.
  • A075465 (program): Rounded average of first n primes.
  • A075484 (program): Length of iteration-list when Collatz-function(A006370) is iterated with initial value 5^n.
  • A075485 (program): Length of iteration list when Collatz-function is iterated with initial value 2^n - 1.
  • A075486 (program): Length of iteration list when Collatz-function is iterated with initial value 2^n + 1.
  • A075487 (program): Length of iteration list when Collatz-function is iterated with initial value 1+3^n.
  • A075488 (program): Length of iteration list when Collatz-function is iterated with initial value -1+3^n.
  • A075490 (program): Sum[phi[n]^j,j = 1..n].
  • A075491 (program): Sum of digits of n minus number of divisors of n.
  • A075492 (program): Sum of digits of n < number of divisors of n.
  • A075493 (program): Numbers k such that (sum of digits of k) > (number of divisors of k).
  • A075494 (program): Squares whose sum of digits exceeds the number of divisors.
  • A075495 (program): Convolution of A075298 with A056594.
  • A075496 (program): a(1)=1, a(n) = sum( k=1,n-1, Max(a(k),a(n-k)) ).
  • A075506 (program): Shifts one place left under 7th-order binomial transform.
  • A075507 (program): Shifts one place left under 8th-order binomial transform.
  • A075508 (program): Shifts one place left under 9th-order binomial transform.
  • A075509 (program): Shifts one place left under 10th-order binomial transform.
  • A075510 (program): Fifth column of triangle A075497.
  • A075511 (program): Sixth column of triangle A075497.
  • A075512 (program): Seventh column of triangle A075497.
  • A075513 (program): Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.
  • A075515 (program): Fifth column of triangle A075498.
  • A075516 (program): Sixth column of triangle A075498.
  • A075518 (program): a(n) = floor(prime(n)/4).
  • A075519 (program): Primes p such that floor(p/4) is prime.
  • A075520 (program): 4*prime(n) + (prime(n) mod 4).
  • A075524 (program): Nonprimes of form 4*p + (p mod 4), p prime.
  • A075526 (program): a(n) = A008578(n+2) - A008578(n+1).
  • A075527 (program): a(n) = A008578(n+3) - A008578(n+1).
  • A075528 (program): Triangular numbers that are half other triangular numbers.
  • A075535 (program): a(1)=1, a(n) = Sum_{k=1..n-1} min(a(k), a(n-k)).
  • A075537 (program): a(1)=1, a(2)=2, then use “merge and minus”: a(n)=merge(a(n-2),a(n-1))-a(n-2)-a(n-1)).
  • A075543 (program): a[n] = a[n-1] + digit sum(n) - 1.
  • A075549 (program): Decimal expansion of 9 - 12*log(2).
  • A075553 (program): Numerators in the Maclaurin series for arctan(1+x).
  • A075554 (program): Denominators in the Maclaurin series for arctan(1+x).
  • A075559 (program): Smallest multiple of n not equal to n ending in (digits of) n.
  • A075561 (program): Domination number for kings’ graph K(n).
  • A075569 (program): a(1)=1, then smallest number >= the previous term such that every partial sum is a prime.
  • A075574 (program): a(1) = 1, then the smallest number (obviously even) greater than the previous term such that every partial sum is prime.
  • A075576 (program): Smaller of two consecutive squares with a prime sum.
  • A075578 (program): Smaller of two successive 4th powers whose sum is a prime.
  • A075581 (program): Let P(n,X) = Product_{i=1..2n+1} (X - 1/cos(Pi*k/(2n+1))); then P(n,X) is a polynomial with integer coefficients. Sequences gives maximum values of absolute values of coefficients of P(n,X).
  • A075614 (program): Let P(k,X) = 4^k*Product_{i=1..k} (X - cos(Pi*i/k)) (which is a polynomial with integer coefficients). Sequence gives maximum values of coefficients of P(n,X).
  • A075643 (program): Group the natural numbers so that the n-th group contains n numbers one each of a multiple of numbers from 1 to n so that the group sum is a multiple of (n+1): (2), (1, 8), (3, 4, 9), (5, 6, 12, 32), (7, 10, 15, 16, 30), (11, 14, 18, 20, 25, 24), … Sequence gives initial terms of groups.
  • A075653 (program): a(n) = n + sopf(n), where sopf is the sum of the distinct prime factors of n (A008472).
  • A075656 (program): n + product of prime factors of n.
  • A075658 (program): Numbers k such that the sum of prime divisors of k (A008472) is composite.
  • A075664 (program): Sum of next n cubes.
  • A075665 (program): Sum of next n 4th powers. i^s, s = 4.
  • A075666 (program): Sum of next n 5th powers.
  • A075667 (program): Sum of next n 6th powers.
  • A075668 (program): Sum of next n 7th powers.
  • A075669 (program): Sum of next n 8th powers.
  • A075670 (program): Sum of next n 9th powers.
  • A075671 (program): Sum of next n 10th powers.
  • A075673 (program): Sum of next n integer interprimes (cf. A024675).
  • A075674 (program): Sum of next n odd interprimes.
  • A075677 (program): Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible.
  • A075680 (program): For odd numbers 2n-1, the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R is defined as R(k) = (3k+1)/2^r, with r as large as possible.
  • A075681 (program): Difference between (n-1)*(n-2)^3 and A003878(n).
  • A075682 (program): First differences of A075681.
  • A075683 (program): 2nd differences of A075681.
  • A075684 (program): For odd numbers 2n-1, the maximum number produced by iterating the reduced Collatz function R defined as R(k) = (3k+1)/2^r, with r as large as possible.
  • A075692 (program): Upper irredundance number for queens graph Q_n on n^2 nodes.
  • A075693 (program): Difference between 10-adic numbers defined in A018248 & A018247.
  • A075694 (program): a(1)=1, then “jump over next prime”: a(n) = 2 nextprime(a(n-1))-a(n-1).
  • A075699 (program): Number of primes in the interval (n,4n].
  • A075704 (program): p and 12*p+1 are both primes.
  • A075709 (program): Upper irredundance number for kings graph K_n on n^2 nodes.
  • A075726 (program): a(n) = Pi * n^2 rounded off.
  • A075727 (program): a(n) = 2 Pi * n rounded off.
  • A075730 (program): Squares of odd semiprimes A046315, odd numbers divisible by exactly 2 primes (counted with multiplicity).
  • A075731 (program): Fibonacci numbers F(k) for k squarefree (A005117).
  • A075732 (program): Fibonacci numbers F(k) for k not squarefree (A013929).
  • A075734 (program): Fibonacci numbers F(k) when k is a product of an even number of distinct primes A030229 (mu(k)=1).
  • A075736 (program): Fibonacci numbers F(k) as k runs through the products of an odd number of distinct primes A030059 (mu(k)=-1).
  • A075738 (program): Squarefree Fibonacci numbers whose indices are also squarefree.
  • A075743 (program): For all numbers of the form 6 +- 1 starting with 5,7,11,13,…, ‘1’ indicates prime and ‘0’ indicates composite.
  • A075745 (program): Numbers n such that 210*n + 13 is prime.
  • A075746 (program): Numbers n such that 210*n-13 is prime.
  • A075747 (program): Numbers n such that 210*n + 17 is prime.
  • A075748 (program): Numbers n such that 210*n-17 is prime.
  • A075749 (program): Numbers k such that 210*k +- 1 are twin primes.
  • A075753 (program): Smallest prime factor of n-th odd triangular number.
  • A075765 (program): a(n) = floor(prime(n)/n) + (prime(n) mod n).
  • A075778 (program): Decimal expansion of the real root of x^3 + x^2 - 1.
  • A075779 (program): Triangle T(n,k) = f(n,k,n-1), n >= 2, 1 <= k <= n-1, where f is given below.
  • A075794 (program): a(n) = the least positive integer k such that phi(k) > phi(1) + … + phi(n).
  • A075795 (program): Number of k, 0<k<=n, such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is equal to 1.
  • A075796 (program): Numbers k such that 5*k^2 + 5 is a square.
  • A075801 (program): Differences between adjacent palindromic nonprime numbers A032350.
  • A075802 (program): Characteristic function of perfect powers, A001597.
  • A075803 (program): Differences between adjacent palindromic squarefree numbers.
  • A075818 (program): Even numbers with exactly 3 prime factors (counted with multiplicity).
  • A075819 (program): Even squarefree numbers with exactly 3 prime factors.
  • A075821 (program): List of possible last two digits (leading zeros omitted) of perfect powers.
  • A075823 (program): Numbers that are not the last two digits (leading zeros omitted) of any perfect power.
  • A075827 (program): Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) =(a(n)*x + b(n))/(c(n)*x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.
  • A075828 (program): Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) =(b(n)*x + a(n))/(c(n)*x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.
  • A075829 (program): Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)*x + c(n))/(d(n)*x + a(n)) (in lowest terms) and a(n), b((n), c(n), d(n) are positive integers.
  • A075830 (program): Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)*x + c(n))/(a(n)*x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.
  • A075835 (program): Numbers n such that 13*n^2 + 4 is a square.
  • A075836 (program): Numbers n such that 10*n^2 + 9 is a square.
  • A075839 (program): Numbers k such that 11*k^2 - 2 is a square.
  • A075841 (program): Numbers k such that 2*k^2 - 9 is a square.
  • A075842 (program): n 1’s followed by n.
  • A075843 (program): Numbers k such that 99*k^2 + 1 is a square.
  • A075844 (program): Numbers n such that 11*n^2 + 4 is a square.
  • A075847 (program): Difference between n^2 and the largest cube <= n^2.
  • A075848 (program): Numbers k such that 2*k^2 + 9 is a square.
  • A075857 (program): Least common multiple of totient and cototient of n.
  • A075858 (program): n followed by n 1’s.
  • A075859 (program): a(n) = n concatenated with n 1’s and n.
  • A075860 (program): a(n) is the fixed point reached by the sum of divisors of n without multiplicity (with the convention a(1)=0).
  • A075861 (program): Least k such that (n-k) divides (n+k).
  • A075862 (program): Numbers m such that the least k such that (m-k) divides (m+k) is prime.
  • A075869 (program): Numbers k such that 5*k^2 - 9 is a square.
  • A075870 (program): Numbers k such that 2*k^2 - 4 is a square.
  • A075871 (program): Numbers k such that 13*k^2 + 1 is a square.
  • A075872 (program): a(n) = binomial(prime(n),n)/prime(n) where prime(n) = n-th prime.
  • A075873 (program): 40*n^2 + 9 is a square.
  • A075875 (program): Triangular numbers that are 3-almost primes.
  • A075876 (program): Values of n for which A075825(n)=1.
  • A075877 (program): Powering the decimal digits of n (left-associative).
  • A075878 (program): Sum of coefficients of (x1)^(2i(1))*(x2)^(2i(2))*(x3)^(2i(3))*(x4)^(2i(4)) for {(i1),(i2),(i3),(i4)}=0,1,2,… : sum(i)=2n in the expansion of (x1+x2+x3+x4)^(2n) where n=1,2,3,…
  • A075879 (program): The (10^n)-th odd-digit number.
  • A075881 (program): a(n) = the sum of the prime factors of Sum_{i=1..n} prime(i).
  • A075882 (program): a(n) = phi(Sum_{i=1,…,n} prime(i)).
  • A075884 (program): Image of n at the second step of the 3x+1 algorithm.
  • A075885 (program): a(n) = 1 + n + n*[n/2] + n*[n/2]*[n/3] + n*[n/2]*[n/3]*[n/4] +… where [x]=floor(x).
  • A075886 (program): Number of cubes at generation n when building fractal cube with edge ratio of 1/2.
  • A075887 (program): a(n) = 1 + n + n[n/2] + n[n/2][n/3] +… + n[n/2][n/3]…[n/n], where [x]=ceiling(x).
  • A075888 (program): Difference of successive primes squared divided by 24, (prime(n+1)^2-prime(n)^2)/24.
  • A075890 (program): Largest term in the prime(n)-th row of Pascal’s triangle, prime(n) being the n-th prime.
  • A075891 (program): Quotient C[p(n),{p(n)+-1}/2]/p(n), where p(n)=n-th prime.
  • A075892 (program): Average of squares of successive primes: a(n) = (prime(n+1)^2 + prime(n)^2)/2, with n >= 2.
  • A075893 (program): Average of three successive primes squared, (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.
  • A075894 (program): Average of four successive primes squared, (prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + prime(n+3)^2)/4, n>=2.
  • A075897 (program): 1 if binary weight of n is 1 or 2, otherwise 0.
  • A075900 (program): G.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).
  • A075906 (program): Seventh column of triangle A075498.
  • A075907 (program): Fourth column of triangle A075499.
  • A075908 (program): Fifth column of triangle A075499.
  • A075909 (program): Sixth column of triangle A075499.
  • A075910 (program): Seventh column of triangle A075499.
  • A075911 (program): Third column of triangle A075500.
  • A075912 (program): Fourth column of triangle A075500.
  • A075913 (program): Fifth column of triangle A075500.
  • A075914 (program): Sixth column of triangle A075500.
  • A075915 (program): Seventh column of triangle A075500.
  • A075916 (program): Third column of triangle A075501.
  • A075917 (program): Fourth column of triangle A075501.
  • A075918 (program): Fifth column of triangle A075501.
  • A075919 (program): Sixth column of triangle A075501.
  • A075920 (program): Seventh column of triangle A075501.
  • A075921 (program): Second column of triangle A075502.
  • A075922 (program): Third column of triangle A075502.
  • A075923 (program): Fourth column of triangle A075502.
  • A075924 (program): Fifth column of triangle A075502.
  • A075925 (program): Sixth column of triangle A075502.
  • A075930 (program): Positions of check bits in code in A075928.
  • A075986 (program): Numerator of 1+1/prime(1)^2+ … + 1/prime(n)^2 where prime(k) is the k-th prime.
  • A075987 (program): Numerator(1+1/prime(1)^3+ … + 1/prime(n)^3) where prime(k) is the k-th prime.
  • A075988 (program): Number of integers k satisfying 1 <= k <= n and 0 < frac(n/k) < 1/2, where frac(n/k) is the fractional part of n/k; i.e., frac(n/k) = n/k - floor(n,k).
  • A075989 (program): Number of k satisfying 1<=k<=n and {n/k} >= 1/2, where {n/k} is the fractional part of n/k, i.e., {n/k} = n/k - floor(n/k).
  • A075993 (program): Triangle read by rows: T(n,m) is the number of integers k such that floor(n/k) = m, n >= 1, k = 1..n.
  • A075994 (program): Irregular triangle T(n,k) = floor(n/k) for k = 1, 2, …, floor(n/2) for n>=2 and T(1,1)=1.
  • A075995 (program): a(n) = Sum_{k=1..floor(n/2)} floor(n/k) for n >= 2, with a(1) = 1.
  • A075997 (program): a(n) = [n/2]-[n/3]+[n/4]-[n/5]+[n/6]-…, where [n/k] = floor(n/k).
  • A075999 (program): Product{[n/k + 1/2]: k=1,2,…,n}, where [x + 1/2] denotes the integer nearest to x.
  • A076000 (program): a(n) = Product_{k=1..n} k/floor(n/k).
  • A076002 (program): Seventh column of triangle A075502.
  • A076003 (program): Third column of triangle A075503.
  • A076004 (program): Fourth column of triangle A075503.
  • A076005 (program): Fifth column of triangle A075503.
  • A076006 (program): Sixth column of triangle A075503.
  • A076007 (program): Seventh column of triangle A075503.
  • A076008 (program): Second column of triangle A075504.
  • A076009 (program): Third column of triangle A075504.
  • A076010 (program): Fourth column of triangle A075504.
  • A076011 (program): Fifth column of triangle A075504.
  • A076012 (program): Sixth column of triangle A075504.
  • A076013 (program): Seventh column of triangle A075504.
  • A076014 (program): Triangle in which m-th entry of n-th row is m^(n-1).
  • A076015 (program): Row sums of triangle A076014.
  • A076024 (program): a(n) = (2^n + 4)*(2^n - 1)/6.
  • A076025 (program): Expansion of g.f.: (1-3*x*C)/(1-4*x*C) where C = (1 - sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers A000108.
  • A076026 (program): Expansion of g.f.: (1-4*x*C)/(1-5*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
  • A076027 (program): Initial members of groups in A076031.
  • A076035 (program): G.f.: 1/(1-4*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
  • A076036 (program): G.f.: 1/(1 - 5*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.
  • A076040 (program): a(n) = (-1)^n * (3^n - 1)/2.
  • A076042 (program): a(0) = 0; thereafter a(n) = a(n-1) + n^2 if a(n-1) < n^2, otherwise a(n) = a(n-1) - n^2.
  • A076049 (program): Numbers k such that the sum of the k-th triangular number and (k+2)-nd triangular number is a triangular number.
  • A076050 (program): Limiting sequence if we start with 2 and successively replace n with 2,3,4,…,n,n+1.
  • A076051 (program): Sum of product of odd numbers <= n and the product of even numbers <= n.
  • A076052 (program): Sum(k=1, n, A006460(k)).
  • A076054 (program): a(n) = Sum_{k=1..n} A006513(k).
  • A076068 (program): Smallest number that can be formed by using the nonzero digits of the numbers 1+n(n-1)/2 through n(n+1)/2.
  • A076074 (program): Initial members of groups in A076077.
  • A076079 (program): Largest multiple of n < the n-th prime.
  • A076080 (program): a(n) = A076079(n)/n.
  • A076092 (program): a(n) = n - 2*Sum_{i=1..n} b(i) (see comment for definition of b(i)).
  • A076095 (program): Initial terms of rows in A076099.
  • A076100 (program): Least common multiple of n numbers starting with n.
  • A076103 (program): Sums of members of groups in A076105.
  • A076108 (program): Least positive n-th power that is the sum of n consecutive integers, or 0 if no such n-th power exists.
  • A076109 (program): Least positive k such that k^n is the sum of n consecutive integers, or 0 if no such k exists.
  • A076110 (program): Triangle (read by rows) in which the n-th row contains first n terms of an arithmetic progression with first term 1 and common difference (n-1).
  • A076111 (program): Product of terms in n-th row of A076110.
  • A076112 (program): Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).
  • A076113 (program): a(n) = n^(n*(n-1)/2).
  • A076118 (program): a(n) = sum_k {n/2<=k<=n} k * (-1)^(n-k) * C(k,n-k).
  • A076121 (program): Complete list of possible cribbage hands.
  • A076127 (program): n-th term is binary string of length t_n with 1’s at positions t_i, where t_n = n-th triangular number.
  • A076128 (program): Difference between the product of numbers up to n and the sum of numbers up to n.
  • A076131 (program): a(n) = 2^n*a(n-1)+1, a(0) = 0.
  • A076136 (program): Numbers n such that Omega(n) = Omega(n-1) + Omega(n-2), where Omega(n) (A001222) denotes the number of prime factors of n, counting multiplicity.
  • A076139 (program): Triangular numbers that are one-third of another triangular number: T(m) such that 3*T(m) = T(k) for some k.
  • A076140 (program): Triangular numbers T(k) that are three times another triangular number: T(k) such that T(k) = 3*T(m) for some m.
  • A076144 (program): Largest squarefree m <= sfn(n) such that m*sfn(n) is also squarefree, where sfn(n) is the n-th squarefree number.
  • A076148 (program): Let b(1)=x, b(2)=y, k*b(k)=(2k-1)*b(k-1) + 3(k+1)*b(k-2); then b(n)=a(n)*x+c(n)/3*y.
  • A076149 (program): Expansion of x^2(3+2x)/(1-x-5x^2-3x^3).
  • A076150 (program): Start of 10 consecutive composite numbers.
  • A076151 (program): (n-1)!*binomial(3*n,n)/(3*(2*n+1)).
  • A076160 (program): Sod_4 - sod_3 + sod_2 - sod_1, where sod_k is the sum of k-th powers of digits of n.
  • A076161 (program): Numbers n such that n + sum of squares of digits of n (A258881) is a prime.
  • A076169 (program): Triangular numbers whose sum of prime factors (with repetition) is also triangular.
  • A076176 (program): a(n) = n!*sum( i+j<=n, 1/i!/j! ) for 0 <= i <= j < n.
  • A076178 (program): a(n) = 2*n^2 - A077071(n).
  • A076182 (program): a(n) = A006666(n) mod 2.
  • A076191 (program): First differences of A001222.
  • A076192 (program): n == 1 mod 10 with property that n through n+9 contain no primes.
  • A076203 (program): Numbers n such that the sum of the digits of 2^n is prime.
  • A076204 (program): Numbers whose cube has a prime sum of digits.
  • A076214 (program): Decimal expansion of C=sum(k>=0,1/2^(2^k-1)).
  • A076217 (program): a(1)=1, a(n) = a(n-1) + n * (sign(n-a(n-1)).
  • A076218 (program): Numbers n such that 2*n^2 - 3*n + 1 is a square.
  • A076221 (program): Triangle read by rows: A(n,k) is the number of x, x<=n, which are coprime to and not equal to k.
  • A076225 (program): Counts of the maximum value in n-th row of A076221.
  • A076236 (program): a(n) = A050435(n) mod A002808(n).
  • A076237 (program): Remainder when 2nd order composite cc[n]=A050435[n] is divided by n.
  • A076238 (program): Remainder when 3rd order composite ccc[n]=A050436[n] is divided by first order composite c[n]=A002808[n].
  • A076239 (program): Remainder when 3rd-order composite ccc(n) = A050436(n) is divided by n.
  • A076240 (program): Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).
  • A076241 (program): Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.
  • A076242 (program): Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).
  • A076243 (program): Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.
  • A076259 (program): Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).
  • A076260 (program): a(n) = 0 if n is a squarefree number, otherwise the distance between the two nearest squarefree numbers around n: A067535(n)-A070321(n).
  • A076264 (program): Number of ternary (0,1,2) sequences without a consecutive ‘012’.
  • A076271 (program): a(1) = 1, a(2) = 2, and for n > 2, a(n) = a(n-1) + gpf(a(n-1)), where gpf = greatest prime factor = A006530.
  • A076272 (program): Largest prime factor of A076271(n): A006530(A076271(n)).
  • A076273 (program): a(0) = 1, a(1) = 2; for n>1, a(n) = prime(n)+prime(n-1)-1.
  • A076274 (program): 2p-1 where p is 1 or a prime.
  • A076290 (program): Sum of the semiprime divisors of n.
  • A076293 (program): Numbers k where the root mean square (RMS) of k and 7 is an integer, i.e., sqrt((k^2 + 7^2)/2) is an integer.
  • A076294 (program): Consider all Pythagorean triples (X,X+7,Z); sequence gives Z values.
  • A076295 (program): Consider all Pythagorean triples (Y-7,Y,Z); sequence gives Y values.
  • A076296 (program): Consider all Pythagorean triples (X,X+7,Z); sequence gives X values.
  • A076297 (program): Prime(n)+ s*n is prime, s=2.
  • A076301 (program): Related to number of labeled partially ordered sets.
  • A076309 (program): a(n) = floor(n/10) - 2*(n mod 10).
  • A076310 (program): a(n) = floor(n/10) + 4*(n mod 10).
  • A076311 (program): a(n) = floor(n/10) - 5*(n mod 10).
  • A076312 (program): a(n) = floor(n/10) + 2*(n mod 10).
  • A076313 (program): a(n) = floor(n/10) - (n mod 10).
  • A076314 (program): a(n) = floor(n/10) + (n mod 10).
  • A076332 (program): Rad(n)+n/rad(n), where rad(n) is the squarefree kernel of n = A007947(n).
  • A076333 (program): Squarefree kernels of nonsquarefree numbers.
  • A076334 (program): Differences between successive squarefree kernels.
  • A076338 (program): a(n) = 512*n + 1.
  • A076339 (program): Primes of the form 512*k+1.
  • A076342 (program): a(n) = A076340(A000040(n)), real part of primes mapped as defined in A076340, A076341.
  • A076354 (program): Numbers n such that 210*n-1 is prime.
  • A076355 (program): Numbers n such that 210*n + 11 is prime.
  • A076358 (program): a(n) = numerator(n!/phi(n!)).
  • A076359 (program): a(n) = denominator(n!/phi(n!)).
  • A076360 (program): a(n) = commutatorsigma,tau = d0(d1(w)) - d1(d0(w)), where d0()=number of, d1()=sum of divisors of n.
  • A076361 (program): Numbers n such that d(sigma(n)) = sigma(d(n)).
  • A076364 (program): Number of distinct terms in dRRS equals 2: A061498(x)=2.
  • A076367 (program): Primes with subscripts from the Bonse sequence.
  • A076368 (program): a(1) = 1; for n > 1, a(n) = prime(n) - prime(n-1) + 1.
  • A076369 (program): n + mu(n), where mu is the Moebius-function (A008683).
  • A076388 (program): a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.
  • A076389 (program): Sum of squares of numbers that cannot be written as t*n + u*(n+1) for nonnegative integers t,u.
  • A076390 (program): Decimal expansion of lemniscate constant B.
  • A076407 (program): Sum of perfect powers <= n.
  • A076409 (program): Sum of the quadratic residues of prime(n).
  • A076410 (program): (Sum of the quadratic residues of prime(n)) / prime(n).
  • A076411 (program): Number of perfect powers < n.
  • A076441 (program): Let u(1) = u(2) = u(3) = 1; u(n) = sign(u(n-1)-u(n-2))*u(n-3), then a(n) = 1+u(n).
  • A076446 (program): Differences of consecutive powerful numbers (definition 1).
  • A076447 (program): Let v(1)=v(2)=v(3)=1, v(n)=(-1)^n*sign(v(n-1)-v(n-2))*v(n-3), then a(n) =1+v(n).
  • A076452 (program): a(n+2) = abs(a(n+1)) - a(n), a(0)=0, a(1)=1.
  • A076453 (program): a(n+2) = abs(a(n+1)) - a(n), a(0)=1, a(1)=0.
  • A076454 (program): Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.
  • A076455 (program): Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly two ways.
  • A076456 (program): Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly three ways.
  • A076457 (program): Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly four ways.
  • A076458 (program): Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly five ways.
  • A076459 (program): Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.
  • A076471 (program): Number of pairs (p,q) of successive primes with p+q<=n.
  • A076479 (program): a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).
  • A076480 (program): n + mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the squarefree kernel (A007947).
  • A076482 (program): Triangle with T(n,k)=n!*(k-1)^k/k! where 1<=k<=n.
  • A076483 (program): a(n) = n!*Sum_{k=1..n} (k-1)^k/k!.
  • A076505 (program): 3 people at a party are saying Hello to each other. Person 1 says Hello. Person 2 counts the times Hello has been said and says Hello twice that number. Person 3 says Hello 3 times the sum of Hello’s and then it is Person 1 again. This is how many Hello’s each person says.
  • A076506 (program): Expansion of x*(1+3*x+12*x^2)/(1-24*x^3).
  • A076507 (program): Three people (P1, P2, P3) are in a circle and are saying Hello to each other. They start with P2 saying “Hello, Hello”. Thereafter Pn says “Hello” for n times the total number of Hello’s so far.
  • A076508 (program): Expansion of 2*x*(1+4*x+8*x^2)/(1-24*x^3).
  • A076509 (program): Expansion of 3*x*(1-x)*(1+2*x+6*x^2)/(1-24*x^3).
  • A076510 (program): Expansion of 3*(1+2*x+6 x^2)/(1-24*x^3).
  • A076511 (program): Numerator of cototient(n)/totient(n).
  • A076512 (program): Denominator of cototient(n)/totient(n).
  • A076520 (program): n appears once if n is the sum of 2 nonzero squares in 1 way, twice if n is the sum of 2 squares in 2 ways, 3 times if n is the sum of 2 squares 3 ways etc.
  • A076523 (program): Maximal number of halving lines for 2n points in plane.
  • A076526 (program): a(n) = r * max(e_1, …, e_r), where n = p_1^e_1 . …. p_r^e_r is the canonical prime factorization of n, a(1) = 0.
  • A076535 (program): a(n) = A064405 (2^m+n) - 2^m (for m large enough this difference appears to be constant).
  • A076536 (program): Image of n at the third step in the 3x+1 Problem: syr(3,n).
  • A076537 (program): Map positive rational numbers to positive integers by diagonal method using c(p,q) = (p + q - 2) * (p + q - 1) / 2 + p where p and q are positive integers. a(n) is an increasing sequence including all c(p,q) where gcd(p,q) > 1.
  • A076538 (program): Numerators a(n) of fractions slowly converging to e: let a(1) = 0, b(n) = n - a(n); if (a(n) + 1) / b(n) < e then a(n+1) = a(n) + 1, else a(n+1)= a(n).
  • A076539 (program): Numerators a(n) of fractions slowly converging to Pi: let a(1) = 0, b(n) = n - a(n); if (a(n) + 1) / b(n) < Pi, then a(n+1) = a(n) + 1, otherwise a(n+1) = a(n).
  • A076540 (program): Number of branches in all ordered trees with n edges.
  • A076541 (program): a(n) = Sum_{k=1..n} C(n,k) mod k.
  • A076543 (program): Sum(k=1 to n) k*sqf(k); sqf(k)=1 if k is squarefree and sqf(k)=-1 otherwise.
  • A076544 (program): mu(n)+sqf(n): mu(n) is Moebius function; sqf(n)=1 if n is squarefree, sqf(n)=-1 otherwise.
  • A076545 (program): sum[k=1 to n] mu(k)+sqf(k): mu(k) is Moebius function; sqf(k)=1 if k is squarefree, sqf(k)=-1 otherwise.
  • A076552 (program): a(n) = (-1)^(n+1)/3/(2n+1)*sum(k=0,n,16^k*B(2k)*C(2n+1,2k)) where B(k) denotes the k-th Bernoulli number.
  • A076555 (program): Greatest prime divisor of n-th prime + 2.
  • A076556 (program): Greatest prime divisor of n-th prime + n.
  • A076557 (program): Greatest prime divisor of n-th prime - n.
  • A076558 (program): a(n) = r * min(e_1, …, e_r), where n = p_1^e_1 . …. p_r^e_r is the canonical prime factorization of n, a(1) = 0.
  • A076559 (program): Greatest prime divisor of n-th interprime: (prime(n) + prime(n+1))/2.
  • A076560 (program): a(1)=1; a(n>1)= greatest prime divisor of (a(n-1) + n).
  • A076561 (program): a(1)=2; a(n>1)= greatest prime divisor of a(n-1) + n.
  • A076562 (program): a(1)=3; a(n>1)= greatest prime divisor of a(n-1) + n.
  • A076563 (program): a(n) = n - greatest prime divisor of n, for n>1.
  • A076565 (program): Greatest prime divisor of 2n+1 (sum of two successive integers).
  • A076566 (program): Greatest prime divisor of 3n+3 (sum of three successive integers).
  • A076567 (program): Greatest prime divisor of 4n+6 (sum of four successive integers).
  • A076568 (program): Greatest prime divisor of 5n+10 (sum of five successive integers).
  • A076569 (program): Greatest prime divisor of 6n+15 (sum of six successive integers).
  • A076570 (program): Greatest prime divisor of sum of first n primes.
  • A076571 (program): Binomial triangle based on factorials.
  • A076577 (program): Sum of squares of divisors d of n such that n/d is odd.
  • A076578 (program): Triangular numbers which are 4-almost primes.
  • A076579 (program): Triangular numbers which are 5-almost primes.
  • A076580 (program): Triangular numbers which are 6-almost primes.
  • A076581 (program): Triangular numbers which are 7-almost primes.
  • A076582 (program): Triangular numbers which are 8-almost primes.
  • A076583 (program): Triangular numbers which are 9-almost primes.
  • A076591 (program): a(1)=1, a(2)=2 a(n)=a(n-1)+(a(n-2) mod n).
  • A076597 (program): Numbers k such that sqrt(k*(k-1)*(k-2)*(k-3)+1) is a prime.
  • A076598 (program): Sum of squares of divisors d of n such that d or n/d is odd.
  • A076601 (program): a(1)=3, a(n_even)=(a(n-1)^2-1)/2; a(n_odd)=a(n-1)+1.
  • A076602 (program): a(1)=7, a(n_even)=(a(n-1)^2-1)/2; a(n_odd)=a(n-1)+1.
  • A076603 (program): a(1)=9, a(n_even)=(a(n-1)^2-1)/2; a(n_odd)=a(n-1)+1.
  • A076604 (program): a(1) = 11, a(n_even) = (a(n-1)^2-1)/2; a(n_odd) = a(n-1)+1.
  • A076605 (program): Largest prime divisor of n^2 - 1.
  • A076606 (program): Min { largest prime factor of n-1, largest prime factor of n+1 }.
  • A076607 (program): a(0)=1 and for n>0: a(n) = if gcd(a(n-1),n)>1 then lcm(a(n-1),n) else a(n-1)+n.
  • A076608 (program): Number of nonprimes k < n such that also n-k is not a prime.
  • A076610 (program): Numbers having only prime factors of form prime(prime); a(1)=1.
  • A076616 (program): Number of permutations of {1,2,…,n} that result in a binary search tree (when elements of the permutation are inserted in that order) of height n-1 (i.e., the second largest possible height).
  • A076618 (program): Least x>1 such that x^d == 1 (mod d) for each divisor d of n.
  • A076619 (program): Least x>1 such that x^d == 1 (mod d) for each divisor d of n, for all nonsquarefree numbers n (cf. A013929).
  • A076621 (program): Least square greater than the product of two successive primes.
  • A076627 (program): a(n) = tau(n)*(n-tau(n)), where tau(n) = number of divisors of n (A000005).
  • A076631 (program): Solve 2^n - 2 = 7(x^2 - x) + (y^2 - y) for (x,y) with x>0, y>0; a(n) = value of y.
  • A076632 (program): Solve 2^n - 2 = 7(x^2 - x) + (y^2 - y) for (x,y) with x>0, y>0; sequence gives value of x.
  • A076634 (program): Coefficient of x^a(n) in (x+1/2)*(x+2/2)*…*(x+n/2) is the largest one.
  • A076637 (program): Numerators of harmonic numbers when these numerators are divisible by squares of primes >= 5 in the case of Wolstenholme’s Theorem.
  • A076638 (program): Denominators of harmonic numbers when the numerators are divisible by squares of primes >= 5 in the case of Wolstenholme’s Theorem.
  • A076639 (program): Numbers that are neither primes nor interprimes.
  • A076640 (program): a(1)=1, a(n) = a(n-phi(n)) + 1.
  • A076642 (program): Coefficient of x^a(n) in (x+1/3!)*(x+2/3!)*…*(x+n/3!) is the largest one.
  • A076644 (program): a(1)=1; for n>1, a(n) = a(n-floor(sqrt(n))) + n.
  • A076645 (program): Counts the ways to write 0 as 1 +- 2 +- 3 +- 4 +- … +- k for some k, where signs alternate except that there is one instance of two consecutive positive terms.
  • A076649 (program): Number of digits required to write the prime factors of n.
  • A076651 (program): Floor( Sqrt( p * (p+2) / 2)) where p is the lesser of the twin primes.
  • A076657 (program): a(n) = (1/24) * binomial(2n,n)*(16^n-binomial(2n,n)^2). Right side of identity involving series A005148.
  • A076662 (program): First differences of A007066.
  • A076663 (program): a(n) = sum of sigma(e) where e ranges over all non-divisors of n that are between 1 and n.
  • A076664 (program): a(n) = Sum_{k=1..n} antisigma(k), where antisigma(i) = sum of the nondivisors of i that are between 1 and i.
  • A076669 (program): Primes of the form n^2*totient(n)+1 (or A053191(n) + 1).
  • A076677 (program): a(0)=a(1)=1 a(n)=a(n-1)+floor(sqrt(a(n-2))).
  • A076678 (program): a(0)=a(1)=1 a(n)=floor(sqrt(a(n-1)))+a(n-2).
  • A076684 (program): Odd terms in A027941.
  • A076685 (program): a(n) = max(core(n),phi(n)) where core(n) is the squarefree part of n.
  • A076686 (program): a(n) = min(core(n),phi(n)) where core(n) is the squarefree part of n.
  • A076690 (program): Nearest integer to average of the smallest and largest prime factors of n (0.5 is rounded to 1).
  • A076694 (program): a(n) = n - sum of the distinct prime factors of n.
  • A076695 (program): Dimension of S2(G0(p)) where p runs through the odd primes >= 13, G0(N) is the Hecke subgroup of SL2(Z), consisting of 2 X 2 matrices (a,b; c,d) with c == 0 (mod N).
  • A076698 (program): a(1) = 2, a(n+1) = smallest squarefree number == 1 (mod a(n)).
  • A076708 (program): Numbers n such that triangular numbers T(n) and T(n+1) sum to another triangular number (that is also a perfect square).
  • A076713 (program): Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits.
  • A076717 (program): a(n) = -Sum_{d|n} (-n/d)^d.
  • A076720 (program): Sum of product of divisors of n and sum of divisors of n.
  • A076721 (program): Difference between product of divisors of n and sum of divisors of n.
  • A076722 (program): Product of product of divisors of n and sum of divisors of n.
  • A076723 (program): Sum_{d divides n} (-d)^d.
  • A076725 (program): a(n) = a(n-1)^2 + a(n-2)^4, a(0) = a(1) = 1.
  • A076726 (program): a(n) = Sum_{k>=0} k^n/2^k.
  • A076727 (program): Primes of the form x^2 + (x+3)^2.
  • A076728 (program): a(n) = (n-1)^2 * n^(n-2).
  • A076729 (program): a(n) = A001147(n+1) * Integral_{x=0..1} (1 + x^2)^n dx.
  • A076731 (program): Table T(n,k) giving number of ways of obtaining exactly 0 correct answers on an (n,k)-matching problem (1 <= k <= n).
  • A076732 (program): Table T(n,k) giving number of ways of obtaining exactly one correct answer on an (n,k)-matching problem (1 <= k <= n).
  • A076733 (program): Largest k such that k! divides C(2n,n).
  • A076736 (program): Let u(1) = u(2) = u(3) = 2, u(n) = (1 + u(n-1)*u(n-2))/u(n-3); then a(n) is the denominator of u(n).
  • A076737 (program): Let u(1)=u(2)=u(3)=2, u(n)=(1+u(n-1)u(n-2))/u(n-3); then a(n) is the numerator of u(n).
  • A076738 (program): Expansion of (1+2*x+6*x^2)/(1-9*x^3).
  • A076739 (program): Number of compositions of n into Fibonacci numbers (1 counted as single Fibonacci number).
  • A076740 (program): a(n) = 2*a(n-1)^2 - a(n-2)^2 with a(0)=0, a(1)=1.
  • A076746 (program): List giving pairs of primes of the form 10k+3 and 10k+7.
  • A076752 (program): a(n) = Sum_{d is a square divisor of n} n/d.
  • A076756 (program): Triangle of coefficients of characteristic polynomial of M_n, the n X n matrix M_(i,j) = min(i,j).
  • A076757 (program): Primes of the form n + pi(n), that is, generated in A077510.
  • A076758 (program): a(n) = n*(n+1)^2*(2+n)*(3+2*n)*(19+8*n)/180.
  • A076765 (program): Partial sums of Chebyshev sequence S(n,8) = U(n,4) = A001090(n+1).
  • A076767 (program): Triangular numbers with square pyramidal indices.
  • A076769 (program): Integers not expressible as the sum of a triangular number and a square.
  • A076770 (program): Even numbers representable as the sum of two odd composites.
  • A076771 (program): Even numbers n representable as the sum of two non-coprime odd composites.
  • A076775 (program): Greatest common divisor of n and the binary representation of n interpreted decimally.
  • A076776 (program): a(0) = 1, a(1) = 2, a(2) = 5; for n > 2, a(n) = a(n-1)*a(n-2).
  • A076782 (program): a(n) = 10^(n^2).
  • A076787 (program): Pisumprimes: prime(k), where k is the sum of the first n digits of Pi.
  • A076788 (program): Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).
  • A076789 (program): Phisumprimes: prime(k), where k is the sum of the first n digits of phi-1 and phi is the golden ratio.
  • A076791 (program): Triangle a(n,k) giving number of binary sequences of length n containing k subsequences 00.
  • A076792 (program): Sum_{d divides n} d^2*(-1)^bigomega(d), where bigomega(n) = A001222(n).
  • A076793 (program): a(n) = Sum_{k=1..n} 2^prime(k).
  • A076795 (program): Partial sums of (2n-1)!!.
  • A076816 (program): Squares modulo triangular numbers: n^2 minus the greatest triangular number smaller than or equal to n^2.
  • A076820 (program): Second-largest distinct prime dividing n (or 1 if n is a power of a prime).
  • A076821 (program): Squares of the differences between consecutive primes.
  • A076824 (program): Let a(1)=a(2)=1, a(n)=(2^ceiling(a(n-1)/2)+1)/a(n-2).
  • A076826 (program): a(n) = 2*(Sum_{k=0..n} A010060(k)) - n, where A010060 is a Thue-Morse sequence.
  • A076835 (program): Coefficients in expansion of Eisenstein series -q*E’_2.
  • A076839 (program): A simple example of the Lyness 5-cycle: a(1) = a(2) = 1; a(n) = (a(n-1)+1)/a(n-2) (for n>2).
  • A076840 (program): a(1) = a(2) = 1; a(n) = (a(n-1) + 1)/a(n-2) (for n>2, n odd), (a(n-1)^2 + 1)/a(n-2) (for n>2, n even).
  • A076844 (program): a(1) = a(2) = a(3) = 1; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).
  • A076849 (program): Let u(1)=1, u(n) = n - abs(u(ceiling(n/2)) - u(floor(n/2))); then a(n) = u(n) - n.
  • A076863 (program): n^(n-2)+(n*(n-1)/2)*(n-1)^(n-3).
  • A076871 (program): Sum of two powerful(1) numbers (A001694).
  • A076872 (program): a(n) = number of numbers <= n that are the sum of two squarefull numbers.
  • A076873 (program): Smallest prime not less than sum of first n primes.
  • A076874 (program): n - floor ( ( 4*n + 1 )^(1/2) ).
  • A076877 (program): a(n) = A020330(n) / n.
  • A076878 (program): a(n) = ceiling(n^(1/n))^n - n.
  • A076883 (program): Let u(0)=1, u(n) = 5/2 * floor(u(n-1)); then a(n) = (2/5)*u(n).
  • A076885 (program): Let u(0)=1, u(1)=1 u(n) = u(n-1) + u(n-2) - n*z where z = (5-sqrt(5))/10 =0.27…, then a(n)=floor(u(n)).
  • A076887 (program): Sum of divisors of nonzero palindromic numbers.
  • A076888 (program): a(n) is the number of divisors of the n-th positive palindromic number.
  • A076890 (program): Number of primes up to n-th palindromic number.
  • A076891 (program): [n/1][n/2][n/3] …[n/n] / n^(tau(n)/2).
  • A076895 (program): a(1) = 1, a(n) = n - a(ceiling(n/2)).
  • A076896 (program): a(1) = 1, a(n) = n-a(floor(2n/3)).
  • A076897 (program): a(1)=1, a(n)=n-a(floor(3n/4)).
  • A076902 (program): a(1)=1, a(n) = floor(n/2) - a(floor(n/2)).
  • A076903 (program): Numerator of coefficients of power series for exp(exp(x)-1).
  • A076904 (program): Denominator of coefficients of power series for exp(exp(x)-1).
  • A076918 (program): a(1) = 1, a(n+1) = A076271(n+1) divided by the highest common factor of A076271(n) and A076271(n+1).
  • A076921 (program): Smallest number such that the highest common factor of pair of successive terms follows the pattern 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, …, i.e., b(2n-1) = b(2n) = n given by A004526.
  • A076926 (program): Smallest multiple of n with n distinct prime divisors.
  • A076927 (program): a(n) = A076926(n)/n.
  • A076928 (program): a(1) = 1, a(n+1)= a(n)*(n+1) divided by the largest prime divisor of n+1.
  • A076929 (program): a(1) = 1, a(n+1)= a(n)*(n+1) divided by the smallest prime divisor of n+1.
  • A076930 (program): Smallest k such that n*k is an n-th power.
  • A076933 (program): Final number obtained when n is divided by its divisors starting from the smallest one in increasing order until one no longer gets an integer.
  • A076934 (program): Smallest integer of the form n/k!.
  • A076936 (program): a(1) = 1; then the smallest number different from its predecessor such that the n-th partial product is an n-th power.
  • A076942 (program): Smallest k > 0 such that nk+1 is a square.
  • A076943 (program): Smallest k > 0 such that n*k + 1 is an n-th power.
  • A076944 (program): Least number such that n*k+1 is an n-th power.
  • A076946 (program): Smallest k such that n*(n+1)*(n+2)…*(n+k) >= n^n.
  • A076947 (program): Smallest k > 0 such that nk+1 is a cube.
  • A076951 (program): Smallest k > 0 such that nk-1 is an n-th power, or 0 if no such number exists.
  • A076952 (program): n-th power associated with A076951, or 0 if no such number exists.
  • A076954 (program): a(n) = Product_{i=1..n} prime(i)^i.
  • A076969 (program): a(1) = 1, a(n+1)= smallest cube greater than the n-th partial sum.
  • A076971 (program): a(1) = 1, a(n+1)= smallest triangular number greater than the n-th partial sum.
  • A076972 (program): a(n) = index of the triangular number A076971(n).
  • A076973 (program): Starting with 2, largest prime divisor of the sum of all previous terms.
  • A076978 (program): Product of the distinct primes dividing the product of composite numbers between consecutive primes.
  • A076981 (program): Smallest k such that n*(n+1)*(n+2)*…*(n+k) is divisible by the product of primes up to n.
  • A076982 (program): Number of triangular numbers that divide the n-th triangular number.
  • A076984 (program): Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.
  • A076986 (program): Smallest squarefree number of the form n*k + 1.
  • A076987 (program): Smallest triangular number of the form n*k + 1 with k>0.
  • A076988 (program): Smallest Fibonacci number of the form n*k + 1 with k>0.
  • A076989 (program): Smallest cube of the form n*k + 1 with k>0.
  • A076994 (program): a(1) = 2, a(n+1) is the largest squarefree number < 2a(n).
  • A076995 (program): a(1) = 4, a(n+1) is the largest composite number < 2a(n).
  • A076998 (program): Difference between cubefree and squarefree components of n.
  • A076999 (program): a(1) = 1, a(n+1) is the largest Fibonacci number <= n*a(n).
  • A077000 (program): a(n) = Fibonacci index of A076999(n).
  • A077005 (program): Smallest k such that prime(n) divides k*prime(n-1) + 1, n > 1.
  • A077008 (program): Legendre symbol (-1,p) where p is the n-th prime.
  • A077011 (program): Triangle in which the n-th row contains all possible products of n-1 of the first n primes in ascending order.
  • A077012 (program): Triangle in which n-th row contains all possible products of n-1 of the first n natural numbers in ascending order.
  • A077017 (program): a(1) = 2, a(n+1) = smallest positive integer divisible by the n-th prime that also has a nontrivial common divisor with a(n).
  • A077018 (program): Closest prime to n (break ties by taking the larger prime).
  • A077020 (program): a(n) is the unique odd positive solution x of 2^n = 7x^2+y^2.
  • A077021 (program): a(n) is the unique odd positive solution y of 2^n = 7x^2 + y^2.
  • A077024 (program): a(n) = Sum_{k=1..n} floor(n/k + 1/2).
  • A077025 (program): Sum{Floor(n/(k + 1/2)): k=1,2,…,n}.
  • A077026 (program): a(n) = Sum_{k=1..n} floor(n/k + 1)-floor(n/k + 1/2).
  • A077028 (program): The rascal triangle, read by rows: T(n,k) (n >= 0, 0 <= k <= n) = k(n-k) + 1.
  • A077029 (program): Rectangle R(i,j) read by ascending antidiagonals: column j has j-1 zeros followed by numbers congruent to 1 mod j-1.
  • A077037 (program): Largest prime < n^3.
  • A077038 (program): Least difference of primes p, q such that p < n^3 < q.
  • A077039 (program): Sums of first n signatured primes (A073579).
  • A077043 (program): “Three-quarter squares”: a(n) = n^2 - A002620(n).
  • A077049 (program): Left summatory matrix, T, by antidiagonals upwards.
  • A077050 (program): Left Moebius transformation matrix, M, by antidiagonals.
  • A077051 (program): Right summatory matrix, T, by antidiagonals.
  • A077052 (program): Right Moebius transformation matrix, M, by antidiagonals.
  • A077063 (program): Squarefree kernel of prime(n) - 1.
  • A077064 (program): Squarefree numbers of form prime - 1.
  • A077065 (program): Semiprimes of form prime - 1.
  • A077066 (program): Squarefree kernel of prime(n) + 1.
  • A077067 (program): Squarefree numbers of form prime + 1.
  • A077068 (program): Semiprimes of the form prime + 1.
  • A077070 (program): Triangle read by rows: T(n,k) is the power of 2 in denominator of coefficients of Legendre polynomials, where n >= 0 and 0 <= k <= n.
  • A077071 (program): Row sums of A077070.
  • A077080 (program): a(n) = phi(sigma(n) + phi(n)) = A000010(A000203(n) + A000010(n)) = A000010(A065387(n)).
  • A077086 (program): Remainder when sigma(n+1) is divided by sigma(n).
  • A077088 (program): a(n) = phi(sigma(n) - phi(n)), where phi is Euler’s totient function and sigma is the sum of divisors function, with a(1) = 0.
  • A077099 (program): a(n) = gcd(A051612(n), A065387(n)), where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).
  • A077101 (program): a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).
  • A077106 (program): Largest integer cube <= n^2.
  • A077107 (program): Least integer cube >= n^2.
  • A077109 (program): Duplicate of A070923.
  • A077110 (program): Nearest integer cube to n^2.
  • A077111 (program): a(n) = A077110(n) - n^2.
  • A077112 (program): a(n)=n^2 times nearest cube to n^2.
  • A077113 (program): Number of integer cubes <= n^2.
  • A077115 (program): Least integer square >= n^3.
  • A077116 (program): n^3 - A065733(n).
  • A077118 (program): Nearest integer square to n^3.
  • A077119 (program): a(n) = A077118(n) - n^3.
  • A077120 (program): n^3 times nearest integer square to n^3.
  • A077121 (program): Number of integer squares <= n^3.
  • A077122 (program): Let M_n be the n X n matrix M_(i,j) = 2^i-2^j then the characteristic polynomial of M_n = x^n-a(n)*x^(n-2).
  • A077126 (program): Sum of even-indexed primes.
  • A077128 (program): Smallest number greater than the previous term which is relatively prime to each of the group of the next n numbers.
  • A077129 (program): Smallest number which is relatively prime to all the numbers between successive odd primes.
  • A077131 (program): Sum of odd-indexed primes.
  • A077133 (program): Difference between the sum of odd-indexed primes and even-indexed primes.
  • A077138 (program): a(0) = 0. If n is odd, a(n) = a(n-1) + n, otherwise a(n) = a(n-1) * n.
  • A077139 (program): a(1) = 1, a(n) = lcm(n, a(n-1)) / gcd(n, a(n-1)).
  • A077140 (program): a(1) = 1 and then add n to the previous term if n is coprime to the previous term, otherwise subtract n from the previous term. a(1) = 1 and a(n) = a(n-1) + n if gcd(n, a(n-1)) = 1, otherwise a(n) = a(n-1) - n.
  • A077146 (program): a(n) = floor((concatenation of next (n+1) numbers) / (concatenation of next n numbers)).
  • A077147 (program): Floor[{concatenation 123 … up to n}/n].
  • A077148 (program): Smallest k such that there are n numbers m relatively prime to n in range n < m < k.
  • A077149 (program): a(1) =3. For n>1, a(n) = smallest k such that there are n numbers m not relatively prime to n in range n < m < k.
  • A077150 (program): Number of composite numbers between n and 2n that are coprime to n.
  • A077152 (program): Smallest k such that there are n primes between n and k.
  • A077153 (program): Smallest k such that there are n composite numbers greater than n and smaller than k.
  • A077163 (program): n-th power of next n numbers.
  • A077166 (program): Final terms of rows in A077164.
  • A077169 (program): Initial terms of rows of A077168.
  • A077193 (program): Number of multiples of n that can be obtained by permuting the digits of number formed by the concatenation of first n numbers.
  • A077218 (program): Sum of numbers of prime factors (counted with multiplicities) of numbers between n-th and (n+1)-st prime.
  • A077221 (program): a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.
  • A077234 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077235 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077236 (program): a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.
  • A077237 (program): Combined Diophantine Chebyshev sequences A054491 and A077234.
  • A077238 (program): Combined Diophantine Chebyshev sequences A077236 and A077235.
  • A077239 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077240 (program): Bisection (even part) of Chebyshev sequence with Diophantine property.
  • A077241 (program): Combined Diophantine Chebyshev sequences A054488 and A077413.
  • A077242 (program): Combined Diophantine Chebyshev sequences A077240 and A077239.
  • A077243 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077244 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077245 (program): Bisection (even part) of Chebyshev sequence with Diophantine property.
  • A077246 (program): Bisection (even part) of Chebyshev sequence with Diophantine property.
  • A077247 (program): Combined Diophantine Chebyshev sequences A077245 and A077243.
  • A077248 (program): Combined Diophantine Chebyshev sequences A077246 and A077244.
  • A077249 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077250 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077251 (program): Bisection (even part) of Chebyshev sequence with Diophantine property.
  • A077252 (program): Sum of digits squared minus sum of digits of n.
  • A077253 (program): Sum of digits squared plus sum of digits of n.
  • A077254 (program): Prime(n)^n mod n.
  • A077259 (program): First member of the Diophantine pair (m,k) that satisfies 5*(m^2 + m) = k^2 + k; a(n) = m.
  • A077260 (program): Triangular numbers that are 1/5 of a triangular number.
  • A077261 (program): Triangular numbers that are 5 times another triangular number.
  • A077262 (program): Second member of the Diophantine pair (m,k) that satisfies 5*(m^2 + m) = k^2 + k; a(n) = k.
  • A077263 (program): Number of (undirected) cycles in the n-th order antiprism graph.
  • A077264 (program): Table of arithmetic sequences, by antidiagonals.
  • A077265 (program): Number of cycles in the n-th order prism graph.
  • A077267 (program): Number of zeros in base-3 expansion of n.
  • A077268 (program): Number of bases in which n requires at least one zero to be written.
  • A077272 (program): a(n) = Sum_{d|n} d^2*2^(d-1)*(n/d-1) for n > 0.
  • A077285 (program): Number of partitions of n with designated summands.
  • A077288 (program): First member of the Diophantine pair (m,k) that satisfies 6(m^2 + m) = k^2 + k: a(n) = m.
  • A077289 (program): Triangular numbers that are 1/6 of another triangular number.
  • A077290 (program): Triangular numbers that are 6 times other triangular numbers.
  • A077291 (program): Second member of Diophantine pair (m,k) that satisfies 6*(m^2 + m) = k^2 + k: a(n) = k.
  • A077309 (program): Concatenation of n numbers starting with n.
  • A077317 (program): a(n) is the n-th prime == 1 (mod n).
  • A077318 (program): Sum of terms in n-th row of A077316.
  • A077319 (program): Average of terms in n-th row of A077316.
  • A077320 (program): Triangle in which n-th row contains n smallest multiples of the n-th prime.
  • A077335 (program): Sum of products of squares of parts in all partitions of n.
  • A077337 (program): Numbers n such that n and R(n) both are squarefree where R(n) (A004086) is the digit reversal of n.
  • A077338 (program): a(n) = smallest multiple of prime(n) such that a(n) +1 is a multiple of prime(n+1).
  • A077339 (program): Triangle in which n-th row contains the first n numbers beginning with n.
  • A077340 (program): Final terms of rows in A077339.
  • A077352 (program): a(n) = (concatenation in ascending order of divisors of 2^n)/2^n.
  • A077354 (program): a(n) = Sum_{i=n+1..2n} prime(i) - Sum_{i=1..n} prime(i).
  • A077373 (program): Fibonacci numbers whose external digits as well as internal digits form a Fibonacci number.
  • A077381 (program): Number of squarefree numbers between successive squares (exclusive).
  • A077385 (program): Triangle read by rows in which n-th row contains 2n-1 terms starting from n^0 to n^(n-1) in increasing order and then in decreasing order to n^0.
  • A077386 (program): Sums of rows of triangle in A077385.
  • A077397 (program): Expansion of (1+31*x-2*x^2-2*x^3)/(1-16*x^2+x^4).
  • A077398 (program): First member of the Diophantine pair (m,k) that satisfies 7*(m^2+m) = k^2+k; a(n)=m.
  • A077399 (program): Triangular numbers that are 1/7 of triangular numbers.
  • A077400 (program): Triangular numbers that are 7 times triangular numbers.
  • A077401 (program): Second member of Diophantine pair (m,k) that satisfies 7*(m^2 + m) = k^2 + k; a(n) = k.
  • A077409 (program): Bisection (even part) of Chebyshev sequence with Diophantine property.
  • A077410 (program): Combined Diophantine Chebyshev sequences A077249 and A077251.
  • A077411 (program): Combined Diophantine Chebyshev sequences A077409 and A077250.
  • A077412 (program): Chebyshev U(n,x) polynomial evaluated at x=8.
  • A077413 (program): Bisection (odd part) of Chebyshev sequence with Diophantine property.
  • A077414 (program): a(n) = n*(n - 1)*(n + 2)/2.
  • A077415 (program): a(n) = n*(n+2)*(n-2)/3.
  • A077416 (program): Chebyshev S-sequence with Diophantine property.
  • A077417 (program): Chebyshev T-sequence with Diophantine property.
  • A077418 (program): Number of divisors of Fibonacci(n+2)-1.
  • A077420 (program): Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.
  • A077421 (program): Chebyshev sequence U(n,11)=S(n,22) with Diophantine property.
  • A077422 (program): Chebyshev sequence T(n,11) with Diophantine property.
  • A077423 (program): Chebyshev sequence U(n,12)=S(n,24) with Diophantine property.
  • A077424 (program): Chebyshev sequence T(n,12) with Diophantine property.
  • A077425 (program): a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).
  • A077429 (program): a(n) = floor(log_10(n^2)).
  • A077430 (program): a(n) = floor(log_10(2*n^2)) + 1.
  • A077431 (program): n repeated in decimal representation, but separated by enough zeros that the square has the pattern (n^2)(2n^2)(n^2).
  • A077432 (program): Squares of the form u’v’w, where in decimal representation u=n^2, v=2*n^2 and w=n^2 possibly with a leading zero.
  • A077433 (program): Number of separating zeros to represent A077431.
  • A077436 (program): Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2).
  • A077438 (program): Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.
  • A077442 (program): 2*a(n)^2 + 7 is a square.
  • A077443 (program): Numbers k such that (k^2 - 7)/2 is a square.
  • A077444 (program): Numbers k such that (k^2 + 4)/2 is a square.
  • A077445 (program): Numbers k such that (k^2 - 8)/2 is a square.
  • A077446 (program): Numbers n such that 2*n^2 + 14 is a square.
  • A077447 (program): Numbers n such that (n^2 - 14)/2 is a square.
  • A077448 (program): Numbers k such that Sum_{d|k} mu(d)*mu(k/d)^2 = +1.
  • A077450 (program): Continued fraction expansion of (29+sqrt(145))/12.
  • A077451 (program): Decimal expansion of (29+sqrt(145))/12.
  • A077457 (program): a(n) = sigma_4(n^4)/sigma_2(n^4).
  • A077458 (program): a(n) = (a(n-1)*a(n-2) + a(n-2)*a(n-3) + 1)/a(n-4).
  • A077459 (program): Numbers k such that k and 3*k have the same digital binary sum.
  • A077463 (program): Number of primes p such that n < p < 2n-2.
  • A077465 (program): Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth’s recurrence.
  • A077467 (program): Sum of binary digits of A077465(n).
  • A077468 (program): Greedy powers of (2/3): sum_{n=1..inf} (2/3)^a(n) = 1.
  • A077469 (program): Greedy powers of (3/4): sum_{n=1..inf} (3/4)^a(n) = 1.
  • A077491 (program): a(n) = smallest k such that 2k has digit sum = n.
  • A077500 (program): Primes of the form 2^r*p^s + 1, where p is an odd prime.
  • A077510 (program): Numbers n such that n + pi(n) is a prime.
  • A077528 (program): a(n) = smallest nontrivial (>1) palindrome == 1 (mod n).
  • A077533 (program): Multiples of 3 using only prime digits (2, 3, 5 and 7).
  • A077534 (program): Multiples of 4 using only prime digits (2, 3, 5 and 7).
  • A077535 (program): Multiples of 6 with only prime digits (2, 3, 5 and 7).
  • A077536 (program): Multiples of 7 using only prime digits (2, 3, 5 and 7).
  • A077537 (program): Sum of next F(n) Fibonacci numbers, where F(n) = n-th Fibonacci number.
  • A077538 (program): First differences of triangular numbers with square pyramidal indices.
  • A077539 (program): a(n) = floor(T(n+1)!*T(n-1)!/(T(n)!)^2), where T(n) = n(n+1)/2 = the n-th triangular number.
  • A077543 (program): Smallest n-digit composite palindrome.
  • A077544 (program): Product of next n numbers + sum of next n numbers.
  • A077545 (program): Primes of the form floor(k*e).
  • A077546 (program): Primes of the form floor(n*Pi).
  • A077552 (program): Consider the following triangle in which the n-th row contains n distinct numbers whose product is the smallest and has the least possible number of divisors. 1 is a member of only the first row. Sequence contains the final term of the rows (the leading diagonal).
  • A077568 (program): a(1,n) as defined in A003148.
  • A077573 (program): Smallest number of the form (10^k -1)/9 == 0 (mod prime(n)). with a(1) = a(3) = 0.
  • A077575 (program): a(n) = A077573(n)/prime(n).
  • A077577 (program): a(n) = Floor[A062273(n)/n].
  • A077582 (program): Sum of terms of n-th row of A077581.
  • A077584 (program): Last term of n-th row of A077583.
  • A077587 (program): a(n) = C(n+1) + n*C(n) where C = A000108 (Catalan numbers).
  • A077588 (program): Maximum number of regions into which the plane is divided by n triangles.
  • A077591 (program): Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals.
  • A077595 (program): Numerator of integral from 0 to 1 of (1 + x^2)^n, in lowest terms.
  • A077597 (program): Coefficient of x in the n-th Moebius polynomial (A074586), M(n,x), which satisfies M(n,-1)=mu(n) the Moebius function of n.
  • A077605 (program): Left summing matrix, S.
  • A077606 (program): Left differencing matrix, D, by antidiagonals.
  • A077607 (program): Convolutory inverse of the factorial sequence.
  • A077608 (program): Number of compositions of n into twin primes (i.e., primes that are members of a twin prime pair, like 3, 5, 7, 11, 13, but not 2 or 23).
  • A077611 (program): Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,…,n}.
  • A077612 (program): Number of adjacent pairs of form (even,even) among all permutations of {1,2,…,n}.
  • A077613 (program): Number of adjacent pairs of form (even,odd) among all permutations of {1,2,…,n}. Also, number of adjacent pairs of form (odd,even).
  • A077616 (program): Binomial transform of n^2*2^n/2.
  • A077623 (program): a(1)=1, a(2)=2, a(3)=4, a(n) = abs(a(n-1)-a(n-2)-a(n-3)).
  • A077624 (program): Largest term in periodic part of continued fraction expansion of square root of 2^n + 1 or 0 if 2^n + 1 is a square.
  • A077625 (program): Largest term in periodic part of continued fraction expansion of square root of -1+2^n or 0 if -1+2^n is square.
  • A077626 (program): Largest term in periodic part of continued fraction expansion of square root of 1+3^n or 0 if 1+3^n is square.
  • A077627 (program): Largest term in periodic part of continued fraction expansion of square root of -1+3^n.
  • A077637 (program): Largest term in periodic part of continued fraction expansion of square root of A051451(n), i.e., sqrt(lcm(1..x)) where x is a prime power from A000961.
  • A077641 (program): Number of squarefree integers in closed interval [n, 2n-1], i.e., among n consecutive numbers beginning with n.
  • A077648 (program): Initial digits of prime numbers.
  • A077649 (program): Initial digit of composite numbers.
  • A077650 (program): Initial decimal digit of sigma(n), the sum of divisors of n.
  • A077651 (program): Initial digit of phi(n), where phi is Euler totient function, A000010.
  • A077653 (program): a(1)=1, a(2)=2, a(3)=2, a(n) = abs(a(n-1)-a(n-2)-a(n-3)).
  • A077654 (program): Composites k such that 2k+1 is also composite.
  • A077655 (program): Number of consecutive successors of n having the same number of prime factors as n (counted with multiplicity).
  • A077656 (program): Numbers having a different number of prime factors as their successors (counted with multiplicity).
  • A077659 (program): a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists.
  • A077660 (program): Sum of terms of n-th row of A077583.
  • A077662 (program): a(n) = n-th positive integer not relatively prime to n, with a(1)=1.
  • A077665 (program): Final term of n-th row of A077664.
  • A077666 (program): Sum of terms of n-th row of A077664.
  • A077677 (program): Squarefree numbers beginning with 1.
  • A077678 (program): Squarefree numbers beginning with 2.
  • A077679 (program): Squarefree numbers beginning with 3.
  • A077680 (program): Squarefree numbers beginning with 4.
  • A077681 (program): Squarefree numbers beginning with 5.
  • A077682 (program): Squarefree numbers beginning with 6.
  • A077683 (program): Squarefree numbers beginning with 7.
  • A077684 (program): Squarefree numbers with 8 as their initial (leftmost) digit.
  • A077685 (program): Squarefree numbers beginning with 9.
  • A077686 (program): 2^(n-1) - (prime(n) mod n).
  • A077694 (program): a(n) = triangular number pertaining to the number f(n) obtained by concatenating first n natural numbers.
  • A077695 (program): a(n) = triangular number pertaining to the number f(n) obtained by concatenating n n times.
  • A077717 (program): Primes which can be expressed as sum of distinct powers of 3.
  • A077718 (program): Primes which can be expressed as sum of distinct powers of 4.
  • A077719 (program): Primes which can be expressed as sum of distinct powers of 5.
  • A077720 (program): Primes which can be expressed as sum of distinct powers of 6.
  • A077725 (program): a(n) = smallest square > 1 which can be expressed as a sum of distinct powers of n.
  • A077726 (program): Smallest number beginning with n and having a digit sum n.
  • A077745 (program): Numerator of integral_{x=1..2} (x^2-1)^n dx.
  • A077750 (program): Least significant digit of A077749(n).
  • A077753 (program): a(1) = 1, a(2) = 2, a(2n) = a(2n-1)*a(2n-2), a(2n+1)= a(2n-1) + a(2n).
  • A077766 (program): Number of primes of form 4k+1 between n^2 and (n+1)^2.
  • A077767 (program): Number of primes of form 4k+3 between n^2 and (n+1)^2.
  • A077768 (program): Number of times that the sum of two squares is an integer between n^2 and (n+1)^2; multiple representations are counted multiply.
  • A077770 (program): Number of ordered pairs of integers (x,y) with n^2 < x^2 + y^2 < (n+1)^2; number of lattice points between circles of radii n and n+1.
  • A077800 (program): List of twin primes {p, p+2}.
  • A077802 (program): Sum of products of parts increased by 1 in hook partitions of n, where hook partitions are of the form h*1^(n-h).
  • A077804 (program): Deficient oblong numbers.
  • A077805 (program): Smallest prime factor of numbers containing in their decimal representation only the digits 0 and 1.
  • A077806 (program): Greatest prime factor of numbers containing in their decimal representation only the digits 0 and 1.
  • A077807 (program): Number of distinct prime factors of numbers containing in their decimal representation only the digits 0 and 1.
  • A077808 (program): Number of prime factors of numbers containing in their decimal representation only the digits 0 and 1 (counted with multiplicity).
  • A077809 (program): Number of divisors of numbers containing in their decimal representation only the digits 0 and 1.
  • A077810 (program): Sum of divisors of numbers containing in their decimal representation only the digits 0 and 1.
  • A077811 (program): Euler’s totient of numbers containing in their decimal representation only the digits 0 and 1.
  • A077812 (program): Squarefree kernel of numbers containing in their decimal representation only the digits 0 and 1.
  • A077814 (program): a(n) = #{0<=k<=n: mod(kn,4)=2}.
  • A077815 (program): 2^phi(n) mod n^2, where phi is Euler’s totient function A000010.
  • A077821 (program): Expansion of (1-x)^(-1)/(1-3*x-3*x^2-3*x^3).
  • A077822 (program): Expansion of (1-x)^(-1)/(1-3*x-3*x^2-2*x^3).
  • A077823 (program): Expansion of (1-x)^(-1)/(1-3*x-2*x^2-3*x^3).
  • A077824 (program): Expansion of (1-x)^(-1)/(1-3*x-2*x^2-2*x^3).
  • A077825 (program): Expansion of (1-x)^(-1)/(1-2*x-3*x^2-3*x^3).
  • A077826 (program): Expansion of (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).
  • A077827 (program): Expansion of (1-x)^(-1)/(1-2*x-2*x^2-2*x^3).
  • A077828 (program): Expansion of 1/(1-3*x-3*x^2-3*x^3).
  • A077829 (program): Expansion of 1/(1-3*x-3*x^2-2*x^3).
  • A077830 (program): Expansion of 1/(1-3*x-2*x^2-3*x^3).
  • A077831 (program): Expansion of 1/(1-3*x-2*x^2-2*x^3).
  • A077832 (program): Expansion of 1/(1-2*x-3*x^2-3*x^3).
  • A077833 (program): Expansion of 1/(1-2*x-3*x^2-2*x^3).
  • A077834 (program): Expansion of 1/(1 - 2*x - 2*x^2 - 3*x^3).
  • A077835 (program): Expansion of 1/(1-2*x-2*x^2-2*x^3).
  • A077836 (program): Expansion of (1-x)/(1-3*x-3*x^2-3*x^3).
  • A077837 (program): Expansion of (1-x)/(1-3*x-3*x^2-2*x^3).
  • A077838 (program): Expansion of (1-x)/(1-3*x-2*x^2-3*x^3).
  • A077839 (program): Expansion of (1 - x)/(1 - 3*x - 2*x^2 - 2*x^3).
  • A077840 (program): Expansion of (1-x)/(1-2*x-3*x^2-3*x^3).
  • A077841 (program): Expansion of (1-x)/(1-2*x-3*x^2-2*x^3).
  • A077842 (program): Expansion of (1-x)/(1-2*x-2*x^2-3*x^3).
  • A077843 (program): Expansion of (1-x)/(1-2*x-2*x^2-2*x^3).
  • A077845 (program): Expansion of (1-x)^(-1)/(1-2*x-2*x^2-x^3).
  • A077846 (program): Expansion of 1/(1 - 3*x + 2*x^3).
  • A077847 (program): Expansion of (1-x)^(-1)/(1-2*x-2*x^2+2*x^3).
  • A077848 (program): Expansion of (1-x)^(-1)/(1-2*x-x^2-2*x^3).
  • A077849 (program): Expansion of (1-x)^(-1)/(1 - 2*x - x^2 - x^3).
  • A077850 (program): Expansion of (1-x)^(-1)/(1 - 2*x - x^2 + x^3).
  • A077851 (program): Expansion of (1-x)^(-1)/(1 - 2*x - 2*x^3).
  • A077852 (program): Expansion of (1-x)^(-1)/(1-2*x-x^3).
  • A077853 (program): Expansion of (1-x)^(-1)/(1-2*x+2*x^3).
  • A077854 (program): Expansion of 1/((1-x)*(1-2*x)*(1+x^2)).
  • A077855 (program): Expansion of (1-x)^(-1)/(1 - 2*x + x^2 - x^3).
  • A077856 (program): Expansion of (1-x)^(-1)/(1-2*x+x^2+x^3).
  • A077857 (program): Expansion of (1-x)^(-1)/(1-2*x+x^2+2*x^3).
  • A077858 (program): Expansion of (1-x)^(-1)/(1-2*x+2*x^2-2*x^3).
  • A077859 (program): Expansion of (1-x)^(-1)/(1-2*x+2*x^2-x^3).
  • A077860 (program): Expansion of 1/((1 - 2*x + 2*x^2)*(1-x)).
  • A077861 (program): Expansion of (1-x)^(-1)/(1-2*x+2*x^2+x^3).
  • A077862 (program): Expansion of (1-x)^(-1)/(1-2*x+2*x^2+2*x^3).
  • A077863 (program): Expansion of (1-x)^(-1)/(1-x-2*x^2-2*x^3).
  • A077864 (program): Expansion of (1-x)^(-1)/(1-x-2*x^2-x^3).
  • A077865 (program): Expansion of (1-x)^(-1)/(1-x-2*x^2+x^3).
  • A077866 (program): Expansion of (1-x)^(-1)/(1 - x - 2*x^2 + 2*x^3).
  • A077867 (program): Expansion of (1-x)^(-1)/(1-x-x^2+2*x^3).
  • A077868 (program): Expansion of 1/(1-x)*(1-x-x^3)).
  • A077869 (program): Expansion of (1-x)^(-1)/(1-x+x^3).
  • A077870 (program): Expansion of (1-x)^(-1)/(1-x+2*x^3).
  • A077871 (program): Expansion of (1-x)^(-1)/(1-x+x^2-2*x^3).
  • A077872 (program): Expansion of 1 / ((1-x)*(1-x+x^2+x^3)).
  • A077873 (program): Expansion of (1-x)^(-1)/(1-x+x^2+2*x^3).
  • A077874 (program): Expansion of (1-x)^(-1)/(1-x+2*x^2-2*x^3).
  • A077875 (program): Expansion of (1-x)^(-1)/(1-x+2*x^2-x^3).
  • A077876 (program): Expansion of (1-x)^(-1)/(1-x+2*x^2).
  • A077877 (program): Expansion of (1-x)^(-1)/(1-x+2*x^2+x^3).
  • A077878 (program): Expansion of (1-x)^(-1)/(1-x+2*x^2+2*x^3).
  • A077879 (program): Expansion of (1-x)^(-1)/(1-2*x^2-2*x^3).
  • A077880 (program): Expansion of (1-x)^(-1)/(1-2*x^2+x^3).
  • A077881 (program): Expansion of (1-x)^(-1)/(1-2*x^2+2*x^3).
  • A077882 (program): Expansion of x/((1-x)*(1-x^2-2*x^3)).
  • A077883 (program): Expansion of (1-x)^(-1)/(1-x^2+x^3).
  • A077884 (program): Expansion of (1-x)^(-1)/(1-x^2+2*x^3).
  • A077885 (program): Expansion of (1-x)^(-1)/(1-2*x^3).
  • A077886 (program): Expansion of (1-x)^(-1)/(1+2*x^3).
  • A077887 (program): Expansion of (1-x)^(-1)/(1+x^2-2*x^3).
  • A077888 (program): Expansion of (1-x)^(-1)/(1+x^2-x^3).
  • A077889 (program): Expansion of 1/( (1-x)*(1 + x^2 + x^3) ).
  • A077890 (program): Expansion of (1-x)^(-1)/(1+x^2+2*x^3).
  • A077891 (program): Expansion of (1-x)^(-1)/(1+2*x^2-2*x^3).
  • A077892 (program): Expansion of (1-x)^(-1)/(1+2*x^2-x^3).
  • A077894 (program): Expansion of (1-x)^(-1)/(1+2*x^2+x^3).
  • A077895 (program): Expansion of (1-x)^(-1)/(1+2*x^2+2*x^3).
  • A077896 (program): Expansion of (1-x)^(-1)/(1+x-2*x^2-2*x^3).
  • A077897 (program): Expansion of (1-x)^(-1)/(1+x-2*x^2-x^3).
  • A077898 (program): Expansion of (1 - x)^(-1)/(1 + x - 2*x^2).
  • A077899 (program): Expansion of (1-x)^(-1)/(1+x-2*x^2+x^3).
  • A077900 (program): Expansion of (1-x)^(-1)/(1+x-2*x^2+2*x^3).
  • A077901 (program): Expansion of (1-x)^(-1)/(1+x-x^2-2*x^3).
  • A077902 (program): Expansion of (1-x)^(-1)/(1 + x - x^2 + x^3).
  • A077903 (program): Expansion of (1-x)^(-1)/(1 + x - x^2 + 2*x^3).
  • A077904 (program): Expansion of (1-x)^(-1)/(1+x-2*x^3).
  • A077905 (program): Expansion of 1/(1 - x^2 - x^3 + x^4).
  • A077906 (program): Expansion of (1-x)^(-1)/(1+x+2*x^3).
  • A077907 (program): Expansion of (1-x)^(-1)/(1+x+x^2-2*x^3).
  • A077908 (program): Expansion of (1-x)^(-1)/(1+x+x^2-x^3).
  • A077909 (program): Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).
  • A077910 (program): Expansion of 1/((1-x)*(1+x+2*x^2-2*x^3)).
  • A077911 (program): Expansion of 1/((1-x)*(1+x+2*x^2-x^3)).
  • A077912 (program): Expansion of 1/(1+x^2-2*x^3).
  • A077913 (program): Expansion of 1/((1-x)*(1+x+2*x^2+x^3)).
  • A077915 (program): Expansion of 1/((1-x)*(1+2*x-2*x^2-2*x^3)).
  • A077916 (program): Expansion of (1-x)^(-1)/(1 + 2*x - 2*x^2 - x^3).
  • A077917 (program): Expansion of (1-x)^(-1)/(1+2*x-2*x^2).
  • A077918 (program): Expansion of (1 - x)^(-1)/(1 + 2*x - 2*x^2 + x^3).
  • A077919 (program): Expansion of (1-x)^(-1)/(1+2*x-2*x^2+2*x^3).
  • A077920 (program): Expansion of (1-x)^(-1)/(1+2*x-x^2-x^3).
  • A077921 (program): Expansion of (1-x)^(-1)/(1+2*x-x^2).
  • A077922 (program): Expansion of (1-x)^(-1)/(1+2*x-x^2+x^3).
  • A077923 (program): Expansion of (1-x)^(-1)/(1+2*x-x^2+2*x^3).
  • A077924 (program): Expansion of (1-x)^(-1)/(1+2*x-2*x^3).
  • A077925 (program): Expansion of 1/((1-x)*(1+2*x)).
  • A077926 (program): Expansion of (1-x)^(-1)/(1+2*x+x^3).
  • A077927 (program): Expansion of (1-x)^(-1)/(1+2*x+2*x^3).
  • A077928 (program): Expansion of (1-x)^(-1)/(1+2*x+x^2-2*x^3).
  • A077929 (program): Expansion of (1-x)^(-1)/(1+2*x+x^2-x^3).
  • A077930 (program): Expansion of (1-x)^(-1)/(1+2*x+x^2+x^3).
  • A077931 (program): Expansion of 1/((1-x)*(1+2*x+x^2+2*x^3)).
  • A077932 (program): Expansion of (1-x)^(-1)/(1+2*x+2*x^2-2*x^3).
  • A077933 (program): Expansion of (1-x)^(-1)/(1+2*x+2*x^2-x^3).
  • A077936 (program): Expansion of 1/(1 - 2*x - 2*x^2 - x^3).
  • A077937 (program): Expansion of 1/(1-2*x-2*x^2+2*x^3).
  • A077938 (program): Expansion of 1/(1-2*x-x^2-2*x^3).
  • A077939 (program): Expansion of 1/(1 - 2*x - x^2 - x^3).
  • A077940 (program): Expansion of 1/(1-2*x+2*x^3).
  • A077941 (program): Expansion of 1/(1-2*x+x^2+x^3).
  • A077942 (program): Expansion of 1/(1-2*x+x^2+2*x^3).
  • A077943 (program): Expansion of 1/(1-2*x+2*x^2-2*x^3).
  • A077944 (program): Expansion of 1/(1-2*x+2*x^2+x^3).
  • A077945 (program): Expansion of 1/(1-2*x+2*x^2+2*x^3).
  • A077946 (program): Expansion of 1/(1 - x - 2*x^2 - 2*x^3).
  • A077947 (program): Expansion of 1/(1 - x - x^2 - 2*x^3).
  • A077948 (program): Expansion of 1/(1-x-x^2+2*x^3).
  • A077949 (program): Expansion of 1/(1-x-2*x^3).
  • A077950 (program): Expansion of 1/(1-x+2*x^3).
  • A077951 (program): Expansion of 1/(1-x+x^2-2*x^3).
  • A077952 (program): Expansion of 1/(1 - x + x^2 + 2*x^3).
  • A077953 (program): Expansion of 1/(1-x+2*x^2-2*x^3).
  • A077954 (program): Expansion of 1/(1-x+2*x^2-x^3) in powers of x.
  • A077955 (program): Expansion of 1/(1-x+2*x^2+x^3).
  • A077956 (program): Expansion of 1/(1-x+2*x^2+2*x^3).
  • A077957 (program): Powers of 2 alternating with zeros.
  • A077958 (program): Expansion of 1/(1-2*x^3).
  • A077959 (program): Expansion of 1/(1+2*x^3).
  • A077961 (program): Expansion of 1 / (1 + x^2 - x^3) in powers of x.
  • A077962 (program): Expansion of 1/(1+x^2+x^3).
  • A077963 (program): Expansion of 1/(1+x^2+2*x^3).
  • A077964 (program): Expansion of 1/(1+2*x^2-2*x^3).
  • A077965 (program): Expansion of 1/(1+2*x^2-x^3).
  • A077966 (program): Expansion of 1/(1+2*x^2).
  • A077967 (program): Expansion of 1/(1+2*x^2+x^3).
  • A077968 (program): Expansion of 1/(1+2*x^2+2*x^3).
  • A077969 (program): Numbers which can be expressed as the sum of two distinct primes in exactly three ways.
  • A077970 (program): Expansion of 1/(1+x-2*x^2+2*x^3).
  • A077971 (program): Expansion of 1/(1+x-x^2-2*x^3).
  • A077972 (program): Expansion of 1/(1+x-x^2+2*x^3).
  • A077973 (program): Expansion of 1/(1+x-2*x^3).
  • A077974 (program): Expansion of 1/(1+x+2*x^3).
  • A077975 (program): Expansion of 1/(1+x+x^2-2*x^3).
  • A077976 (program): Expansion of 1/(1+x+x^2+2*x^3).
  • A077977 (program): Expansion of 1/(1+x+2*x^2-2*x^3).
  • A077978 (program): Expansion of 1/(1+x+2*x^2-x^3).
  • A077979 (program): Expansion of 1/(1+x+2*x^2+x^3).
  • A077980 (program): Expansion of 1/(1 + x + 2*x^2 + 2*x^3).
  • A077981 (program): Expansion of 1/(1+2*x-2*x^2-2*x^3).
  • A077983 (program): Expansion of 1/(1 + 2*x - 2*x^2 + x^3).
  • A077984 (program): Expansion of 1/(1+2*x-2*x^2+2*x^3).
  • A077985 (program): Expansion of 1/(1 + 2*x - x^2).
  • A077986 (program): Expansion of 1/(1 + 2*x - x^2 + x^3).
  • A077987 (program): Expansion of 1/(1+2*x-x^2+2*x^3).
  • A077988 (program): Expansion of 1/(1+2*x-2*x^3).
  • A077989 (program): Expansion of 1/(1+2*x+x^2-2*x^3).
  • A077990 (program): Expansion of 1/(1+2*x+x^2-x^3).
  • A077991 (program): Expansion of 1/(1+2*x+2*x^2-2*x^3).
  • A077992 (program): Expansion of 1/(1+2*x+2*x^2-x^3).
  • A077993 (program): Expansion of 1/(1+2*x+2*x^2+2*x^3).
  • A077995 (program): Expansion of (1 - x)/(1 - 2*x - 2*x^2 - x^3).
  • A077996 (program): Expansion of (1-x)/(1-2*x-x^2-2*x^3).
  • A077997 (program): Expansion of (1-x)/(1-2*x-x^2-x^3).
  • A077998 (program): Expansion of (1-x)/(1-2*x-x^2+x^3).
  • A077999 (program): Expansion of (1-x)/(1-2*x-2*x^3).
  • A078001 (program): Expansion of (1-x)/(1-2*x+x^2+x^3).
  • A078002 (program): Expansion of (1-x)/(1-2*x+x^2+2*x^3).
  • A078003 (program): Expansion of (1-x)/(1-2*x+2*x^2-2*x^3).
  • A078004 (program): Expansion of (1-x)/(1-2*x+2*x^2+x^3).
  • A078005 (program): Expansion of (1-x)/(1-2*x+2*x^2+2*x^3).
  • A078006 (program): Expansion of (1-x)/(1-x-2*x^2-2*x^3).
  • A078007 (program): Expansion of (1-x)/(1-x-2*x^2-x^3).
  • A078008 (program): Expansion of (1-x)/( (1+x)*(1-2*x) ).
  • A078009 (program): a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^k*N(n,k) where N(n,k) = C(n,k)*C(n,k+1)/n are the Narayana numbers (A001263).
  • A078010 (program): Expansion of (1-x)/(1 - x - x^2 - 2*x^3).
  • A078011 (program): Expansion of (1-x)/(1-x-x^2+2*x^3).
  • A078012 (program): Expansion of (1 - x) / (1 - x - x^3) in powers of x.
  • A078013 (program): Expansion of (1-x)/(1-x+x^3).
  • A078014 (program): Expansion of (1-x)/(1-x+2*x^3).
  • A078015 (program): Expansion of (1-x)/(1-x+x^2-2*x^3).
  • A078016 (program): Expansion of (1-x)/(1-x+x^2+x^3).
  • A078017 (program): Expansion of (1-x)/(1-x+x^2+2*x^3).
  • A078018 (program): a(n) = Sum_{k=0..n} 6^k*N(n,k), with a(0)=1, where N(n,k) = C(n,k) * C(n,k+1)/n are the Narayana numbers (A001263).
  • A078019 (program): Expansion of (1-x)/(1-x+2*x^2-x^3).
  • A078020 (program): Expansion of (1-x)/(1-x+2*x^2).
  • A078021 (program): Expansion of (1-x)/(1-x+2*x^2+x^3).
  • A078022 (program): Expansion of (1-x)/(1-x+2*x^2+2*x^3).
  • A078023 (program): Expansion of (1-x)/(1-2*x^2-2*x^3).
  • A078024 (program): Expansion of (1-x)/(1-2*x^2-x^3).
  • A078025 (program): Expansion of (1-x)/(1-2*x^2+2*x^3).
  • A078026 (program): Expansion of (1-x)/(1-x^2-2*x^3).
  • A078027 (program): Expansion of (1 - x)/(1 - x^2 - x^3).
  • A078028 (program): Expansion of (1-x)/(1-x^2+2*x^3).
  • A078029 (program): Expansion of (1-x)/(1-2*x^3).
  • A078030 (program): Expansion of (1-x)/(1+2*x^3).
  • A078031 (program): Expansion of (1-x)/(1 + x^2 - x^3).
  • A078032 (program): Expansion of (1-x)/(1+x^2+x^3).
  • A078033 (program): Expansion of (1-x) / (1+x^2+2*x^3).
  • A078034 (program): Expansion of (1-x)/(1+2*x^2-2*x^3).
  • A078035 (program): Expansion of (1-x)/(1+2*x^2-x^3).
  • A078036 (program): Expansion of (1-x)/(1+2*x^2+x^3).
  • A078037 (program): Expansion of (1-x)/(1+2*x^2+2*x^3).
  • A078038 (program): Expansion of (1-x)/(1+x-2*x^2-x^3).
  • A078039 (program): Expansion of (1-x)/(1+x-2*x^2+x^3).
  • A078040 (program): Expansion of (1-x)/(1+x-2*x^2+2*x^3).
  • A078041 (program): Expansion of (1-x)/(1+x-x^2-2*x^3).
  • A078042 (program): Expansion of (1-x)/(1+x-x^2+x^3).
  • A078043 (program): Expansion of (1 - x)/(1 + x - x^2 + 2*x^3).
  • A078044 (program): Expansion of (1-x)/(1+x+2*x^3).
  • A078045 (program): Expansion of (1-x)/(1+x+x^2-2*x^3).
  • A078046 (program): Expansion of (1-x)/(1 + x + x^2 - x^3).
  • A078047 (program): Expansion of (1-x)/(1+x+x^2+2*x^3).
  • A078048 (program): Expansion of (1-x)/(1+x+2*x^2-2*x^3).
  • A078049 (program): Expansion of (1-x)/(1+x+2*x^2-x^3).
  • A078050 (program): Expansion of (1-x)/(1+x+2*x^2).
  • A078051 (program): Expansion of (1-x)/(1+x+2*x^2+x^3).
  • A078052 (program): Expansion of (1-x)/(1+x+2*x^2+2*x^3).
  • A078053 (program): Expansion of (1-x)/(1+2*x-2*x^2-2*x^3).
  • A078054 (program): Expansion of (1-x)/(1+2*x-2*x^2+x^3).
  • A078055 (program): Expansion of (1-x)/(1+2*x-2*x^2+2*x^3).
  • A078056 (program): Expansion of (1-x)/(1+2*x-x^2-x^3).
  • A078057 (program): Expansion of (1+x)/(1-2*x-x^2).
  • A078058 (program): Expansion of (1-x)/(1+2*x-x^2+x^3).
  • A078059 (program): Expansion of (1-x)/(1+2*x-x^2+2*x^3).
  • A078060 (program): Expansion of (1-x)/(1+2*x-2*x^3).
  • A078061 (program): Expansion of (1-x)/(1+2*x+x^3).
  • A078062 (program): Expansion of (1-x)/(1+2*x+2*x^3).
  • A078063 (program): Expansion of (1-x)/(1+2*x+x^2-2*x^3).
  • A078064 (program): Expansion of (1-x)/(1+2*x+x^2-x^3).
  • A078065 (program): Expansion of (1-x)/(1+2*x+x^2+x^3).
  • A078066 (program): Expansion of (1-x)/(1+2*x+x^2+2*x^3).
  • A078067 (program): Expansion of (1-x)/(1+2*x+2*x^2-2*x^3).
  • A078068 (program): Expansion of (1-x)/(1+2*x+2*x^2-x^3).
  • A078069 (program): Expansion of (1-x)/(1+2*x+2*x^2).
  • A078070 (program): Expansion of (1-x)/(1 + 2*x + 2*x^2 + x^3).
  • A078071 (program): Expansion of (1-x)/(1+2*x+2*x^2+2*x^3).
  • A078100 (program): 1/6 of the number of ways of 3-coloring a 4 X n grid.
  • A078107 (program): Numbers n such that it is not possible to arrange the numbers from 1 to n in a chain with adjacent links summing to a square.
  • A078111 (program): a(n) = floor((n+2)^(n+2)/n^n).
  • A078112 (program): Coefficients a(n) in the unique expansion sin(1) = Sum[a(n)/n!, n>=1], where a(n) satisfies 0<=a(n)<n.
  • A078113 (program): Let u(1)=u(2)=1, u(3)=n, u(k) = (1/2)*abs(2*u(k-1) -u(k-2)-u(k-3)); sequence gives values of n such that sum(k>=1, u(k)) is an integer.
  • A078119 (program): sigma(n) - phi(n+2), where sigma = sum of divisors (A000203) and phi = Euler totient function (A000010).
  • A078126 (program): Negative determinant of n X n matrix M_{i,j}=1 if i=j or i+j=1 (mod 2).
  • A078128 (program): Number of ways to write n as sum of cubes>1.
  • A078134 (program): Number of ways to write n as sum of squares > 1.
  • A078135 (program): Numbers which cannot be written as a sum of squares > 1.
  • A078136 (program): Numbers having exactly one representation as sum of squares>1.
  • A078137 (program): Numbers which can be written as sum of squares>1.
  • A078138 (program): Primes which can be written as sum of squares > 1.
  • A078140 (program): Convolutory inverse of signed lower Wythoff sequence.
  • A078147 (program): First differences of sequence of nonsquarefree numbers, A013929.
  • A078152 (program): a(n) = A055086(n) - A000005(n).
  • A078153 (program): a(n) = A051201(n) - A000203(n).
  • A078157 (program): A078152(2^n).
  • A078159 (program): a(n) = A055086(2^n).
  • A078160 (program): a(n) = A055086(n!).
  • A078163 (program): a(n)=A051201[n^2].
  • A078171 (program): a(n)=A055086[A000040(n)].
  • A078172 (program): a(n)=A051201[A000040(n)].
  • A078174 (program): Numbers with an integer arithmetic mean of distinct prime factors.
  • A078175 (program): Numbers with an integer arithmetic mean of all prime factors.
  • A078177 (program): Composite numbers with an integer arithmetic mean of all prime factors.
  • A078181 (program): a(n) = Sum_{d|n, d==1(mod 3)} d.
  • A078182 (program): a(n) = Sum_{d|n, d=2 mod 3} d.
  • A078191 (program): a(n) = concatenation of n n times divided by n.
  • A078192 (program): a(n) = floor(concatenation of n down to 1 divided by n).
  • A078193 (program): In the following triangle the n-th row contains n n-digit (or (n-1)-digit) numbers whose concatenation (with a 0 prefixed for (n-1)-digit numbers) gives a substring of the cyclic concatenation of 1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,…: 1; 12 34; 123 456 789; 1234 5678 9012 3456; 12345 67890 12345 67890 12345; … Sequence contains the final terms of rows.
  • A078222 (program): a(1) = 2, a(n+1) > a(n) is the smallest multiple of a(n) using only even digits.
  • A078258 (program): a(n) = numerator(N), where N = 0.123…n (concatenation of 1 to n after decimal point).
  • A078261 (program): Smallest integer multiple of the decimal number N = 0.246…up to 2n (decimal point followed by concatenation of 2 through 2n of first n even numbers).
  • A078262 (program): Sum of the forward and reverse concatenations of 1 to n.
  • A078263 (program): Product of the forward and reverse concatenations of 1 to n.
  • A078264 (program): Integer part of the geometric mean of the forward and reverse concatenations of 1 to n.
  • A078267 (program): Smallest k such that k*N is an integer where N is obtained by placing the string “n” after a decimal point.
  • A078268 (program): Smallest integer which is an integer multiple of the number N obtained by placing the string “n” after a decimal point.
  • A078285 (program): Least nontrivial multiple of the n-th prime beginning with 1.
  • A078287 (program): Least nontrivial multiple of the n-th prime beginning with 3.
  • A078299 (program): Numbers which can be expressed as the sum of two distinct primes in exactly four ways.
  • A078306 (program): a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2.
  • A078307 (program): a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3.
  • A078308 (program): a(n) = Sum_{d divides n} d^(n/d + 1).
  • A078309 (program): Numbers that are congruent to {1, 4, 7} mod 10.
  • A078310 (program): a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).
  • A078311 (program): Smallest prime factor of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078312 (program): Greatest prime factor of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078313 (program): Number of distinct prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078314 (program): Total number of prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078316 (program): Maximal exponent in prime factorization of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078317 (program): Number of divisors of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078318 (program): Sum of divisors of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078319 (program): Sum of distinct prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078320 (program): Sum of all prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078321 (program): Euler’s totient of n*rad(n)+1, where rad=A007947 (squarefree kernel).
  • A078322 (program): a(n) = rad(n*rad(n)+1), where rad = A007947 (squarefree kernel).
  • A078323 (program): Arithmetic derivative of n*rad(n)+1, where rad = A007947 (squarefree kernel).
  • A078330 (program): Primes p such that mu(p-1) = -1, where mu is the Moebius function; that is, p-1 is squarefree and has an odd number of prime factors.
  • A078338 (program): Let u(1)=u(2)=u(3)=1 and u(n)=(-1)^n*sign(u(n-1)-u(n-2))*u(n-3), then a(n)=sum(k=1,n,u(k)).
  • A078339 (program): Let u(1)=u(2)=u(3)=1 and u(n)=(-1)^n*sign(u(n-1)-u(n-2))*u(n-3); then a(n)=sum(k=1,n,sum(i=1,k,u(i)) - 3*(n-1).
  • A078342 (program): Number of positive integers less than n that are coprime to all primes less than or equal to the square root of n.
  • A078343 (program): a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
  • A078344 (program): a(1)=1; a(2)=2; a(3)=3; a(n) = sum(k=3,n-1,a(k) + a(k-1) + a(k-2) ).
  • A078346 (program): a(1) = 1; a(n) = Sum_{k=1..n-1} a(floor((n-1)/k)).
  • A078349 (program): Number of primes in sequence h(m) defined by h(1) = n, h(m+1) = Floor(h(m)/2).
  • A078358 (program): Non-oblong numbers: Complement of A002378.
  • A078359 (program): Number of ways to write n as sum of a positive square and a positive cube.
  • A078360 (program): Numbers having a unique representation as sum of a positive square and a positive cube.
  • A078362 (program): A Chebyshev S-sequence with Diophantine property.
  • A078363 (program): A Chebyshev T-sequence with Diophantine property.
  • A078364 (program): A Chebyshev S-sequence with Diophantine property.
  • A078365 (program): A Chebyshev T-sequence with Diophantine property.
  • A078366 (program): A Chebyshev S-sequence with Diophantine property.
  • A078367 (program): A Chebyshev T-sequence with Diophantine property.
  • A078368 (program): A Chebyshev S-sequence with Diophantine property.
  • A078369 (program): A Chebyshev T-sequence with Diophantine property.
  • A078370 (program): a(n) = 4*(n+1)*n + 5.
  • A078371 (program): a(n) = (2*n+5)*(2*n+1).
  • A078375 (program): Smallest prime factor of numbers which can be written as sum of a positive square and a positive cube.
  • A078376 (program): Greatest prime factor of numbers which can be written as sum of a positive square and a positive cube.
  • A078377 (program): Number of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.
  • A078378 (program): Total number of prime factors of numbers which can be written as sum of a positive square and a positive cube.
  • A078380 (program): Maximal exponent in prime factorization of numbers which can be written as sum of a positive square and a positive cube.
  • A078381 (program): Number of divisors of numbers which can be written as sum of a positive square and a positive cube.
  • A078382 (program): Sum of divisors of numbers which can be written as sum of a positive square and a positive cube.
  • A078383 (program): Sum of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.
  • A078384 (program): Sum of all prime factors of numbers which can be written as sum of a positive square and a positive cube.
  • A078385 (program): Euler’s totient of numbers which can be written as sum of a positive square and a positive cube.
  • A078386 (program): Squarefree kernels of numbers which can be written as sum of a positive square and a positive cube.
  • A078387 (program): Moebius’s mu of numbers which can be written as sum of a positive square and a positive cube.
  • A078388 (program): Arithmetic derivative of numbers which can be written as sum of a positive square and a positive cube.
  • A078390 (program): Composite numbers which can be written as sum of a positive square and a positive cube.
  • A078391 (program): Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).
  • A078392 (program): Sum of GCD’s of parts in all partitions of n.
  • A078393 (program): Squarefree numbers which can be written as sum of a positive square and a positive cube.
  • A078400 (program): Iterated sum-of-digits of A078403(n).
  • A078401 (program): Triangle read by rows: T(n,k) = number of numbers <= k that are coprime to n, 1<=k<=n.
  • A078402 (program): Numbers k such that k^2 + 5 is prime.
  • A078403 (program): Primes such that digital root (A038194) is prime.
  • A078406 (program): Number of ways to partition 4*n into distinct positive integers.
  • A078407 (program): Number of ways to partition 4*n+2 into distinct positive integers.
  • A078408 (program): Number of ways to partition 2n+1 into distinct positive integers.
  • A078409 (program): Number of ways to partition 4*n+1 into distinct positive integers.
  • A078411 (program): Expansion of Molien series for a certain 4-D group of order 48.
  • A078414 (program): a(n) = (a(n-1)+a(n-2))/7^k, where 7^k is the highest power of 7 dividing a(n-1)+a(n-2).
  • A078417 (program): Numbers k such that h(k) = h(k+1), where h(k) is the length of k, f(k), f(f(k)), …, 1 in the Collatz (or 3x + 1) problem. (The earliest “1” is meant.)
  • A078427 (program): Sum of all the decimal digits of numbers from 1 to 10^n.
  • A078428 (program): Partial sums of A035187.
  • A078429 (program): Number of integers k among 1..n for which gcd(k,n) is a cube.
  • A078430 (program): Sum of gcd(k^2,n) for 1 <= k <= n.
  • A078435 (program): Number of composites <= n^2.
  • A078436 (program): Triangle read by rows in which n-th row counts multisets associated with hook partitions.
  • A078439 (program): a(n) = Sum_{k=1..n} gcd(k,n)*mu(gcd(k,n))^2.
  • A078442 (program): a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.
  • A078444 (program): Floor of geometric mean of consecutive primes.
  • A078445 (program): Primes in A060620, i.e., primes which are integer parts of averages of initial primes.
  • A078446 (program): a(1)=a(2)=1; a(n)=a(n-2)/2 if a(n-2) is even, a(n)=a(n-1)+a(n-2) otherwise.
  • A078447 (program): Final terms of rows of A078448.
  • A078449 (program): a(n) = sum of terms in n-th row of A078448.
  • A078450 (program): a(n) = product of terms in n-th row of A078448.
  • A078458 (program): Total number of factors in a factorization of n into Gaussian primes.
  • A078461 (program): a(n) = 0 if n is divisible by the square of odd prime, a(n) = 1 if n is an odd squarefree number, a(n) = 2 otherwise.
  • A078462 (program): Partial sums of A035185.
  • A078463 (program): a(n) = A000594(n+1) - A000594(n).
  • A078465 (program): Primonacci numbers: a(n)=a(n-2)+a(n-3)+a(n-5)+a(n-7)+a(n-11)+…+a(n-p(k))+… until n > p(k), where p(k) is the k-th prime. a(1)=a(2)=1.
  • A078467 (program): a(n) = a(n-1) + a(n-4); first four terms are 0,1,2,3.
  • A078468 (program): Distinct compositions of the complete graph with one edge removed (K^-_n).
  • A078469 (program): Number of different compositions of the ladder graph L_n.
  • A078471 (program): Sum of all odd divisors of all positive integers <= n.
  • A078474 (program): a(1)=a(2)=a(3)=1, a(n)=n-a(a(n-1))-a(a(n-2))-a(a(n-3)).
  • A078476 (program): Time taken to get n people from one side of a bridge to the other where (a) the only flashlight must be carried when crossing; (b) only one or two people may cross at the same time; (c) a pair crosses at the speed of the slowest member; and (d) the k-th person’s speed requires k seconds to cross the bridge.
  • A078477 (program): Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).
  • A078480 (program): Number of permutations p of {1,2,…,n} such that |p(i)-i| != 1 for all i.
  • A078483 (program): G.f.: -2*x/(1 - 5*x - sqrt(1-4*x) + x*sqrt(1-4*x) + 2*x^2).
  • A078484 (program): G.f.: -x*(1-2*x+2*x^2)/(2*x^3-4*x^2+4*x-1).
  • A078485 (program): Number of irreducible indecomposable permutations of degree n.
  • A078488 (program): First differences of coefficients of g.f. (1-x)^24.
  • A078489 (program): a(n)=j such that binomial(n,j)<binomial(n-1,j-2).
  • A078492 (program): No-prime decades.
  • A078493 (program): One-prime decades.
  • A078494 (program): Primes occurring only once in their decade.
  • A078495 (program): a(n) = (a(n-1) * a(n-6) + a(n-3) * a(n-4)) / a(n-7) (a variant of Somos-7).
  • A078499 (program): Two-prime decades.
  • A078501 (program): a(n) = sum(k=1,n^2, A078446(k)).
  • A078503 (program): a(n) = binomial(phi(n+1),phi(n)).
  • A078504 (program): a(n) = binomial(sigma(n+1), sigma(n)).
  • A078509 (program): Number of permutations p of {1,2,…,n} such that p(i)-i != 1 and p(i)-i != 2 for all i.
  • A078513 (program): a(0)=0, a(1)=1, a(n)=a(n-1)+a(n-2)+a(n-3) if a(n-1) is even, a(n)=a(n-1)+a(n-2) if a(n-1) is odd.
  • A078516 (program): Sum of balls on the lawn for the s=4 tennis ball problem.
  • A078522 (program): Numbers n such that (n+1)*(2*n+1) is a perfect square.
  • A078531 (program): Coefficients of power series that satisfies A(x)^2 - 4*x*A(x)^3 = 1, A(0)=1.
  • A078532 (program): Coefficients of power series that satisfies A(x)^3 - 9*x*A(x)^4 = 1, A(0)=1.
  • A078533 (program): Coefficients of power series that satisfies A(x)^4 - 16x*A(x)^5 = 1, A(0)=1.
  • A078534 (program): Coefficients of power series that satisfies A(x)^5 - 25x*A(x)^6 = 1, A(0)=1.
  • A078535 (program): Coefficients of power series that satisfies A(x)^6 - 36x*A(x)^7 = 1, A(0)=1.
  • A078545 (program): Largest prime dividing tau(n), the number of divisors of n.
  • A078551 (program): Largest prime dividing sigma(2,n).
  • A078552 (program): Largest prime dividing sigma(3,n).
  • A078553 (program): Largest prime dividing sigma(4,n).
  • A078554 (program): Largest prime dividing sigma(5,n).
  • A078558 (program): GCD of sigma(p#) and phi(p#) where p# = A002110(n) is the product of the first n primes.
  • A078565 (program): Number of zeros in the binary expansion of n!.
  • A078567 (program): Number of arithmetic subsequences of [1..n] with length > 1.
  • A078570 (program): Number of distinct prime factors of the average of n-th twin prime pair.
  • A078571 (program): Total number of prime factors of the average of n-th twin prime pair.
  • A078572 (program): Minimal exponent in prime factorization of the average of n-th twin prime pair.
  • A078573 (program): Maximal exponent in prime factorization of the average of n-th twin prime pair.
  • A078574 (program): Number of divisors of the average of n-th twin prime pair.
  • A078575 (program): Sum of divisors of the average of n-th twin prime pair.
  • A078576 (program): Sum of distinct prime factors of the average of n-th twin prime pair.
  • A078577 (program): Sum of all prime factors of the average of n-th twin prime pair.
  • A078578 (program): Euler’s totient of the average of n-th twin prime pair.
  • A078579 (program): Squarefree kernel of the average of n-th twin prime pair.
  • A078580 (program): Moebius function mu of the average of n-th twin prime pair.
  • A078581 (program): Arithmetic derivative of the average of n-th twin prime pair.
  • A078584 (program): a(n) = prime(2n) - prime(2n-1).
  • A078585 (program): Decimal expansion of Sum_{n>=0} 1/4^(2^n).
  • A078586 (program): a(n) is the product of the first n primes of the form 4k+3.
  • A078588 (program): a(n) = 1 if the integer multiple of phi nearest n is greater than n, otherwise 0, where phi = (1+sqrt(5))/2.
  • A078589 (program): a(1)=0, a(2)=1, a(n) = abs(abs(a(n-1)) - a(n-2) - n + 1).
  • A078599 (program): Product of squarefree divisors of n.
  • A078606 (program): Constant c(p) used in determining divisibility by the n-th prime, p=A000040(n), for n>=4.
  • A078607 (program): Least positive integer x such that 2*x^n > (x+1)^n.
  • A078608 (program): a(n) = ceiling(2/(2^(1/n)-1)).
  • A078609 (program): Least positive integer x such that 2*x^n>(x+3)^n.
  • A078612 (program): Number of transitions necessary for a Turing machine to compute the differences between consecutive primes (primes written in unary), when using the instruction table below.
  • A078613 (program): Same numbers of distinct prime factors of forms 4*k+1 and 4*k+3.
  • A078614 (program): Differences of A072633.
  • A078615 (program): a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).
  • A078616 (program): a(n) = Sum_{k=0..n} A010815(k).
  • A078617 (program): Floor(average of first n squares).
  • A078618 (program): a(n) = floor(average of first n cubes).
  • A078619 (program): Floor(average of first n factorials).
  • A078620 (program): Floor(average of first n Fibonacci numbers).
  • A078621 (program): Values of n such that Sum[ -(-1)^(k) n/k (n-1)/(k+1),{k,1,n}] (n!!) is an integer.
  • A078623 (program): Number of matched parentheses and brackets of length n, where a closing bracket will close any remaining open parentheses back to the matching open bracket (as in some versions of LISP).
  • A078625 (program): Primenomial primes: primes generated by polynomials of degree n with sequentially increasing prime coefficients. This is for n = 3 or 2x^3 + 3x^2 + 5x + 7.
  • A078627 (program): Write n in binary; repeatedly sum the “digits” until reaching 1; a(n) = 1 + number of steps required.
  • A078632 (program): Number of geometric subsequences of [1,…,n] with integral successive-term ratio and length > 1.
  • A078633 (program): Smallest number of sticks of length 1 needed to construct n squares with sides of length 1.
  • A078636 (program): rad{n(n+1)}.
  • A078637 (program): a(n) = rad(n(n+1)(n+2)), where rad(m) is the largest squarefree number dividing m (see A007947).
  • A078638 (program): rad(n(n+1)(n+2)(n+3)).
  • A078640 (program): Number of numbers between 1 and n-1 that are coprime to n(n+1)(n+2).
  • A078641 (program): Number of numbers between 1 and n-1 inclusive that are coprime to n(n+1)(n+2)(n+3).
  • A078642 (program): Numbers with two representations as the sum of two Fibonacci numbers.
  • A078644 (program): a(n) = tau(2*n^2)/2.
  • A078649 (program): Numbers n such that A000002(n)=A000002(n+1) where A000002 is the Kolakoski sequence.
  • A078650 (program): 2-A000002(n) where A000002 is the Kolakoski sequence.
  • A078651 (program): Number of geometric subsequences of [1,…,n] with integral successive-term ratio and length >= 1.
  • A078653 (program): a(n) = prime(k) where k = n-th prime congruent to 1 mod 4.
  • A078654 (program): a(n) = prime(k) where k = n-th prime congruent to 3 mod 4.
  • A078655 (program): a(n) = prime(k) where k = n-th prime congruent to 1 mod 6.
  • A078656 (program): a(n) = prime(k) where k = n-th prime congruent to 1 mod 10.
  • A078677 (program): Write n in binary; repeatedly sum the “digits” until reaching 1; a(n) = sum of these sums (including ‘1’ and n itself).
  • A078678 (program): Number of binary strings with n 1’s and n 0’s avoiding zigzags, that is avoiding the substrings 101 and 010.
  • A078684 (program): a(n) = 3^floor(n^2/4).
  • A078685 (program): Minimum value of |prime(n) - 2^x|.
  • A078687 (program): Number of x>=0 such that prime(n)-2^x is prime.
  • A078688 (program): Continued fraction expansion of e^(1/4).
  • A078689 (program): Continued fraction expansion of e^(1/3).
  • A078694 (program): Numbers n such that floor(2*Pi*n) is prime.
  • A078695 (program): a(n+1) = a(n)+greatest prime divisor of a(n-1).
  • A078696 (program): a(n+1)=a(n)+a(n-1) if a(n-1) odd, a(n+1)=a(n)+a(n-1)/2 if a(n-1) even.
  • A078700 (program): Number of symmetric ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.
  • A078701 (program): Least odd prime factor of n, or 1 if no such factor exists.
  • A078703 (program): Number of ways of subtracting twice a triangular number from a perfect square to obtain the integer n.
  • A078704 (program): Integer part of the square root of phi(n).
  • A078705 (program): Integer part of the square root of sigma(n).
  • A078706 (program): a(n) = smallest integer >=a(n-1) such that sum of first n terms is prime.
  • A078707 (program): Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.
  • A078708 (program): Sum of divisors d of n such that n/d is not congruent to 0 mod 3.
  • A078709 (program): a(n) = floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
  • A078711 (program): Sequence is S(infinity), where S(1)={1,2,3}, S(n+1)=S(n)S’(n) and S’(n) is obtained from S(n) by changing last term using the cyclic permutation 1->2->3->1.
  • A078712 (program): Series expansion of (-3-2*x)/(1+x-x^3) in powers of x.
  • A078713 (program): Sum of squares of the distances between successive divisors of n.
  • A078716 (program): Sequence has period 9 and differences between successive terms are 4, -3, 4, -3, 4, -3, 4, -3, -4.
  • A078718 (program): a(n) = (-1)^n*(2*n - 1)*CatalanNumber(n - 2) for n >= 2, a(n) = n for n = 0, 1.
  • A078719 (program): Number of odd terms among n, f(n), f(f(n)), …., 1 for the Collatz function (that is, until reaching “1” for the first time), or -1 if 1 is never reached.
  • A078721 (program): a(n) = prime(n*(n+1)/2 + 1).
  • A078722 (program): a(n) = prime(n*(n+1)/2+2).
  • A078723 (program): a(n) = prime(n*(n+1)/2 + n).
  • A078724 (program): a(n) = prime(n*(n+1)/2+3).
  • A078725 (program): a(n) = prime(n*(n+1)/2+4).
  • A078730 (program): Sum of products of two successive divisors of n.
  • A078734 (program): Start with 1,2, concatenate 2^k previous terms and change last term as follows: 1->2, 2->3, 3->1.
  • A078736 (program): Numerators of convergents to sqrt(e).
  • A078737 (program): Denominators of convergents to sqrt(e).
  • A078746 (program): a(n) = prime(2*n*(n+1)+1).
  • A078747 (program): Expansion of Sum_{k>0} k*phi(k)*x^k/(1+x^k).
  • A078750 (program): Least m not less than 2*n such that m + n = m OR n (logical ‘or’, bitwise).
  • A078760 (program): Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n.
  • A078761 (program): Sum of the digits of all n-digit numbers.
  • A078762 (program): Numbers n such that n + sigma(n) is prime.
  • A078763 (program): List primes between (2n-1)^2 and (2n)^2.
  • A078764 (program): List primes between (2n)^2 and (2n+1)^2.
  • A078766 (program): Number of primes less than n*phi(n).
  • A078767 (program): Let f(n) = A003434(n) be the number of iterations of phi needed to reach 1. Then a(n) = max(f(1), f(2), …, f(n)).
  • A078770 (program): a(n) = the least positive integer k such that k^2 + k + N is prime, where N is the n-th positive odd integer.
  • A078771 (program): a(n) = A008475(n) - A001414(n).
  • A078772 (program): a(n) = phi(n-p) where p is largest prime < n, a(1) = a(2) = 1 by convention.
  • A078773 (program): a(n) is the largest prime less than or equal to phi(n), a(1) = a(2) = 0.
  • A078774 (program): n*phi(n*phi(n)).
  • A078779 (program): Union of S, 2S and 4S, where S = odd squarefree numbers (A056911).
  • A078782 (program): Nonprimes (A018252) with prime (A000040) subscripts.
  • A078784 (program): Primes on axis of Ulam square spiral (with rows … / 7 8 9 / 6 1 2 / 5 4 3 / … ) with origin at (1).
  • A078787 (program): a(n) = 101*n + 1.
  • A078788 (program): Smallest m such that (n-1)*m+1 mod n = 0, or 0 if no such number exists.
  • A078789 (program): Expansion of (1-4*x+2*x^2)/(1-7*x+13*x^2-4*x^3).
  • A078794 (program): a(n) = (-1)^(n+1) * Sum_{k=0..n} 16^k * B(2k) * C(2n+1,2k) where B(k) is the k-th Bernoulli number.
  • A078795 (program): Concatenate first n triangular numbers.
  • A078796 (program): a(n) = 2*ceiling(n*tau) - ceiling(n*sqrt(5)) where tau=(1+sqrt(5))/2 is the Golden ratio.
  • A078802 (program): Triangular array T given by T(n,k) = number of 01-words of length n containing k 1’s, no three of which are consecutive.
  • A078803 (program): Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}.
  • A078809 (program): Number of divisors of the average of consecutive odd primes.
  • A078812 (program): Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
  • A078817 (program): Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).
  • A078818 (program): a(n) = 30*binomial(2n,n)/(n+3).
  • A078819 (program): 140*C(2n,n)/(n+4).
  • A078820 (program): 20*C(2n,n)*(2n+1)/(n+4).
  • A078830 (program): Numbers having in binary representation exactly one binary substring representing a prime.
  • A078832 (program): Smallest prime contained as binary substring in binary representation of n>1, a(1)=1.
  • A078835 (program): Sum of the divisors of the average of consecutive odd primes.
  • A078836 (program): a(n) = n*2^(n-6).
  • A078837 (program): a(n)=sum(k=1,p(n)-1, floor(k^3/p(n))) where p(n) denotes the n-th prime.
  • A078838 (program): a(n)=sum(k=1,(p(n)-1)*(p(n)-2),floor((k*p(n))^(1/3))) where p(n) denotes the n-th prime.
  • A078841 (program): Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.
  • A078860 (program): Least positive residues [mod 210] representing those residue classes which can be lesser of prime pairs from A029710.
  • A078865 (program): Difference of consecutive primorial numbers divided by 4.
  • A078876 (program): a(n) = n^4*(n^4-1)/240.
  • A078879 (program): Triangle read by rows: T(n,k) = ceiling(k^2 / n), 0 < k <= n.
  • A078880 (program): The sequence starting with 2 that equals its own run length sequence.
  • A078881 (program): Size of the largest subset S of {1,2,3,…,n} with the property that if i and j are distinct elements of S then i XOR j is not in S, where XOR is the bitwise exclusive-OR operator.
  • A078885 (program): Decimal expansion of Sum {n>=0} 1/3^(2^n).
  • A078886 (program): Decimal expansion of Sum {n>=0} 1/5^(2^n).
  • A078887 (program): Decimal expansion of Sum {n>=0} 1/6^(2^n).
  • A078888 (program): Decimal expansion of Sum {n>=0} 1/7^(2^n).
  • A078889 (program): Decimal expansion of Sum {n>=0} 1/8^(2^n).
  • A078890 (program): Decimal expansion of Sum {n>=0} 1/9^(2^n).
  • A078891 (program): Concatenate first n triangular numbers in reverse order.
  • A078892 (program): Numbers n such that phi(n) - 1 is prime, where phi is Euler’s totient function (A000010).
  • A078896 (program): Number of times the smallest prime factor of n is a factor in all numbers <=n; a(1)=1.
  • A078897 (program): Number of times the greatest prime factor of n is a factor in all numbers <=n; a(1)=1.
  • A078903 (program): a(n) = n^2 - Sum_{u=1..n} Sum_{v=1..u} valuation(2*v, 2).
  • A078904 (program): a(n) = 4a(n-1) + 3n with a(0) = 0.
  • A078905 (program): The q expansion of Lambda^5, a Hauptmodul for Gamma_1(5).
  • A078907 (program): Expansion of modular function j/256 in powers of m=k^2=lambda(t).
  • A078916 (program): a(n) = prime(n) + 2*n.
  • A078917 (program): Primes of the form prime(k) + 2*k.
  • A078918 (program): a(n) = (a(n-1) + a(n-3)) * a(n-2) / a(n-4). a(1) = a(2) = a(3) = a(4) = 1.
  • A078919 (program): Partial products of A079069.
  • A078921 (program): Signed variant of A077012.
  • A078922 (program): a(n) = 11*a(n-1) - a(n-2) with a(1)=1, a(2) = 10.
  • A078929 (program): Least k > 0 such that A000002(n+k) = A000002(n).
  • A078932 (program): Number of compositions (ordered partitions) of n into powers of 3.
  • A078934 (program): Smallest semiperimeter of integral rectangle with area n*(n+1)/2.
  • A078935 (program): Largest divisor of n*(n+1)/2 that is <= sqrt(n*(n+1)/2).
  • A078936 (program): Smallest divisor of n*(n+1)/2 that is >= sqrt(n*(n+1)/2).
  • A078938 (program): Cube of lower triangular matrix of A056857 (successive equalities in set partitions of n).
  • A078939 (program): Fourth power of lower triangular matrix of A056857 (successive equalities in set partitions of n).
  • A078940 (program): Row sums of A078938.
  • A078941 (program): Flipping burnt pancakes. Maximum number of spatula flips to sort a stack of n pancakes of different sizes, each burnt on one side, so that the smallest ends up on top, …, the largest at the bottom and each has its burnt side down.
  • A078942 (program): Flipping burnt pancakes. Given a sorted stack of n burnt pancakes of different sizes (smallest on top, …, largest at the bottom), each with its burnt side up, a(n) is the number of spatula flips needed to restore them to their initial order but with the burnt sides down.
  • A078944 (program): First column of A078939, the fourth power of lower triangular matrix A056857.
  • A078945 (program): Row sums of A078939.
  • A078971 (program): Numbers n such that C(4n,n)/(3n+1) (A002293) is not divisible by 4.
  • A078978 (program): Sequence is S(infinity), where S(1)={1,2,3,4}, S(n+1)=S(n)S’(n) and S’(n) is obtained from S(n) by changing last term using the cyclic permutation 1->2->3->4->1.
  • A078979 (program): a(n) = A078711(n) - 1.
  • A078986 (program): Chebyshev T(n,19) polynomial.
  • A078987 (program): Chebyshev U(n,x) polynomial evaluated at x=19.
  • A078988 (program): Chebyshev sequence with Diophantine property.
  • A078989 (program): Chebyshev sequence with Diophantine property.
  • A078995 (program): a(n) = Sum_{k=0..n} C(4*k,k)*C(4*(n-k),n-k).
  • A078998 (program): Choose a(n) so that a(1)+a(2)+…+a(n) = concatenation of n first natural numbers.
  • A078999 (program): Coefficients A_n for the s=4 tennis ball problem.
  • A079000 (program): a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) is odd”.
  • A079001 (program): Digital equivalents of letters A, B, C, …, Z on touch-tone telephone keypad.
  • A079003 (program): Least k >= 3 such that Fibonacci(k) == -1 (mod 3^n).
  • A079004 (program): Least x>=3 such that F(x)==1 (mod 3^n) where F(x) denotes the x-th Fibonacci number (A000045).
  • A079006 (program): Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.
  • A079027 (program): a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i)=6, m(i,j)=i/j.
  • A079028 (program): a(0) = 1, a(n) = (n+4)*4^(n-1) for n >= 1.
  • A079033 (program): First differences of A023203.
  • A079034 (program): Determinant of M(n), the n X n matrix defined by m(i,i)=1, m(i,j)=i-j.
  • A079044 (program): Numbers k such that Sum_{j=0..k} sin(j/Pi) < 0.
  • A079050 (program): Sum of the digits of LookAndSay(n).
  • A079054 (program): a(n) = -1 if the closest prime to prime(n) is prime(n-1); = 1 if the closest prime to prime(n) is prime(n+1); = 0 if prime(n-1) and prime(n+1) are equally close to prime(n).
  • A079055 (program): Numbers of prime pairs (p,q), p<=q, such that (p+q) divides n.
  • A079057 (program): a(n) = Sum_{k=1..n} bigomega(tau(k)).
  • A079065 (program): In prime factorization of n replace odd primes with 3.
  • A079067 (program): Number of primes less than greatest prime factor of n but not dividing n.
  • A079068 (program): Largest prime less than greatest prime factor of n but not dividing n, or 1 if no such prime exists.
  • A079069 (program): a(n) = a(n-2) * a(n-3) + a(n-3), n>3. a(1) = a(2) = a(3) = 1.
  • A079073 (program): Sum of numbers < n having in binary representation the same number of 1’s as n.
  • A079078 (program): a(0) = 1, a(1) = 2; for n > 1, a(n) = prime(n)*a(n-2).
  • A079079 (program): a(n) = (prime(n)+1)*(prime(n+1)+1)/4.
  • A079080 (program): a(n) = gcd((prime(n)+1)*(prime(n+1)+1)/4, prime(n)*prime(n+1)+1).
  • A079081 (program): Numerator of (prime(n)+1)*(prime(n+1)+1)/(4*(prime(n)*prime(n+1)+1).
  • A079084 (program): Greatest prime factor of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079085 (program): Number of distinct prime factors of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079086 (program): Total number of prime factors of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079087 (program): Maximal exponent in prime factorization of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079088 (program): Number of divisors of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079089 (program): Sum of divisors of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079090 (program): Sum of distinct prime factors of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079091 (program): Sum of all prime factors of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079092 (program): Euler’s totient of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079093 (program): Squarefree kernel of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079094 (program): Arithmetic derivative of (prime(n)+1)*(prime(n+1)+1)/4.
  • A079097 (program): Mix odd numbers and squares.
  • A079099 (program): Number of 0’s in n primorial or 0’s in A002110(n).
  • A079100 (program): Number of 1’s in n# (n primorial) = 1’s in A002110(n).
  • A079102 (program): a(2n) = 2^n, a(2n+1) = 2^(2n).
  • A079110 (program): Number of 2’s in n# (n primorial) = 2’s in A002110(n).
  • A079111 (program): Numbers n such that 2*n+1 or n itself is prime.
  • A079112 (program): Numbers in binary representation with odd length.
  • A079113 (program): Number of 3’s in n# (n primorial) = 3’s in A002110(n).
  • A079114 (program): Least squarefree number > n that is coprime to n.
  • A079115 (program): Least k such that n+k is squarefree and coprime to n.
  • A079123 (program): Number of 4’s in n# (n primorial) = 4’s in A002110(n).
  • A079127 (program): Number of 5’s in n# (n primorial) = 5’s in A002110(n).
  • A079130 (program): Primes such that iterated sum-of-digits (A038194) is a square.
  • A079131 (program): Primes such that iterated sum-of-digits (A038194) is odd.
  • A079132 (program): Primes such that iterated sum-of-digits (A038194) is even.
  • A079133 (program): Number of 6’s in n# (n primorial) = 6’s in A002110(n).
  • A079134 (program): Number of 8’s in n# (n primorial) = 8’s in A002110(n).
  • A079138 (program): Primes of the form k^2 + 7.
  • A079143 (program): Numbers divisible by prime ceilings of their square roots + 1.
  • A079147 (program): Primes p such that p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 2.
  • A079148 (program): Primes p such that p-1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 2.
  • A079150 (program): Primes p such that p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 3.
  • A079162 (program): a(n) = 5a(n-2) - 2a(n-4).
  • A079163 (program): Number of 9’s in n# (n primorial) = 9’s in A002110(n).
  • A079164 (program): Twin-primorial numbers: running products of twin primes.
  • A079165 (program): a(n) = (4n-2)*a(n-1)+a(n-2) with a(0)=1 and a(1)=2.
  • A079167 (program): Weighted roundness of n. If n = p_1^e_1…p_k^e_k, then a(n) = e_1 + 2*e_2 + … + k*e_k. Note that p_i < p_j, i < j is assumed.
  • A079168 (program): Weighted quadratic roundness of n. If n=p_1^e_1…p^k_e^k, then a(n) = e_1 + (2^2 * e_2) + … + (k^2 * e_k). Note that p_i<p_j, i<j, is assumed.
  • A079169 (program): Difference between A079168(n) and A079167(n).
  • A079189 (program): Number of anti-commutative closed binary operations (groupoids) on a set of order n.
  • A079228 (program): Least number > n with greater squarefree kernel than that of n.
  • A079229 (program): Least k>0 such that rad(n+k) > rad(n), where rad is the squarefree kernel (A007947).
  • A079247 (program): Number of pairs (p,q), 0 <= p < q, such that p+q divides n.
  • A079248 (program): Sum of q in all pairs (p,q), 0 <= p < q, p+q divides n.
  • A079249 (program): Sum of p in all pairs (p,q), 0<=p<q, p+q divides n.
  • A079250 (program): Even numbers in A079000.
  • A079251 (program): Complement of A079000.
  • A079252 (program): Even numbers not in A079000.
  • A079253 (program): a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) is even”.
  • A079255 (program): a(n) is taken to be the smallest positive integer greater than a(n-1) such that the condition “n is in the sequence if and only if a(n) is odd and a(n+1) is even” can be satisfied.
  • A079259 (program): a(n) is taken to be the smallest positive integer greater than a(n-1) such that the condition “n is in the sequence if and only if a(n) and a(n+1) are both odd” can be satisfied.
  • A079260 (program): Characteristic function of primes of form 4n+1 (1 if n is prime of form 4n+1, 0 otherwise).
  • A079261 (program): Characteristic function of primes of form 4n+3 (1 if n is prime of form 4n+3, 0 otherwise).
  • A079262 (program): Octanacci numbers: a(0)=a(1)=…=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).
  • A079272 (program): a(n) = ((2n+1)*3^n - 1)/2.
  • A079273 (program): Octo numbers (a polygonal sequence): a(n) = 5*n^2 - 6*n + 2 = (n-1)^2 + (2*n-1)^2.
  • A079275 (program): Number of divisors of n that are semiprimes with distinct factors.
  • A079276 (program): Multiplicative inverse in the finite field F(prime(n)) of the product of the first n-1 primes modulo prime(n).
  • A079277 (program): Largest integer k < n such that any prime factor of k is also a prime factor of n.
  • A079280 (program): Number of log-concave paths of length n starting from the origin (0,0) with steps from {N=(0,1), E=(1,0) and S=(0,-1)} that stay in the second octant and never touch the line y=x except possibly at the beginning or the end.
  • A079282 (program): Diagonal sums of triangle A055249.
  • A079284 (program): Diagonal sums of triangle A008949.
  • A079285 (program): First differences of A079284.
  • A079289 (program): For even n, a(n) = a(n-2) + a(n-1) + 2^(n/2-2), n>2. For odd n, a(n) = a(n-2) + a(n-1).
  • A079291 (program): Squares of Pell numbers.
  • A079295 (program): (D(p)-6)/(12p) where D(p) denotes the denominator of the 2p-th Bernoulli number and p runs through the primes.
  • A079297 (program): Triangle read by rows: the k-th column is an arithmetic progression with difference 2k-1 and the top entry is the hexagonal number k*(2*k-1) (A000384).
  • A079309 (program): a(n) = C(1,1) + C(3,2) + C(5,3) + … + C(2*n-1,n).
  • A079311 (program): Integer part of length of diagonal of integral rectangle with area n and smallest semiperimeter.
  • A079314 (program): Number of first-quadrant cells (including the two boundaries) born at stage n of the Holladay-Ulam cellular automaton.
  • A079315 (program): Number of cells that change from OFF to ON at stage n of the cellular automaton described in A079317.
  • A079316 (program): Number of first-quadrant cells (including the two boundaries) That are ON at stage n of the cellular automaton described in A079317.
  • A079317 (program): Number of ON cells after n generations of cellular automaton on square grid in which cells which share exactly one edge with an ON cell change their state.
  • A079318 (program): a(0) = 1; for n > 0, a(n) = (3^(A000120(n)-1) + 1)/2.
  • A079319 (program): a(0) = 1; for n > 1, a(n) = 4*a(n-1) - (2^n-1).
  • A079322 (program): Composite numbers of the form 1^1 * 2^2 * 3^3 * 4^4 * … * n^n + 11.
  • A079326 (program): a(n) = the largest number m such that if m monominoes are removed from an n X n square then an L-tromino must remain.
  • A079328 (program): Let f(n)=A001359(n) be the smaller member of the n-th pair of twin primes. Then a(n) is the average of f(n) and f(n+1).
  • A079329 (program): Let g(n)=A006512(n) be the larger member of the n-th pair of twin primes. Then a(n) is the average of g(n) and g(n+1).
  • A079340 (program): Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.
  • A079343 (program): Period 6: repeat [0, 1, 1, 2, 3, 1]; also F(n) mod 4, where F(n) = A000045(n).
  • A079344 (program): F(n) mod 8, where F(n) = A000045(n) is the n-th Fibonacci number.
  • A079345 (program): Fibonacci(n) mod 16.
  • A079351 (program): a(1)=3; for n > 1, a(n) is the smallest integer greater than a(n-1) consistent with the condition “n is in the sequence if and only if a(n) is congruent to 0 (mod 5)”.
  • A079352 (program): a(1)=1, then a(n)=3*a(n-1) if n is already in the sequence, a(n)=2*a(n-1) otherwise.
  • A079356 (program): a(1)=1; a(n) = a(n-1) - 2 if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.
  • A079360 (program): Sequence of sums of alternating increasing powers of 2.
  • A079362 (program): Sequence of sums of alternating powers of 3.
  • A079364 (program): Composite numbers having two composite neighbors.
  • A079390 (program): spf(n) * spf(n+1) * spf(n+2), where spf=A020639 (smallest prime factor).
  • A079395 (program): a(n) = prime(n)^11.
  • A079396 (program): Ramanujan’s tau function squared.
  • A079398 (program): a(0)=0, a(1)=1, a(2)=1, a(3)=1, a(n) = a(n-3) + a(n-4) for n > 3.
  • A079409 (program): Array T(m,n) (m>=0, n>=0) read by antidiagonals: T(0, 0) = 1, T(0, n) = 0 if n > 0, T(m, n) = T(m-1, n - T(m-1, n)) + T(m-1, n - T(m-1, n-1)) if m > 0.
  • A079414 (program): a(n) = 4*n^4 - 3*n^2.
  • A079416 (program): a(n) = round(prime(n)/n).
  • A079418 (program): Numbers n such that prime(n)/n < prime(n-1)/(n-1).
  • A079419 (program): Primes p such that p/i(p) < prime(i(p)-1)/(i(p)-1), where i(p)=A049084(p).
  • A079424 (program): A bisection of A024675. Cf. A058296.
  • A079429 (program): a(0) = 2, a(1) = 3, a(2) = 5; a(n) = a(n-1) + [a(n-1)-a(n-2)] * [a(n-2)-a(n-3)].
  • A079445 (program): phi(n)*(n-phi(n))-1 is prime.
  • A079446 (program): Integers k such that phi(k)*(k-phi(k))+1 is prime.
  • A079450 (program): a(n) = 2^(n-1)*u(n) where u(1)=1 and u(n) = frac(3/2*u(n-1)) + 1.
  • A079458 (program): Number of Gaussian integers in a reduced system modulo n.
  • A079460 (program): Let r(n) be the real positive root of Sum_{k=1..n} x^k = 1, then a(n) = round(1/(r(n) - 1/2)).
  • A079472 (program): Number of perfect matchings on an n X n L-shaped graph.
  • A079475 (program): Number described by n using the “Look and Say” rule.
  • A079476 (program): First prime greater than or equal to phi(n^2).
  • A079477 (program): First prime after phi(p_n^2).
  • A079484 (program): a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.
  • A079489 (program): Series reversion of x(1-x^2)/(1+x^2)^2 expanded in odd powers of x.
  • A079491 (program): Numerator of Sum_{k=0..n} binomial(n,k)/2^(k*(k-1)/2).
  • A079495 (program): Let b=3. Sum of squares of digits in base b gives 0 (mod b).
  • A079496 (program): a(0) = a(1) = 1; thereafter a(2*n+1) = 2*a(2*n) - a(2*n-1), a(2*n) = 4*a(2*n-1) - a(2*n-2).
  • A079497 (program): a(1)=1; for n > 2, a(n) is the smallest integer > a(n-1) such that frac(sqrt(5)*a(n)) < frac(sqrt(5)*a(n-1)).
  • A079498 (program): Numbers whose sum of digits in base b gives 0 (mod b), for b = 3.
  • A079501 (program): Number of compositions of the integer n with strictly smallest part in the first position.
  • A079503 (program): a(n) = (n-1)^3*((n-2)^2 - 2*(n-3)).
  • A079504 (program): a(n) = 8*n^3*((2*n-1)^2 - 4*n + 4).
  • A079505 (program): The last number for which a determinant of base-n numbers is nonzero.
  • A079508 (program): Triangle T(n,k) (n >= 2, k >= 1) of Raney numbers read by rows.
  • A079511 (program): a(n) = constant arising in game of n-times nim.
  • A079514 (program): Second column of triangular array in A079513.
  • A079523 (program): Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.
  • A079524 (program): Expansion of (x + b*x^2 - b*x^3)/((1 - x^2)*(1 - x)^2) with b=2.
  • A079528 (program): a(n) = sigma(n) - ceiling(n + sqrt n).
  • A079529 (program): sigma(n) - ceiling(n + sqrt n) as n runs through the composite numbers A002808.
  • A079530 (program): a(n) = phi(n) - ceiling(sqrt(n)).
  • A079531 (program): a(n) = phi(n) - ceiling(n^(2/3)).
  • A079532 (program): a(n) = floor(n - sqrt(n)) - phi(n).
  • A079533 (program): Floor(n - sqrt(n)) - phi(n) as n runs through the composite numbers (A002808).
  • A079535 (program): a(n) = phi(n)*d(n) - n.
  • A079536 (program): a(n) = phi(n)*d(n) - sigma(n).
  • A079537 (program): a(n) = phi(2*n+1)*d(2*n+1) - sigma(2*n+1).
  • A079540 (program): a(n) = phi(n) + d(n)*(n - phi(n)) - sigma(n).
  • A079545 (program): Primes of the form x^2 + y^2 + 1 with x,y >= 0.
  • A079546 (program): a(n) = sigma(n) - 4*phi(n).
  • A079547 (program): a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.
  • A079551 (program): a(n) = Sum_{primes p <= n} d(p-1), where d() = A000005.
  • A079552 (program): Record values in A079551.
  • A079553 (program): a(n) = floor( d(n^2) / d(n) ), where d() = A000005.
  • A079559 (program): Number of partitions of n into distinct parts of the form 2^j-1, j=1,2,….
  • A079563 (program): a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=7.
  • A079578 (program): Least coprime to n, greater than n+1.
  • A079579 (program): Totally multiplicative with p -> (p-1)*p, p prime.
  • A079581 (program): Consider pairs (r,s) such that the polynomial (x^r+1) divides (x^s+1) and 1 <= r < s. This sequence gives the s values; A079673 gives the r values.
  • A079583 (program): a(n) = 3*2^n - n - 2.
  • A079584 (program): Number of ones in the binary expansion of n!.
  • A079585 (program): Decimal expansion of c = (7-sqrt(5))/2 = 2.3819660112501…
  • A079588 (program): a(n) = (n+1)*(2*n+1)*(4*n+1).
  • A079589 (program): a(n) = C(5n+1,n).
  • A079590 (program): C(6n+1,n).
  • A079593 (program): Primes equal to floor(Pi*x) where x is prime.
  • A079598 (program): a(n) = 2^(4n+1) - 2^(2n).
  • A079609 (program): a(p) = (b(p)-2)/p where b(k) denotes the k-th Bell number and p runs through the prime numbers.
  • A079610 (program): a(n) = (5*n+1)*(5*n+3)*(5*n+5).
  • A079612 (program): Largest number m such that a^n == 1 (mod m) whenever a is coprime to m.
  • A079615 (program): Product of all distinct prime factors of all composite numbers between n-th prime and next prime.
  • A079617 (program): Occurrences of prime factorization templates, unordered.
  • A079621 (program): Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).
  • A079623 (program): a(1)=a(2)=1, a(3)=4, a(n)=abs(a(n-1)-a(n-2)-a(n-3)).
  • A079624 (program): a(1)=a(2)=1, a(3)=6, a(n)=abs(a(n-1)-a(n-2)-a(n-3)).
  • A079628 (program): Array of coefficients of P(n,x) = det (M(n,x)) where M(n,x) is the n X n matrix m(i,j)=x if i>j m(i,j)=1-x if i<=j.
  • A079632 (program): a(n) = floor(n/floor(sqrt(n)))-floor(sqrt(n)).
  • A079635 (program): Sum of (2 - p mod 4) for all prime factors p of n (with repetition).
  • A079643 (program): a(n) = floor(n/floor(sqrt(n))).
  • A079644 (program): n (mod sqrtint(n)).
  • A079645 (program): Numbers n such that Integer part of the cube root of n divides n.
  • A079651 (program): Prime numbers using only the straight digits 1, 4 and 7.
  • A079662 (program): a(n) = the number of occurrences of 1 in all compositions of n without 2’s = # of occurrences of the integer k in compositions of n+k-1 without 2’s (k > 2).
  • A079667 (program): a(n) = (1/2) * Sum_{d divides n} abs(n/d-d).
  • A079675 (program): a(1)=1; a(n)=sum(u=1,n-1,sum(v=1,u,sum(w=1,v,sum(x=1, w,sum(y=1,x,a(y)))))).
  • A079678 (program): a(n) = a(n,m) = sum(k=0,n,binomial(m*k,k)*binomial(m*(n-k),n-k)) for m=5.
  • A079679 (program): a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=6.
  • A079680 (program): Number of 1’s in n!.
  • A079684 (program): Number of 3’s in n!.
  • A079688 (program): Number of 4’s in n!.
  • A079690 (program): Number of 5’s in n!.
  • A079691 (program): Number of 6’s in n!.
  • A079692 (program): Number of 7’s in n!.
  • A079693 (program): Number of 8’s in n!.
  • A079696 (program): Numbers one more than composite numbers.
  • A079698 (program): Values of the odd part of A005277(n).
  • A079701 (program): A congruence property: a(p)=(A026375(p)-3)/(2p) where p runs through the primes.
  • A079702 (program): Numbers 2p where p is prime and 2p + 1 is composite.
  • A079704 (program): a(n) = 2*prime(n)^2.
  • A079705 (program): 3p^2 where p runs through the primes.
  • A079707 (program): In prime factorization of n replace odd primes with their prime predecessor.
  • A079709 (program): Numbers m such that the squarefree kernel of m is less than the squarefree kernel of m+1.
  • A079710 (program): Numbers m such that the squarefree kernel of m is larger than the squarefree kernel of m+1.
  • A079714 (program): Number of 2’s in n!.
  • A079715 (program): a(n) = Pi(n) - Pi(sqrt(n)) + 1.
  • A079719 (program): a(n) = n + floor[sum_k{k<n}a(k)/2] with a(0)=0.
  • A079725 (program): Sum of composite numbers less than n-th prime.
  • A079727 (program): a(n) = 1 + C(2,1)^3 + C(4,2)^3 + … + C(2n,n)^3.
  • A079728 (program): sum(k=0,p,binomial(2*k,k)) (mod p) where p runs through the primes.
  • A079733 (program): Primes of the form x^2 + y^2 + 3 (x,y nonnegative).
  • A079739 (program): Primes of the form x^2 + y^2 + 2 (x,y nonnegative).
  • A079747 (program): Numbers k such that gpf(k-1) < gpf(k) < gpf(k+1), where gpf(k) is the greatest prime factor of k (A006530).
  • A079748 (program): Largest k such that the greatest prime factors from n to n+k are monotonically increasing.
  • A079750 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of comparisons required to find j in step L2.2’.
  • A079751 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of cases where the j search loop runs beyond j=n-3.
  • A079752 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Algorithm L lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of times the search for an element exchangeable with a_j has to be started.
  • A079753 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives total executions of step L3.1’.
  • A079754 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of times l has to be repeatedly decreased in step L3.1’.
  • A079755 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Knuth’s Algorithm L (lexicographic permutation generation).
  • A079756 (program): Operation count to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchanges in reversal step.
  • A079757 (program): Periodic sequence 1 0 -2 3 -2 0,…
  • A079760 (program): Denominator in the expression for a(n) in A079759.
  • A079772 (program): Let C(n) be the n-th composite number; then a(n) is the smallest number > C(n) and not coprime to C(n).
  • A079773 (program): a(n) = 2*a(n-1)+15*a(n-2) with n>0, a(0)=0, a(1)=1.
  • A079777 (program): a(0) = 0, a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) (mod n).
  • A079779 (program): a(n) = smallest prime > n*prime(n).
  • A079780 (program): a(n) = largest prime <= n*prime(n).
  • A079781 (program): Initial term of n-th row of triangle in A079784.
  • A079782 (program): Final term of n-th row of triangle in A079784.
  • A079783 (program): Sum of n-th row of triangle in A079784.
  • A079784 (program): Triangle read by rows in which the n-th row contains the smallest set of n consecutive numbers such that the r-th number in the n-th row is divisible by n-r+1. The first term of the n-th row must exceed n.
  • A079806 (program): Number of even numbers that can be formed by permuting the digits of n.
  • A079810 (program): Sums of diagonals (upward from left to right) of the triangle shown in A079809.
  • A079813 (program): n 0’s followed by n 1’s.
  • A079816 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={1}.
  • A079821 (program): Smallest n-digit multiple of the n-th prime.
  • A079822 (program): Smallest n-digit multiple of the n-th composite number.
  • A079824 (program): Sum of numbers in n-th upward diagonal of triangle in A079823.
  • A079827 (program): a(n) = floor((n + reverse(n))/2).
  • A079838 (program): a(1) = 1 and then smallest multiple of a(n) which has no nonzero digit in common with a(n).
  • A079858 (program): E.g.f. 1/(cos(2*x) - sin(2*x)).
  • A079859 (program): a(n) = n*2^(n-4).
  • A079861 (program): a(n) is the number of occurrences of 7’s in the palindromic compositions of 2*n-1, or also, the number of occurrences of 8’s in the palindromic compositions of 2*n.
  • A079862 (program): a(i) = the number of occurrences of 9’s in the palindromic compositions of n=2*i-1 = the number of occurrences of 10’s in the palindromic compositions of n=2*i.
  • A079863 (program): a(n) is the number of occurrences of 11s in the palindromic compositions of m=2*n-1 = the number of occurrences of 12s in the palindromic compositions of m=2*n.
  • A079878 (program): a(1)=1, then a(n)=2*a(n-1) if 2*a(n-1)<=n, a(n)=2*a(n-1)-n otherwise.
  • A079882 (program): A run of 2^n 1’s followed by a run of 2^n 2’s, for n=0, 1, 2, …
  • A079884 (program): Number of comparisons required to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2.
  • A079885 (program): Number of index tests required to create all permutations of n distinct elements using the “streamlined” version of Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2.
  • A079896 (program): Discriminants of indefinite binary quadratic forms.
  • A079898 (program): a(1) = 1; a(n) = tau(n) - tau(n-1)* a(n-1) if n > 1.
  • A079899 (program): a(1) = 1; a(n) = Fibonacci(n) - Fibonacci(n-1)* a(n-1) if n > 1.
  • A079901 (program): Triangle of powers, T(n,k) = n^k, 0<=k<=n, read by rows.
  • A079903 (program): a(n) = (9n^4 - 18n^3 + 18n^2 - 9n + 2)/2.
  • A079904 (program): Triangle read by rows: T(n, k) = n*k, 0<=k<=n.
  • A079905 (program): a(1)=1; then a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+1 for n>1.
  • A079908 (program): Solution to the Dancing School Problem with 3 girls and n+3 boys: f(3,n).
  • A079909 (program): Solution to the Dancing School Problem with 4 girls and n+4 boys: f(4,n).
  • A079921 (program): Solution to the Dancing School Problem with n girls and n+2 boys: f(n,2).
  • A079922 (program): Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3).
  • A079929 (program): a(n)=(3*n+1)!/(n!*3^n).
  • A079935 (program): a(n) = 4*a(n-1) - a(n-2) with a(1) = 1, a(2) = 3.
  • A079944 (program): A run of 2^n 0’s followed by a run of 2^n 1’s, for n=0, 1, 2, …
  • A079945 (program): Partial sums of A079882.
  • A079946 (program): Binary expansion of n has form 11**…*0.
  • A079947 (program): Partial sums of A030300.
  • A079948 (program): First differences of A079000.
  • A079949 (program): Special values of Hermite polynomials.
  • A079950 (program): Triangle of n-th prime modulo twice primes less n-th prime.
  • A079951 (program): Number of primes p with prime(n) == 1 (modulo 2*p).
  • A079952 (program): Number of primes less than prime(n)/2.
  • A079953 (program): Smallest prime p such that prime(n) mod 2*p = prime(n).
  • A079954 (program): Partial sums of A030301.
  • A079955 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,2,3}.
  • A079956 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,1,4}.
  • A079957 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,1,3}.
  • A079958 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={3,4}.
  • A079959 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={2,4}.
  • A079960 (program): Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={2,3}.
  • A079961 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={1,4}.
  • A079962 (program): Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={1,3}.
  • A079963 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={1,2}.
  • A079964 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,4}.
  • A079965 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,3}.
  • A079966 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,2}.
  • A079967 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={4}.
  • A079968 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={3}.
  • A079969 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={2}.
  • A079971 (program): Number of compositions (ordered partitions) of n into parts 1, 2, and 5.
  • A079972 (program): Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=4, I={1,2}.
  • A079973 (program): Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=4, I={0,3}.
  • A079974 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=4, I={0,2}.
  • A079975 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=4, I={3}.
  • A079976 (program): G.f.: 1/(1-x-x^2-x^4-x^5).
  • A079977 (program): Fibonacci numbers interspersed with zeros.
  • A079978 (program): Characteristic function of multiples of three.
  • A079979 (program): Characteristic function of multiples of six.
  • A079980 (program): Number of permutations of length 2n satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..2n, with k=3, r=3, I={-2,0,1,2}. There is no one such permutation of length 2n+1.
  • A079981 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,0,1,2}.
  • A079986 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,0,2}.
  • A079998 (program): The characteristic function of the multiples of five.
  • A080013 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={0,1}.
  • A080014 (program): Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={1}.
  • A080015 (program): Expansion of theta_3(q) / theta_3(q^2) in powers of q.
  • A080020 (program): Primes of the form 9k^2 + 3k + 367, where k can be negative.
  • A080023 (program): log_phi(n) is closer to an integer than is log_phi(m) for any m with 2<=m<n, where phi=(1+sqrt(5))/2 is the golden ratio.
  • A080027 (program): Let s(1) = 1; let s(2m) = {s(2m-1),m+1,s(2m-1)}, s(2m+1) = {s(2m),s(2m)}; sequence gives limit of s(k) for large k.
  • A080028 (program): Let s(1) = 1; let s(2m+1) = {s(2m),m+1,s(2m)}, s(2m) = {s(2m-1),s(2m-1)}; sequence gives limit of s(k) for large k.
  • A080029 (program): a(n) is taken to be the smallest positive integer not already present which is consistent with the condition “n is a member of the sequence if and only if a(n) is a multiple of 3”.
  • A080030 (program): a(n) is taken to be the smallest positive integer not already present which is consistent with the condition “n is a member of the sequence if and only if a(n) is congruent to 1 mod 3”.
  • A080031 (program): a(n) is taken to be the smallest positive integer not already present which is consistent with the condition “n is a member of the sequence if and only if a(n) is congruent to 2 mod 3”.
  • A080036 (program): a(n) = n + round(sqrt(2*n)) + 1.
  • A080037 (program): a(0)=2; for n > 0, a(n) = n + floor(sqrt(4n-3)) + 2.
  • A080038 (program): Start with a(1)=3; apply 3 -> 343, 4 -> 3443; iterate.
  • A080039 (program): a(n) = floor((1+sqrt(2))^n).
  • A080040 (program): a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0)=2, a(1)=2.
  • A080041 (program): a(n)=floor((1+sqrt(3))^n).
  • A080042 (program): a(n) = 4*a(n-1)+3*a(n-2) for n>1, a(0)=2, a(1)=4.
  • A080043 (program): a(n)=floor((2+sqrt(7))^n).
  • A080047 (program): Operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of times l has to be repeatedly decreased in step L3.
  • A080048 (program): Operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of loop repetitions in reversal step.
  • A080049 (program): Operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth’s The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchange operations in step L4.
  • A080054 (program): G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)).
  • A080063 (program): n mod (spf(n)+1), where spf(n) is the smallest prime dividing n (A020639).
  • A080064 (program): Numbers n such that n == 1 modulo (spf(n)+1), where spf(m) is the smallest prime dividing m (A020639).
  • A080065 (program): Numbers n such that n == 3 modulo (spf(n)+1), where spf(m) is the smallest prime dividing m (A020639).
  • A080066 (program): First differences of A000966 (number of zeros that n! will never end in).
  • A080075 (program): Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.
  • A080076 (program): Proth primes: primes of the form k*2^m + 1 with odd k < 2^m, m >= 1.
  • A080079 (program): Least number causing the longest carry sequence when adding numbers <= n to n in binary representation.
  • A080081 (program): Beatty sequence for (3+sqrt(13))/2.
  • A080084 (program): Number of prime factors in the factorial of the n-th prime, counted with multiplicity.
  • A080085 (program): Number of factors of 2 in the factorial of the n-th prime, counted with multiplicity.
  • A080086 (program): Number of factors of 3 in the factorial of the n-th prime, counted with multiplicity.
  • A080087 (program): Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.
  • A080093 (program): Let sum(k>=0, k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=x(n).
  • A080095 (program): Let sum(k>=0, k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=z(n).
  • A080096 (program): a(1)=a(2)=1; a(3)=2; a(n)=abs(a(n-1)-a(n-2)-a(n-3)).
  • A080097 (program): a(n) = Fibonacci(n+2)^2 - 1.
  • A080098 (program): Triangle T(n,k) = n OR k, 0 <= k <= n, bitwise logical OR, read by rows.
  • A080099 (program): Triangle T(n,k) = n AND k, 0<=k<=n, bitwise logical AND, read by rows.
  • A080100 (program): a(n) = 2^(number of 0’s in binary representation of n).
  • A080101 (program): Number of prime powers in all composite numbers between n-th prime and next prime.
  • A080102 (program): Smallest prime power in all composite numbers between n-th prime and next prime, a(n)=1 if no such prime power exists.
  • A080103 (program): Greatest prime power in all composite numbers between n-th prime and next prime, a(n)=1 if no such prime power exists.
  • A080108 (program): a(n) = Sum_{k=1..n} k^(n-k)*binomial(n-1,k-1).
  • A080109 (program): Square of primes of the form 4k+1 (A002144).
  • A080116 (program): Characteristic function of A014486. a(n) = 1 if n’s binary expansion is totally balanced, otherwise zero.
  • A080121 (program): a(n) is the smallest k > 0 such that n^2^k + (n+1)^2^k is prime, or -1 if no such k exists.
  • A080141 (program): (3^(n-1))*(n!)^2.
  • A080143 (program): a(n) = F(3)*F(n)*F(n+1) + F(4)*F(n+1)^2 - F(4) if n even, F(3)*F(n)*F(n+1) + F(4)*F(n+1)^2 if n odd, where F(n) is the n-th Fibonacci number (A000045).
  • A080144 (program): a(n) = F(4)*F(n)*F(n+1) + F(5)*F(n+1)^2 if n odd, a(n) = F(4)*F(n)*F(n+1) + F(5)*F(n+1)^2 - F(5) if n even, where F(n) is the n-th Fibonacci number (A000045).
  • A080145 (program): a(n) = Sum_{m=1..n} Sum_{i=1..m} F(i)*F(i+1) where F(n)=Fibonacci numbers A000045.
  • A080147 (program): Positions of primes of the form 4*k+1 (A002144) among all primes (A000040).
  • A080148 (program): Positions of primes of the form 4*k+3 (A002145) among all primes (A000040).
  • A080149 (program): Numbers k such that k^2 + 1 and k^2 + 3 are both prime.
  • A080150 (program): Let m = Wonderful Demlo number A002477(n); a(n) = square of the sum of digits of m.
  • A080151 (program): Let m = Wonderful Demlo number A002477(n); a(n) = sum of digits of m.
  • A080159 (program): Triangular array of ways of drawing k non-intersecting chords between n points on a circle; i.e., Motzkin polynomial coefficients.
  • A080163 (program): Sum of an infinite series: a(n) = Sum_{k>=0} ((k+1)*(k+2))^n/(16*(2^k)).
  • A080165 (program): Primes having initial digits “10” in binary representation.
  • A080166 (program): Primes having initial digits “11” in binary representation.
  • A080167 (program): Primes beginning with ‘10’ and ending with ‘01’ in binary representation.
  • A080168 (program): Primes beginning and ending with ‘11’ in binary representation.
  • A080169 (program): Numbers that are cubes of primes of the form 4k+1 (A002144).
  • A080170 (program): Numbers k such that gcd(C(2*k,k), C(3*k,k), C(4*k,k), …, C((k+1)*k,k) ) = 1.
  • A080171 (program): a(n)=na(n-1)-(n-1)^2a(n-2), a(0)=1, a(1)=1.
  • A080175 (program): Fourth power of primes of the form 4k+1 (A002144).
  • A080180 (program): a(1)=1, a(n+1)=a(n)+spf(Sum(a(i): 1<=i<=n)), where spf=A020639 (smallest prime factor).
  • A080181 (program): a(1)=A080180(1), a(n)=a(n-1)+A080180(n) for n>1.
  • A080183 (program): a(1)=A080182(1), a(n)=a(n-1)+A080182(n) for n>1.
  • A080193 (program): 5-smooth numbers which are not 3-smooth.
  • A080194 (program): 7-smooth numbers which are not 5-smooth.
  • A080195 (program): 11-smooth numbers which are not 7-smooth.
  • A080197 (program): 13-smooth numbers: numbers whose prime divisors are all <= 13.
  • A080204 (program): Number of fixed points under n-fold inflation for the substitution rule a->abc, b->ab, c->b that underlies the Kolakoski (3,1) sequence.
  • A080211 (program): a(n) = binomial(n, smallest prime factor of n).
  • A080212 (program): Binomial(n, smallest odd prime factor of n).
  • A080213 (program): a(n) = binomial(n, greatest prime factor of n).
  • A080214 (program): Binomial(greatest prime factor of n, smallest prime factor of n).
  • A080215 (program): Binomial(greatest prime factor of n, smallest odd prime factor of n).
  • A080216 (program): a(n) is the largest value taken by binomial(n,j) mod j for j in [1..n].
  • A080218 (program): Monotonically increasing sequence such that every positive integer n appears if and only if d(n) doesn’t (d(n)=number of divisors of n, A000005).
  • A080224 (program): Number of abundant divisors of n.
  • A080225 (program): Number of perfect divisors of n.
  • A080226 (program): Number of deficient divisors of n.
  • A080227 (program): a(n) = n*a(n-1) + (1/2)*(1+(-1)^n), a(0)=0.
  • A080232 (program): Triangle T(n,k) of differences of pairs of consecutive terms of triangle A071919.
  • A080233 (program): Triangle T(n,k) obtained by taking differences of consecutive pairs of row elements of Pascal’s triangle A007318.
  • A080234 (program): Triangle whose rows are the differences of consecutive pairs of row elements of A080232.
  • A080236 (program): Triangle of differences of consecutive pairs of row elements of triangle A080233.
  • A080237 (program): Start with 1 and apply the process: k-th run is 1, 2, 3, …, a(k-1)+1.
  • A080238 (program): Largest squarefree number dividing sum of cubes of divisors of n.
  • A080239 (program): Antidiagonal sums of triangle A035317.
  • A080240 (program): Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n} for n >= 0, B_0=0, B_1=1 and for n >= 2, B_n = 2B_{n-1}+(-1)^{A_n}. Sequence gives A_n.
  • A080242 (program): Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)*P(n-1,x) + (-x)^(n+1).
  • A080243 (program): Signed super-Catalan or little Schroeder numbers.
  • A080244 (program): Signed generalized Fibonacci numbers.
  • A080245 (program): Inverse of coordination sequence array A113413.
  • A080246 (program): Signed version of A035607.
  • A080247 (program): Formal inverse of triangle A080246. Unsigned version of A080245.
  • A080250 (program): Expansion of 1/((1-x)(1-4x)(1-10x)(1-20x)).
  • A080251 (program): Paired decomposition of tetrahedral numbers A000292 arranged as number triangle.
  • A080252 (program): a(n) = n*a(n-1)+4*a(n-2)-4*(n-2)*a(n-3).
  • A080253 (program): a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).
  • A080256 (program): Sum of numbers of distinct and of all prime factors of n.
  • A080257 (program): Numbers having at least two distinct or a total of at least three prime factors.
  • A080258 (program): Either 4th power of a prime, or product of a prime and the square of a different prime.
  • A080259 (program): Numbers whose squarefree kernel is not a primorial number, i.e., A007947(a(n)) is not in A002110.
  • A080260 (program): a(n)=1+(1/12)(n*(n+1)*(n+3)*(4-n)).
  • A080267 (program): a(n) = Sum_{d divides n} d*2^(n-n/d).
  • A080275 (program): a(n)=(-1)^n(1 - (1/12)n(n + 1)(12 - n + n^2)).
  • A080276 (program): Variation on Connell sequence (A001614). In this one, a(1)=1, terms a(n) onwards are generated in “blocks” as the next a(n-1) odd numbers > a(n-1) if the previous block ends with a(n-1) even and the next a(n-1) even numbers > a(n-1) if the previous block ends with a(n-1) odd.
  • A080277 (program): Partial sums of A038712.
  • A080278 (program): a(n) = (3^(v_3(n) + 1) - 1)/2, where v_3(n) = highest power of 3 dividing n = A007949(n).
  • A080291 (program): a(n)=(-1)^(n+1)n^n(n+(1/12)(n^2-1)).
  • A080299 (program): A014486-encoding of plane binary trees (Stanley’s d) whose interior zigzag-tree (Stanley’s c, i.e., tree obtained by discarding the outermost edges of the binary tree) is isomorphic to a valid plane binary tree (Stanley’s d).
  • A080300 (program): Global ranking function for totally balanced binary sequences.
  • A080303 (program): Rewrite 0->100 in the binary expansion of n.
  • A080304 (program): Numerator of n^mu(n), where mu is the Moebius function (A008683).
  • A080305 (program): Denominator of n^mu(n), where mu is the Moebius function (A008683).
  • A080307 (program): Multiples of the Fermat numbers 2^(2^n)+1.
  • A080308 (program): Non-multiples of Fermat numbers 2^(2^n)+1.
  • A080310 (program): Rewrite 0->100 in the binary expansion of n (but leaving single zero as zero) and append 10 to the right.
  • A080322 (program): Determinant of the n X n tridiagonal matrix M with the elements on the diagonals equal to 1, except M(n,n-1)=M(n-1,n)=n.
  • A080323 (program): a(n) = mu(n)^n, where mu is the Moebius function (A008683).
  • A080324 (program): Union of even squarefree numbers (A001747) and squarefree numbers for which the number of prime factors is even (A030229).
  • A080332 (program): G.f.: Product_{n>0} (1 - x^n)^3 * (1 - x^(2*n - 1))^2 = Sum_{n in Z} (6*n + 1) * x^(n*(3*n + 1)/2).
  • A080333 (program): Partial sums of A080278.
  • A080334 (program): n^2 read backwards, for n = 51, 50, 49, …
  • A080335 (program): Diagonal in square spiral or maze arrangement of natural numbers.
  • A080336 (program): Partial sums of A007001.
  • A080339 (program): Characteristic function of {1} union {primes}: 1 if n is 1 or a prime, else 0.
  • A080340 (program): First known infinite sequence containing no odd integer of the form 2^m+p (p prime).
  • A080341 (program): Sum of the first n terms that are congruent to 1, 4 or 5 mod 6 (A047259).
  • A080342 (program): Number of weighings required to identify a single bad coin out of n coins, using a two-pan balance.
  • A080343 (program): a(n) = round(sqrt(2*n)) - floor(sqrt(2*n)).
  • A080344 (program): Partial sums of A023969.
  • A080352 (program): Partial sums of A080343.
  • A080353 (program): a(1)=5; for n>1, a(n)=a(n-1)+1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
  • A080354 (program): First differences of A080353.
  • A080355 (program): a(1)=1; thereafter, a(n+1) = a(n) + 2^(prime(n)-1).
  • A080356 (program): Number of twin primes between n and 2n: a(n) = number of j in range n <= j <= 2*n such that j and j+2 are primes.
  • A080358 (program): Value of Vandermonde determinant for the first n prime numbers: V[p(1),…,p(n)].
  • A080359 (program): The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.
  • A080363 (program): Nonprime numbers k such that the largest prime divisor of k is unitary.
  • A080364 (program): Composite numbers whose least prime factor appears with multiplicity 1.
  • A080367 (program): Largest unitary prime divisor of n or a(n)=0 if no such prime divisor exists.
  • A080368 (program): a(n) is the least unitary prime divisor of n, or 0 if no such prime divisor exists.
  • A080378 (program): Residues mod 4 of the n-th difference between consecutive primes.
  • A080381 (program): Triangle read by rows: gcd(binomial(n,floor(n/2)), binomial(n,i), i=0..n; greatest common divisor of binomial coefficients and corresponding central binomial coefficient.
  • A080382 (program): Triangle read by rows: T(n,k) = C(n,floor(n/2))/gcd(C(n,floor(n/2)),C(n,k)), k=0..n; central binomial coefficient is divided by greatest common divisor of binomial coefficients and corresponding central binomial coefficient.
  • A080396 (program): Largest squarefree numbers dividing the binomial coefficients C(n,k) read by row, 0<=k<=n. Squarefree kernel of Pascal triangle.
  • A080397 (program): Largest squarefree number dividing central binomial coefficient A000984(n).
  • A080398 (program): Largest squarefree number dividing sum of divisors of n.
  • A080399 (program): Largest squarefree number dividing sum of squares of divisors of n.
  • A080400 (program): Largest squarefree number dividing phi(n).
  • A080401 (program): A001157(n) is squarefree: sum of squares of divisors of n is squarefree.
  • A080402 (program): A001157(n) is not squarefree: sum of squares of divisors of n is not squarefree.
  • A080404 (program): a(n)=A007947[A055932(n)]; the sequence consists of primorial numbers;.
  • A080405 (program): Number of distinct primes dividing n-th Catalan number.
  • A080412 (program): Exchange rightmost two binary digits of n > 1; a(0)=0, a(1)=2.
  • A080413 (program): Take the rightmost three binary digits of n (for n<4 padded with leading zeros) and rotate left 1 digit.
  • A080414 (program): Take the rightmost three binary digits of n (for n<4 padded with leading zeros) and rotate right 1 digit.
  • A080418 (program): Generalized Pascal triangle.
  • A080419 (program): Triangle of generalized Chebyshev coefficients.
  • A080420 (program): a(n) = (n+1)*(n+6)*3^n/6.
  • A080421 (program): (n+1)(n+2)(n+9)3^n/18.
  • A080422 (program): (n+1)(n+2)(n+3)(n+12)3^n/72.
  • A080423 (program): (n+1)(n+2)(n+3)(n+4)(n+15)3^n/360.
  • A080424 (program): a(n) = 3*a(n-1) + 18*a(n-2), a(0)=0, a(1)=1.
  • A080425 (program): Period 3: repeat [0, 2, 1].
  • A080426 (program): a(1)=1, a(2)=3; all terms are either 1 or 3; each run of 3’s is followed by a run of two 1’s; and a(n) is the length of the n-th run of 3’s.
  • A080428 (program): First differences of A079255.
  • A080445 (program): a(1) = 1, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1).
  • A080446 (program): a(1) = 2, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1).
  • A080447 (program): a(1) = 3, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1).
  • A080448 (program): a(1) = 4, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1). Also a(n) is not divisible by 10.
  • A080449 (program): a(1) = 5, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1). Also a(n) is not divisible by 10.
  • A080450 (program): a(1) = 6, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1). Also a(n) is not divisible by 10.
  • A080453 (program): a(1) = 9, a(n) = smallest (nontrivial) multiple of a(n-1) containing n digits, a(n) not equal to 10*a(n-1).
  • A080455 (program): a(1)=1; for n>1, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.
  • A080456 (program): a(1) = a(2) = 2; for n > 2, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.
  • A080457 (program): a(1)=3; for n>1, a(n)=a(n-1) if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A080458 (program): a(1)=4; for n>1, a(n)=a(n-1) if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A080460 (program): a(1) = 2; for n > 1, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.
  • A080467 (program): Multiples of 11 in which the even positioned digits from left are odd and the odd positioned ones are even.
  • A080468 (program): a(n) = A080578(n)-2n.
  • A080473 (program): a(n)= sum of the products of taking (n-1) numbers from the next n numbers. The next n numbers can be grouped like this (1), (2,3), (4,5,6), (7,8,9,10),… and a(n) is the (sum of the reciprocals of all members) multiplied by (the product of all members).
  • A080474 (program): a(n)= product of sum of taking (n-1) numbers from the next n numbers. The next n numbers can be grouped like this (1), (2,3), (4,5,6), (7,8,9,10),… . Let N be the sum of all the members of the n-th group. Let k be a member and f(k) = N - k. Then a(n) = the product of all f(k) for k taking all member values.
  • A080476 (program): Floor( geometric mean of next n numbers ).
  • A080480 (program): Largest number formed by using all the digits (with multiplicity) of next n numbers.
  • A080481 (program): Concatenation of next n numbers (a(1) = 0).
  • A080482 (program): A080481(n)/3.
  • A080483 (program): Reverse concatenation of next n numbers with a(1) = 0.
  • A080484 (program): A080483(n)/3.
  • A080486 (program): a(1) = 1, a(n) = smallest multiple of a(n-1) (not equal to 10*a(n-1)) obtained by inserting digits anywhere in a(n-1).
  • A080488 (program): a(1) =3, a(n) = smallest multiple of a(n-1) (not equal to 10*a(n-1)) obtained by inserting digits anywhere in a(n-1).
  • A080492 (program): a(1) =7, a(n) = smallest multiple of a(n-1) (not equal to 10*a(n-1)) obtained by inserting digits anywhere in a(n-1).
  • A080493 (program): a(1) =8, a(n) = smallest multiple of a(n-1) (not equal to 10*a(n-1)) obtained by inserting digits anywhere in a(n-1).
  • A080494 (program): a(1) =9, a(n) = smallest multiple of a(n-1) (not equal to 10*a(n-1)) obtained by inserting digits anywhere in a(n-1).
  • A080497 (program): a(n) = (n-p_1)(n-p_2)…(n-p_k) where p_k is the k-th prime and is also the largest prime < n.
  • A080498 (program): a(n) = (n-c_1)(n-c_2)…(n-c_k) where c_k is the k-th composite number and is also the largest composite number < n.
  • A080500 (program): a(n) = (n-1)(n-4)(n-9)…(n-k^2) where k^2 < n <= (k+1)^2.
  • A080511 (program): Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose arithmetic mean is an integer.
  • A080512 (program): a(n) = n if n is odd, a(n) = 3*n/2 if n is even.
  • A080513 (program): a(n) = round(n/2) + 1 = ceiling(n/2) + 1 = floor((n+1)/2) + 1.
  • A080522 (program): Leading diagonal of triangle in A080521.
  • A080523 (program): a(n) = n^n - n(n-1)/2.
  • A080525 (program): First column of triangle in A080524.
  • A080526 (program): Final entry in n-th row of triangle in A080524.
  • A080527 (program): Expansion of e.g.f. exp(3*cosh(x))/e^3 (even powers only).
  • A080529 (program): Number of nucleons in longest known radioactive decay series ending with Lead 206 (“uranium series”), reversed.
  • A080530 (program): Number of nucleons in longest known radioactive decay series ending with Lead 207 (“actinium series”), reversed.
  • A080531 (program): Number of nucleons in longest known radioactive decay series ending with Lead 208 (“thorium series”), reversed.
  • A080532 (program): Number of nucleons in longest known radioactive decay series ending with Lead 209, reversed.
  • A080534 (program): Number of protons in longest known radioactive decay series ending with Lead 207 (“actinium series”), reversed.
  • A080538 (program): Number of neutrons in longest known radioactive decay series ending with Lead 207 (“actinium series”), reversed.
  • A080540 (program): Number of neutrons in longest known radioactive decay series ending with Lead 209, reversed.
  • A080541 (program): In binary representation: keep the first digit and left-rotate the others.
  • A080542 (program): In binary representation: keep the first digit and rotate right the others.
  • A080543 (program): In binary representation: keep the first digit and rotate left the others twice.
  • A080544 (program): In binary representation: keep the first digit and rotate right the others twice.
  • A080545 (program): Characteristic function of {1} union {odd primes}: 1 if n is 1 or an odd prime, else 0.
  • A080565 (program): Binary expansion of n has form 11**…*1.
  • A080566 (program): Partial sums of A079000.
  • A080567 (program): 1 + Sum_{k=2..n} 2^((prime(k)-1)/2).
  • A080568 (program): Sum of the Fibonacci numbers A000045 and the factorials A000142.
  • A080572 (program): Number of ordered pairs (i,j), 0 <= i,j < n, for which (i & j) is nonzero, where & is the bitwise AND operator.
  • A080578 (program): a(1)=1; for n > 1, a(n) = a(n-1) + 1 if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.
  • A080579 (program): a(1)=1; for n>1, a(n)=a(n-1)+1 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A080580 (program): a(1)=1; for n>1, a(n)=a(n-1)+2 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A080584 (program): A run of 3*2^n 0’s followed by a run of 3*2^n 1’s, for n=0, 1, 2, …
  • A080585 (program): Partial sums of A080584.
  • A080586 (program): A run of 3*2^n 1’s followed by a run of 3*2^n 2’s, for n=0, 1, 2, …
  • A080587 (program): Partial sums of A080586.
  • A080590 (program): a(1)=1; for n>1, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A080594 (program): Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z in reverse lexicographic order; sequence gives y values.
  • A080595 (program): Consider the standard game of Nim with 3 heaps and make a list of the losing positions (x,y,z) with x <= y <= z in reverse lexicographic order; sequence gives z values.
  • A080596 (program): a(1)=1; for n >= 2, a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+3.
  • A080599 (program): Expansion of e.g.f.: 2/(2-2*x-x^2).
  • A080600 (program): a(n) = ceiling(n*(3 + sqrt(13))/2).
  • A080604 (program): Triangular array of hypotenuses of right triangles with integer legs: T(n,k) = round(sqrt(n^2 + k^2)), 1 <= k <= n.
  • A080609 (program): Binomial transform of central Delannoy numbers A001850.
  • A080610 (program): Partial sums of Jacobsthal gap sequence.
  • A080612 (program): Numbers n such that 1/p(2n+1)*sum(k=1,n,p(2k+1)-p(2k)) >= 1/p(2*n)*sum(k=1,n,p(2k)-p(2k-1)) where p(k) denotes the k-th prime.
  • A080633 (program): a(1)=3; for n > 1, a(n) is the smallest integer greater than a(n-1) consistent with the condition “n is in the sequence if and only if a(n) is congruent to 1 (mod 4)”.
  • A080637 (program): a(n) is the smallest positive integer which is consistent with the sequence being monotonically increasing and satisfying a(1)=2, a(a(n)) = 2n+1 for n > 1.
  • A080639 (program): a(1) = 1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition “for n>1, n is a member of the sequence if and only if a(n) is even”.
  • A080643 (program): a(0)=0; for n>0, a(n) = 4^n - 2*3^(n-1).
  • A080645 (program): a(1) = 1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition “for n>1, if n is a member of the sequence then a(n) is even”.
  • A080646 (program): a(1) = 3; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition “if n is a member of the sequence then a(n) is divisible by 3”.
  • A080647 (program): Sum of prime factors of phi(n).
  • A080649 (program): Sum of prime factors of sigma(n).
  • A080652 (program): a(1)=2; for n>1, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
  • A080653 (program): a(1) = 2; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) such that the condition “a(a(n)) is always even” is satisfied.
  • A080663 (program): a(n) = 3*n^2 - 1.
  • A080664 (program): Numbers n such that n-th Catalan number is squarefree.
  • A080667 (program): a(1)=3; for n>1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.
  • A080668 (program): Numbers of the form n!+n^3.
  • A080671 (program): Numbers having divisors 2 or 3 or 5.
  • A080672 (program): Numbers having divisors 2 or 3 or 5 or 7.
  • A080674 (program): a(n) = (4/3)*(4^n - 1).
  • A080675 (program): a(n) = (5*4^n - 8)/6.
  • A080676 (program): a(1) = 1; for n>1, a(n) is the smallest number > a(n-1) such that the first n terms of the sequence contain a total of a(n) digits.
  • A080677 (program): a(n) = n + 1 - A004001(n).
  • A080679 (program): Lexicographically earliest de Bruijn cycle of length 16 (repeated indefinitely)
  • A080680 (program): Integer part of the square root of the n-th prime of the form 4k+1.
  • A080681 (program): 17-smooth numbers: numbers whose prime divisors are all <= 17.
  • A080683 (program): 23-smooth numbers: numbers whose prime divisors are all <= 23.
  • A080684 (program): Number of 13-smooth numbers <= n.
  • A080685 (program): Number of 17-smooth numbers <= n.
  • A080686 (program): Number of 19-smooth numbers <= n.
  • A080689 (program): Powers of 10 that reach …,7,8,4,2,1 under the mapping: if n is even divide by 2 else add 1.
  • A080692 (program): a(n)=(-1)^(n+1)*det(M(n)) where M(n) is the n X n matrix M(i,j)=min(abs(i-j),i).
  • A080696 (program): Piptorial numbers = product of first n pips or prime-indexed primes.
  • A080697 (program): a(n) = n * prime(prime(n)).
  • A080698 (program): Product of twin-prime-indexed primes and their lower bound twin prime.
  • A080699 (program): Product of twin-prime-indexed primes and their upper bound twin prime.
  • A080700 (program): Product of upper bound twin-prime-indexed primes and their lower bound twin prime.
  • A080701 (program): Product of upper bound twin-prime-indexed primes and their upper bound twin prime.
  • A080702 (program): a(1)=3; for n>1, a(n) = smallest number > a(n-1) such that the condition “if n is in the sequence then a(n) is even” is satisfied.
  • A080703 (program): a(1)=5; for n>1, a(n) = smallest number > a(n-1) such that the condition “if n is in the sequence then a(n) is a multiple of 4” is satisfied.
  • A080704 (program): a(1)=2; for n>1, if n is in the sequence then a(n) is the smallest even integer > a(n-1), otherwise a(n) = a(n-1) + 3.
  • A080705 (program): Numbers n such that A080704(n+1) - A080704(n) = 1.
  • A080706 (program): Powers of 3 that reach …,7,8,4,2,1 under the mapping: if n is even divide by 2 else add 1.
  • A080707 (program): a(1)=5; for n>1, a(n) = smallest number > a(n-1) such that the condition “n is in the sequence if and only if a(n) is a multiple of 3” is satisfied.
  • A080709 (program): Take sum of squares of digits of previous term, starting with 4.
  • A080710 (program): a(0) = 1; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) is a multiple of 3”.
  • A080711 (program): a(0) = 2; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) is a multiple of 3”.
  • A080712 (program): a(0) = 4; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) is a multiple of 3”.
  • A080717 (program): Let S(0) = 1; obtain S(k) from S(k-1) by applying 1 -> 2, 2 -> 31, 3 -> 332; sequence is S(0), S(1), S(2), …
  • A080720 (program): a(0) = 5; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) is a multiple of 3”.
  • A080722 (program): a(0) = 0; for n > 0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a term of the sequence if and only if a(n) == 1 (mod 3)”.
  • A080723 (program): a(0) = 1; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) == 1 mod 3”.
  • A080724 (program): a(0) = 2; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) == 1 mod 3”.
  • A080725 (program): a(1) = 2; for n>1, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) == 1 mod 3”.
  • A080726 (program): a(0) = 0; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) == 2 mod 3”.
  • A080727 (program): a(0) = 1; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) == 2 mod 3”.
  • A080728 (program): a(0) = 3; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition “n is a member of the sequence if and only if a(n) == 2 mod 3”.
  • A080731 (program): a(1)=1; a(2)=6; for n > 2, a(n) is taken as the smallest positive integer greater than a(n-1) such that the condition “n is a member of the sequence if and only if a(n) is odd” is satisfied.
  • A080734 (program): a(1)=1, then a(n)=a(n-1)+2 if the final decimal digit of a(n) is 0, a(n)=a(n-1)+3 otherwise.
  • A080735 (program): a(1)=1, then a(n)=2*a(n-1) if a(n-1) is prime, a(n)=a(n-1)+1 otherwise.
  • A080736 (program): Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler’s function A000010.
  • A080737 (program): a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.
  • A080746 (program): Inverse Aronson transform of lower Wythoff sequence A000201.
  • A080750 (program): a(n) = largest number greater than a(n-1) such that the first n terms of the sequence contain a total of a(n) base-10 digits.
  • A080751 (program): a(n) is smallest number greater than a(n-1) such that the sequence contains a total of a(n) base 10 digits + commas through n terms (assuming one comma between each pair of terms).
  • A080752 (program): a(1)=1; a(2)=8; for n > 2, a(n) is smallest integer greater than a(n-1) that satisfies the condition “n is in the sequence if and only if a(n) is odd.”.
  • A080753 (program): a(1)=2; for n > 1, a is the second-smallest positive integer greater than a(n-1) such that the condition “n is in the sequence if and only if a(n) is odd” is satisfied.
  • A080754 (program): a(n) = ceiling(n*(1+sqrt(2))).
  • A080755 (program): a(n) = ceiling(n*(1+1/sqrt(2))).
  • A080756 (program): A positive integer n is in this sequence if it has infinitely many multiples that have exactly n positive divisors each.
  • A080757 (program): First differences of Beatty sequence A022838(n) = floor(n sqrt(3)).
  • A080763 (program): Exchange 1’s and 2’s in the eta-sequence A006337.
  • A080764 (program): First differences of A049472, floor(n/sqrt(2)).
  • A080765 (program): Integers m such that m+1 divides lcm(1 through m).
  • A080770 (program): a(n)=[e*(n+3)! ]-(n+3)(n+2)(n+1)(n)*[e*(n-1)! ].
  • A080773 (program): In binary representation: sum of number of 1’s in prime factors of n (with repetition).
  • A080774 (program): Numbers with two prime factors: (4*i+1)*(4*j+3).
  • A080775 (program): Number of n X n monomial matrices whose nonzero entries are unit Hurwitz quaternions.
  • A080776 (program): Oscillating sequence which rises to 2^(k-1) in k-th segment (k>=1) then falls back to 0.
  • A080782 (program): a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
  • A080787 (program): a(1)=a(2)=1; a(n) = a(n-1) + last decimal digit of a(n-2).
  • A080791 (program): Number of nonleading 0’s in binary expansion of n.
  • A080795 (program): Number of minimax trees on n nodes.
  • A080799 (program): Number of divide by 2 and add 1 operations required to reach …,7,8,4,2,1 when started at n.
  • A080800 (program): Similar to A080799 but count only division steps.
  • A080801 (program): Similar to A080799 but count only addition steps.
  • A080804 (program): Least number of connected subgraphs of the binary cube GF(2)^n such that every vertex of GF(2)^n lies in at least one of the subgraphs and no two vertices lie in the same set of subgraphs (such a collection is called an identifying set).
  • A080806 (program): Positive integer values of n such that 6n^2-5 is a square.
  • A080813 (program): Lexicographically largest overlap-free binary sequence.
  • A080817 (program): Leading diagonal of triangle in A080818.
  • A080819 (program): Row sums from triangle in A080818.
  • A080820 (program): Least m such that m^2 >= n*(n+1)/2.
  • A080827 (program): Rounded up staircase on natural numbers.
  • A080833 (program): E.g.f.: exp( x/(1 - x - x^2) ).
  • A080834 (program): E.g.f. exp( x/(1 - x - x^2) ) / (1 - x - x^2).
  • A080838 (program): Orchard crossing number of complete bipartite graph K_{1,n}.
  • A080843 (program): Tribonacci word: limit S(infinity), where S(0) = 0, S(1) = 0,1, S(2) = 0,1,0,2 and for n >= 0, S(n+3) = S(n+2) S(n+1) S(n).
  • A080844 (program): G.f. is F^2, where F is g.f. for Fibonacci word (A003849).
  • A080845 (program): G.f. is 1/F, where x*F is g.f. for Fibonacci word (A003849).
  • A080846 (program): Fixed point of the morphism 0->010, 1->011, starting from a(1) = 0.
  • A080847 (program): a(n) = mu(n)+2, where mu is the Moebius function (A008683).
  • A080848 (program): a(n) = n*(mu(n)+2), where mu is the Moebius function (A008683).
  • A080849 (program): (mu(n)+2)*n^2, where mu is the Moebius function (A008683).
  • A080851 (program): Square array of pyramidal numbers, read by antidiagonals.
  • A080852 (program): Square array of 4D pyramidal numbers, read by antidiagonals.
  • A080853 (program): Square array of generalized polygonal numbers, read by antidiagonals.
  • A080854 (program): Numbers which can be expressed as the sum of two distinct primes in exactly five ways.
  • A080855 (program): a(n) = (9*n^2 - 3*n + 2)/2.
  • A080856 (program): a(n) = 8*n^2 - 4*n + 1.
  • A080857 (program): (25*n^2 - 15*n + 2)/2.
  • A080859 (program): a(n) = 6*n^2 + 4*n + 1.
  • A080860 (program): 10*n^2 + 5*n + 1.
  • A080861 (program): 15*n^2 + 6*n + 1.
  • A080862 (program): Numbers which can be expressed as the sum of two distinct primes in exactly six ways.
  • A080863 (program): Numbers n such that spf(n)-2 = spf(n+2), where spf=A020639 (smallest prime factor).
  • A080864 (program): Numbers n such that spf(n)+2 = spf(n-2), where spf=A020639 (smallest prime factor).
  • A080871 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 3, given a(0)=a(1)=1, a(2)=4.
  • A080872 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.
  • A080873 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 5, given a(0)=a(1)=1, a(2)=2.
  • A080874 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 5, given a(0)=a(1)=1, a(2)=3.
  • A080875 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 5, given a(0)=a(1)=1, a(2)=6.
  • A080876 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0) = 1, a(1) = 1, and a(2) = 1.
  • A080877 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.
  • A080878 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=3.
  • A080879 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=6.
  • A080880 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=2.
  • A080881 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=10.
  • A080882 (program): a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.
  • A080883 (program): Distance of n to next square.
  • A080884 (program): Sum of (0,1) versions of Thue-Morse word (A010060) and Fibonacci word (A003849).
  • A080885 (program): Boolean AND of (0,1) versions of Thue-Morse word (A010060) and Fibonacci word (A003849).
  • A080886 (program): Boolean OR of (0,1) versions of Thue-Morse word (A010060) and Fibonacci word (A003849).
  • A080887 (program): Boolean XOR of (0,1) versions of Thue-Morse word (A010060) and Fibonacci word (A003849).
  • A080888 (program): Number of compositions into Fibonacci numbers (1 counted as two distinct Fibonacci numbers).
  • A080889 (program): Expansion of 1/(1+Sum_{k>0} (-x)^Fibonacci(k)).
  • A080891 (program): Period 5: repeat [0, 1, -1, -1, 1].
  • A080893 (program): E.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.
  • A080894 (program): Expansion of the exponential series exp( x M(x) ) = exp( (1-sqrt(1-2x-3x^2))/(2x) ), where M(x) is the ordinary generating series of the Motzkin numbers A001006.
  • A080896 (program): Expansion of the exponential series exp( x * T(x) ) = exp( x / sqrt(1 - 2*x - 3*x^2) ), where T(x) is the ordinary generating series of the central trinomial coefficients (A002426).
  • A080902 (program): a(1)=1, a(n)=a(n-1)+2 if (n and n+3 are in the sequence), a(n)=a(n-1)+3 otherwise.
  • A080903 (program): a(1)=1; for n>1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
  • A080907 (program): Numbers whose aliquot sequence terminates in a 1.
  • A080920 (program): a(n) = 2a(n-1) + 35a(n-2), a(0)=0, a(1)=1.
  • A080921 (program): a(n) = 2*a(n-1) + 48*a(n-2), a(0)=0, a(1)=1.
  • A080923 (program): First differences of A003946.
  • A080924 (program): Jacobsthal gap sequence.
  • A080925 (program): Binomial transform of Jacobsthal gap sequence (A080924).
  • A080926 (program): Partial sums of A080925.
  • A080928 (program): Triangle T(n,k) read by rows: T(n,k) = Sum_{i=0..n} C(n,2i)*C(2i,k).
  • A080929 (program): Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).
  • A080930 (program): a(n) = 2^(n-3)*(n+2)*(n+3)*(n+4)/3.
  • A080937 (program): Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.
  • A080938 (program): Number of Catalan paths (nonnegative, starting and ending at 0, step +-1) of 2*n steps with all values less than or equal to 7.
  • A080940 (program): Smallest proper divisor of n which is a suffix of n in binary representation; a(n) = 0 if no such divisor exists.
  • A080950 (program): Number of numbers that differ from n in binary representation by exactly one edit-operation: deletion, insertion, or substitution.
  • A080951 (program): Sequence associated with recurrence a(n) = 2*a(n-1) + k*(k+2)*a(n-2).
  • A080952 (program): a(n) = 2^(n-4)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)/15.
  • A080954 (program): E.g.f. exp(5x)/(1-x).
  • A080955 (program): Square array of numbers related to the incomplete gamma function, read by antidiagonals.
  • A080956 (program): a(n) = (n+1)*(2-n)/2.
  • A080957 (program): Expansion of (5 - 9*x + 6*x^2)/(1-x)^4.
  • A080958 (program): a(n) = n!*(2/1 - 3/2 + 4/3 - … + s*(n+1)/n), where s = (-1)^(n+1).
  • A080960 (program): Third binomial transform of A010685 (period 2: repeat 1,4).
  • A080961 (program): Fourth binomial transform of A010686 (period 2: repeat 1,5).
  • A080962 (program): 5th binomial transform of the periodic sequence (1,6,1,1,6,1…).
  • A080966 (program): Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.
  • A080978 (program): a(n) = 2*A006046(n) + 1.
  • A080982 (program): Smallest k such that the k-th triangular number has n^2 as divisor.
  • A080983 (program): Smallest triangular number having n^2 as divisor.
  • A080995 (program): Characteristic function of generalized pentagonal numbers A001318.
  • A080996 (program): Special values of the hypergeometric function 3F1: a(n) = binomial(n,2) * hypergeom([1,-n+1,-n+2],[3],1).
  • A081002 (program): a(n) = Fibonacci(4n) + 1, or Fibonacci(2n-1)*Lucas(2n+1).
  • A081003 (program): a(n) = Fibonacci(4n+1) + 1, or Fibonacci(2n+1)*Lucas(2n).
  • A081004 (program): a(n) = Fibonacci(4n+2) + 1, or Fibonacci(2n+2)*Lucas(2n).
  • A081005 (program): a(n) = Fibonacci(4n+3) + 1, or Fibonacci(2n+1)*Lucas(2n+2).
  • A081006 (program): a(n) = Fibonacci(4n) - 1, or Fibonacci(2n+1)*Lucas(2n-1).
  • A081007 (program): a(n) = Fibonacci(4n+1) - 1, or Fibonacci(2n)*Lucas(2n+1).
  • A081008 (program): a(n) = Fibonacci(4n+2) - 1, or Fibonacci(2n)*Lucas(2n+2).
  • A081009 (program): a(n) = Fibonacci(4n+3) - 1, or Fibonacci(2n+2)*Lucas(2n+1).
  • A081010 (program): a(n) = Fibonacci(4n+1) + 2, or Fibonacci(2n-1)*Lucas(2n+2).
  • A081011 (program): a(n) = Fibonacci(4n+3) + 2, or Fibonacci(2n+3)*Lucas(2n).
  • A081012 (program): a(n) = Fibonacci(4n+1) - 2, or Fibonacci(2n+2)*Lucas(2n-1).
  • A081013 (program): a(n) = Fibonacci(4*n+3) - 2, or Fibonacci(2*n)*Lucas(2*n+3).
  • A081014 (program): a(n) = Lucas(4*n+1) + 1, or Lucas(2*n)*Lucas(2*n+1).
  • A081015 (program): a(n) = Lucas(4n+3) + 1, or 5*Fibonacci(2n+1)*Fibonacci(2n+2).
  • A081016 (program): a(n) = (Lucas(4*n+3) + 1)/5, or Fibonacci(2*n+1)*Fibonacci(2*n+2), or A081015(n)/5.
  • A081017 (program): a(n) = Lucas(4n+1) - 1, or 5*Fibonacci(2n)*Fibonacci(2n+1).
  • A081018 (program): a(n) = (Lucas(4n+1)-1)/5, or Fibonacci(2n)*Fibonacci(2n+1), or A081017(n)/5.
  • A081019 (program): a(n) = Lucas(4n+3) - 1, or Lucas(2n+1)*Lucas(2n+2).
  • A081020 (program): Even order Taylor coefficients at x = 0 of exp(-x^2/(x^2-2)), odd order coefficients being equal to zero.
  • A081021 (program): Even order Taylor coefficients at x = 0 of exp( (sqrt(2)-sqrt(-2*x^2+2))/(-2*x^2+2)^(1/2) ), odd order coefficients being equal to zero.
  • A081026 (program): Variation on Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = smallest (n odd) or largest (n even) number > a(n-1) that is a unique sum of two distinct earlier terms.
  • A081031 (program): Positions of white keys on piano keyboard, starting with A0 = the 1st key.
  • A081032 (program): Positions of black keys on piano keyboard, starting with A0 = the 1st key.
  • A081033 (program): 6th binomial transform of the periodic sequence (1,7,1,1,7,1…).
  • A081034 (program): 7th binomial transform of the periodic sequence (1,8,1,1,8,1…).
  • A081035 (program): 8th binomial transform of the periodic sequence (1,9,1,1,9,1…).
  • A081036 (program): 9th binomial transform of the periodic sequence (1,10,1,1,10,1…).
  • A081037 (program): Inverse binary transform of A027656.
  • A081038 (program): 3rd binomial transform of (1,2,0,0,0,0,0,0,…).
  • A081039 (program): 4th binomial transform of (1,3,0,0,0,0,0,…..).
  • A081040 (program): 5th binomial transform of (1,4,0,0,0,0,…).
  • A081041 (program): 6th binomial transform of (1,5,0,0,0,0,0,0,…).
  • A081042 (program): 7th binomial transform of (1,6,0,0,0,0,0,0,…).
  • A081043 (program): 8th binomial transform of (1,7,0,0,0,0,0,…).
  • A081044 (program): 9th binomial transform of (1,8,0,0,0,0,0,0,…..).
  • A081045 (program): 10th binomial transform of (1,9,0,0,0,0,0,…).
  • A081046 (program): Difference of the first two Stirling numbers of the first kind.
  • A081047 (program): Difference of Stirling numbers of the first kind.
  • A081048 (program): Signed Stirling numbers of the first kind.
  • A081049 (program): Generalized Stirling numbers of the first kind.
  • A081050 (program): Generalized Stirling numbers of the first kind.
  • A081051 (program): Stirling numbers of the first kind.
  • A081052 (program): Difference of Stirling numbers of the first kind.
  • A081055 (program): Number of partitions of 2n in which no parts are multiples of 4.
  • A081056 (program): Number of partitions of 2n+1 in which no parts are multiples of 4.
  • A081057 (program): E.g.f.: Sum_{n>=0} a(n)*x^n/n! = {Sum_{n>=0} F(n+1)*x^n/n!}^2, where F(n) is the n-th Fibonacci number.
  • A081061 (program): Union of 3-smooth numbers and prime powers.
  • A081062 (program): Neither 3-smooth numbers nor prime powers.
  • A081063 (program): Number of numbers <= n that are 3-smooth or prime powers.
  • A081065 (program): Numbers n such that n^2 = (1/3)*(n+floor(sqrt(3)*n*floor(sqrt(3)*n))).
  • A081066 (program): Even order Taylor expansion coefficients at x=0 of exp(exp(x^2/2)-1), odd order coefficients being equal to zero.
  • A081067 (program): Lucas(4n+2)+2, or 5*Fibonacci(2n+1)^2.
  • A081068 (program): a(n) = (Lucas(4*n+2) + 2)/5, or Fibonacci(2*n+1)^2, or A081067(n)/5.
  • A081069 (program): Lucas(4n)+2, or Lucas(2n)^2.
  • A081070 (program): Lucas(4n)-2, or 5*Fibonacci(2n)^2.
  • A081071 (program): a(n) = Lucas(4*n+2)-2, or Lucas(2*n+1)^2.
  • A081072 (program): Fibonacci(4n) + 3, or Fibonacci(2n+2)*Lucas(2n-2).
  • A081073 (program): Fibonacci(4n+2)+3, or Fibonacci(2n-1)*Lucas(2n+3).
  • A081074 (program): Fibonacci(4n)-3, or Fibonacci(2n-2)*Lucas(2n+2).
  • A081075 (program): a(n) = Fibonacci(4n+2) - 3.
  • A081076 (program): a(n) = Lucas(4n) + 3, or 5*Fibonacci(2n-1)*Fibonacci(2n+1).
  • A081077 (program): a(n) = Lucas(4*n+2) + 3, or Lucas(2*n)*Lucas(2*n+2).
  • A081078 (program): a(n) = Lucas(4n) - 3, or Lucas(2n-1)*Lucas(2n+1).
  • A081079 (program): Lucas(4n+2) - 3, or 5*Fibonacci(2n)*Fibonacci(2n+2).
  • A081083 (program): Numbers n such that rad(n+1)=rad(n)+1, where rad(m)=A007947(m) is the squarefree kernel of m.
  • A081085 (program): Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.
  • A081091 (program): Primes of the form 2^i + 2^j + 1, i > j > 0.
  • A081092 (program): Primes having in binary representation a prime number of 1’s.
  • A081094 (program): 4th differences of partition numbers A000041.
  • A081095 (program): 5th differences of partition numbers A000041.
  • A081105 (program): 5th binomial transform of (1,1,0,0,0,0,…..).
  • A081106 (program): 6th binomial transform of (1,1,0,0,0,0,…).
  • A081107 (program): 7th binomial transform of (1,1,0,0,0,0,…….).
  • A081108 (program): 8th binomial transform of (1,1,0,0,0,0,………).
  • A081109 (program): 9th binomial transform of (1,1,0,0,0,0,0,….).
  • A081113 (program): Number of paths of length n-1 a king can take from one side of an n X n chessboard to the opposite side.
  • A081114 (program): Triangle read by rows: T(n,k) = n*T(n-1,k) + n - k starting at T(n,n)=0.
  • A081115 (program): (p^2 - 1)/12 where p > 3 runs through the primes.
  • A081117 (program): Differences of Beatty sequence for cube root of 2.
  • A081118 (program): Triangle of first n numbers per row having exactly n 1’s in binary representation.
  • A081122 (program): 10th binomial transform of (1,1,0,0,0,0,……).
  • A081123 (program): a(n) = floor(n/2)!.
  • A081124 (program): Binomial transform of floor(n/2)!.
  • A081125 (program): a(n) = n! / floor(n/2)!.
  • A081126 (program): Duplicate of A018191.
  • A081127 (program): 11th binomial transform of (0,1,0,0,0,0,0,……).
  • A081128 (program): 12th binomial transform of (0,1,0,0,0,0,0,0,…).
  • A081129 (program): Differences of Beatty sequence for cube root of 3.
  • A081131 (program): a(n) = n^(n-2) * binomial(n,2).
  • A081132 (program): a(n) = (n+1)^n*binomial(n+2,2).
  • A081133 (program): a(n) = n^n*binomial(n+2, 2).
  • A081134 (program): Distance to nearest power of 3.
  • A081135 (program): 5th binomial transform of (0,0,1,0,0,0, …).
  • A081136 (program): 6th binomial transform of (0,0,1,0,0,0, …).
  • A081138 (program): 8th binomial transform of (0,0,1,0,0,0, …).
  • A081139 (program): 9th binomial transform of (0,0,1,0,0,0,…).
  • A081140 (program): 10th binomial transform of (0,0,1,0,0,0,…).
  • A081141 (program): 11th binomial transform of (0,0,1,0,0,0,…).
  • A081142 (program): 12th binomial transform of (0,0,1,0,0,0,…).
  • A081143 (program): 5th binomial transform of (0,0,0,1,0,0,0,0,……).
  • A081144 (program): Starting at 1, four-fold convolution of A000400 (powers of 6).
  • A081147 (program): Differences of Beatty sequence for square root of 5.
  • A081168 (program): Differences of Beatty sequence for square root of 10.
  • A081172 (program): Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = 1, a(2) = 0.
  • A081178 (program): a(0)=1; for n>=1, a(n) = sum(7^k*N(n,k), k=0..n), where N(n,k)=(1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
  • A081179 (program): 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,…).
  • A081180 (program): 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,…).
  • A081181 (program): Staircase on Pascal’s triangle.
  • A081182 (program): 5th binomial transform of (0,1,0,2,0,4,0,8,0,16,…).
  • A081183 (program): 6th binomial transform of (0,1,0,2,0,4,0,8,0,16,…).
  • A081184 (program): 7th binomial transform of (0,1,0,2,0,4,0,8,0,16,…).
  • A081185 (program): 8th binomial transform of (0,1,0,2,0,4,0,8,0,16,…).
  • A081186 (program): 4th binomial transform of (1,0,1,0,1,…), A059841.
  • A081187 (program): 5th binomial transform of (1,0,1,0,1,…), A059841.
  • A081188 (program): 6th binomial transform of (1,0,1,0,1,…..), A059841.
  • A081189 (program): 7th binomial transform of (1,0,1,0,1,…), A059841.
  • A081190 (program): 8th binomial transform of (1,0,1,0,1,…..), A059841.
  • A081192 (program): 10th binomial transform of (1,0,1,0,1,……), A059841.
  • A081193 (program): a(n) = 6*a(n-1)-8*a(n-2) for n>1, a(0)=1, a(1)=9.
  • A081194 (program): a(n) = 8*a(n-1) -15*a(n-2), a(0)=1, a(1)=16.
  • A081195 (program): a(n) = 10*a(n-1)-24*a(n-2) for n>1, a(0)=1, a(1)=25.
  • A081196 (program): a(n) = (n+4)^n*binomial(n+2,2).
  • A081197 (program): Diagonal sums of A081130.
  • A081199 (program): 5th binomial transform of (0,1,0,1,…), A000035.
  • A081200 (program): 6th binomial transform of (0,1,0,1,0,1,…), A000035.
  • A081201 (program): 7th binomial transform of (0,1,0,1,0,1,….), A000035.
  • A081202 (program): 8th binomial transform of (0,1,0,1,0,1,….), A000035.
  • A081203 (program): 9th binomial transform of (0,1,0,1,0,1,…..), A000035.
  • A081204 (program): Staircase on Pascal’s triangle.
  • A081205 (program): Staircase on Pascal’s triangle.
  • A081207 (program): Main diagonal of number square A081206.
  • A081209 (program): a(n) = Sum_{k=0..n} (-1)^(n-k)*n^k.
  • A081215 (program): a(n) = (n^(n+1)+(-1)^n)/(n+1)^2.
  • A081216 (program): a(n) = (n^n-(-1)^n)/(n+1).
  • A081217 (program): Greatest squarefree number not exceeding n-th prime power.
  • A081219 (program): One sixtieth the product of primitive Pythagorean triangles’ sides whose odd values differ by 2.
  • A081221 (program): Number of consecutive numbers >= n having at least one square divisor > 1.
  • A081223 (program): Smallest k such that floor(k*gamma) begins with n (gamma=0.5772156649…).
  • A081239 (program): #{(i,j): mu(i)*mu(j) = 0, 1<=i,j<=n}, where mu=A008683 (Moebius function).
  • A081241 (program): Position in B of reversal of n-th term of B, where B is the logic-binary sequence, A007931.
  • A081242 (program): Left-to-right binary enumeration.
  • A081243 (program): a(n) = Mod[n+(Mod[Prime[n],3]-1),10]
  • A081245 (program): Number of days in months in the Haab year of Mayan/Mesoamerican calendars.
  • A081249 (program): Partial sums of A081134.
  • A081250 (program): Numbers k such that A081249(m)/m^2 has a local minimum for m = k.
  • A081251 (program): Numbers n such that A081249(m)/m^2 has a local maximum for m = n.
  • A081252 (program): Partial sums of A053646.
  • A081253 (program): Numbers k such that A081252(m)/m^2 has a local minimum for m = k.
  • A081254 (program): Numbers k such that A081252(m)/m^2 has a local maximum for m = k.
  • A081256 (program): Greatest prime factor of n^3 + 1.
  • A081257 (program): a(n) is the greatest prime factor of (n^3 - 1).
  • A081259 (program): a(n) is the smallest k such that C(3n,n) divides k!.
  • A081260 (program): a(1)=4; for n>1, a(n) is taken to be the third-smallest integer greater than a(n-1) such that the condition “n is a member of the sequence if and only if a(n) is odd” is satisfied.
  • A081261 (program): Start with a(1)=4; apply 4 -> 665, 5 -> 56665, 6 -> 566665; iterate.
  • A081265 (program): Triangle of coefficients of the polynomials a(n, x) = 2*a(n-1, x)+ x^2*a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.
  • A081266 (program): Staggered diagonal of triangular spiral in A051682.
  • A081267 (program): Diagonal of triangular spiral in A051682.
  • A081268 (program): Diagonal of triangular spiral in A051682.
  • A081270 (program): Diagonal of triangular spiral in A051682.
  • A081271 (program): Vertical of triangular spiral in A051682.
  • A081272 (program): Downward vertical of triangular spiral in A051682.
  • A081275 (program): Shallow diagonal of triangular spiral in A051682.
  • A081276 (program): Floor(n^3/8).
  • A081277 (program): Square array of unsigned coefficients of Chebyshev polynomials of the first kind.
  • A081278 (program): Binomial transform of Chebyshev polynomial coefficients A001793.
  • A081279 (program): Binomial transform of Chebyshev coefficients A001794.
  • A081280 (program): Binomial transform of Chebyshev coefficients A006974.
  • A081282 (program): Generalized centered polygonal numbers.
  • A081283 (program): An interleaved sequence of pyramidal and polygonal numbers.
  • A081284 (program): An interleaved sequence of pyramidal and polygonal numbers.
  • A081288 (program): a(n) is the minimal i such that A000108(i) > n.
  • A081289 (program): a(0) = 0, a(n) = A081293(A081288(n)-1).
  • A081290 (program): a(0) = 0, and for n >=1, a(n) = the largest Catalan number <= n.
  • A081291 (program): Complement of A072795.
  • A081293 (program): a(n) = A000108(n) + A014137(n).
  • A081294 (program): Expansion of (1-2*x)/(1-4*x).
  • A081295 (program): a(n) = (-1)^(n+1)* coefficient of x^n in Sum_{k>=1} x^k/(1+2*x^k).
  • A081297 (program): Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).
  • A081298 (program): Main diagonal of the square array A081297.
  • A081299 (program): Diagonal of square array A081297.
  • A081300 (program): Diagonal of square array A081297.
  • A081301 (program): Subdiagonal of square array A081297.
  • A081302 (program): Subdiagonal of square array A081297.
  • A081303 (program): gpf(m) - 2*spf(m), where gpf(m) is the greatest and spf(m) is the smallest prime factor of m (A006530, A020639).
  • A081304 (program): Number of numbers m <= n with prime factors less than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).
  • A081305 (program): Number of numbers m <= n with at least one prime factor greater than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).
  • A081306 (program): Numbers n with prime factors less than 2*spf(n), where spf(m) is the smallest prime factor of m (A020639).
  • A081307 (program): a(n) = (n+1)*tau(n) - sigma(n).
  • A081320 (program): Largest 3-smooth divisor of n-th Fibonacci number.
  • A081321 (program): a(n) = (2/3)*(2*n+1)*(2*n-1)!*binomial(3*n,2*n).
  • A081324 (program): Twice a square but not the sum of 2 distinct squares.
  • A081325 (program): sigma(n^2) modulo 4.
  • A081326 (program): Number of partitions of n into two 3-smooth numbers.
  • A081329 (program): Numbers having no representation as sum of two 3-smooth numbers.
  • A081332 (program): Numbers having a unique partition into two 3-smooth numbers.
  • A081334 (program): sigma(2*n^2) modulo 4.
  • A081335 (program): a(n) = (6^n + 2^n)/2.
  • A081336 (program): a(n) = (7^n + 3^n)/2.
  • A081337 (program): (8^n+4^n)/2.
  • A081338 (program): (9^n+5^n)/2.
  • A081339 (program): Numbers n such that sigma(n^2) modulo 4 = 1.
  • A081340 (program): (5^n+(-1)^n)/2.
  • A081341 (program): Expansion of exp(3*x)*cosh(3*x).
  • A081342 (program): a(n) = (8^n + 2^n)/2.
  • A081343 (program): a(n) = (10^n + 4^n)/2.
  • A081345 (program): First row in maze arrangement of natural numbers A081344.
  • A081346 (program): First column in maze arrangement of natural numbers A081344.
  • A081347 (program): First column in maze arrangement of natural numbers.
  • A081348 (program): First row in maze arrangement of natural numbers.
  • A081350 (program): First column in maze array of natural numbers A081349.
  • A081351 (program): First row in square maze array of natural numbers A081349.
  • A081352 (program): Main diagonal of square maze arrangement of natural numbers A081349.
  • A081353 (program): Diagonal of square maze arrangement of natural numbers A081349.
  • A081354 (program): Numbers k such that sigma(k^2) modulo 4 = 3.
  • A081358 (program): E.g.f.: log((1+x) / (1-x)) / (2*(1-x)).
  • A081360 (program): Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.
  • A081362 (program): Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q.
  • A081367 (program): E.g.f.: exp(2*x)/sqrt(1-2*x).
  • A081369 (program): Binomial(n^2, n) reduced mod n^2.
  • A081371 (program): Binomial coefficients C[n,j] reduced modulo j, j=1,…n; read by rows, j=0 is omitted because of mod[n,0].
  • A081372 (program): Binomial coefficients C(n,j) reduced modulo j, j=1..n; read by rows, j=0 is omitted because of n mod 0. a(n) is the number of zero residues counted in n-th row.
  • A081374 (program): Size of “uniform” Hamming covers of distance 1, that is, Hamming covers in which all vectors of equal weight are treated the same, included or excluded from the cover together.
  • A081386 (program): Number of unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of those prime factors in C(2n,n), whose exponent equals one.
  • A081387 (program): Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.
  • A081391 (program): Numbers k such that the central binomial coefficient C(2k,k) has only one prime divisor whose exponent equals one.
  • A081392 (program): Numbers k such that the central binomial coefficient C(k, floor(k/2)) has only one prime divisor whose exponent is greater than one.
  • A081396 (program): Number of common prime factors (ignoring multiplicity) of sigma(n) = A000203(n) and phi(n) = A000010(n).
  • A081399 (program): Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).
  • A081400 (program): a(n) = d(n) - bigomega(n) - A005361(n).
  • A081401 (program): Pseudologarithm (A056239) of n!: a(n) = A056239(A000142(n)).
  • A081402 (program): a(n) = A008475(n!).
  • A081403 (program): a(n) = A008475(n^2).
  • A081404 (program): a(n) = A008475(prime(n)-1).
  • A081405 (program): a(n) = (n+1)*a(n-2) with a(0) = a(1) = 1.
  • A081406 (program): a(n) = (n+1)*a(n-3), a(0)=a(1)=a(2)=1 for n>1.
  • A081407 (program): 4th-order non-linear (“factorial”) recursion: a(0)=a(1)=a(2)=a(3)=1, a(n) = (n+1)*a(n-4).
  • A081408 (program): a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.
  • A081410 (program): a(n) = a(n-1) + a(n-2) + n (mod 3), with a(1)=a(2)=1.
  • A081411 (program): Partial product of prime gaps: a(n) = a(n-1)*(prime(n+1) - prime(n)).
  • A081412 (program): Largest prime divisors of differences between consecutive primes.
  • A081417 (program): A000720 applied to Pascal-triangle: Pi[C(n,j)], j,0..n and n=0,1,2,…
  • A081421 (program): Quotient after one division by 2 of numbers of the form 3^(2n) + 5^(2n).
  • A081422 (program): Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.
  • A081423 (program): Subdiagonal of array of n-gonal numbers A081422.
  • A081431 (program): RevBinary(RevDecimal(n)), where RevBinary(m) is the binary reversal of m (A030101) and RevDecimal(m) is the decimal reversal of m (A004086).
  • A081432 (program): RevDecimal(RevBinary(n)), where RevDecimal(m) is the decimal reversal of m (A004086) and RevBinary(m) is the binary reversal of m (A030101).
  • A081435 (program): Diagonal in array of n-gonal numbers A081422.
  • A081436 (program): Fifth subdiagonal in array of n-gonal numbers A081422.
  • A081437 (program): Diagonal in array of n-gonal numbers A081422.
  • A081438 (program): Diagonal in array of n-gonal numbers A081422.
  • A081439 (program): Expansion of exp(2*x)*cosh(x/sqrt(1 - x^2)).
  • A081440 (program): Expansion of e.g.f.: exp(x)*cosh(x/sqrt(1 - x^2)).
  • A081441 (program): a(n) = (2*n^3 - n^2 - n + 2)/2.
  • A081442 (program): Expansion of e.g.f.: cosh(x/sqrt(1-x^2)) (even powers).
  • A081443 (program): Binomial transform of expansion of cosh(sinh(x)).
  • A081445 (program): Smallest squares such that partial sums of the sequence plus 11 are primes.
  • A081446 (program): a(n) = sqrt( A081445(n) )/6.
  • A081447 (program): Smallest squares such that partial sums of the sequence plus 5 are primes.
  • A081448 (program): a(n) = sqrt( A081447(n) )/6.
  • A081449 (program): Smallest squares such that partial sums of the sequence plus 17 are primes.
  • A081450 (program): a(n) = sqrt(A081449(n))/6.
  • A081458 (program): Table T(m,n) = (3^m + 5^n)/2, for m, n = 0, 2, 4, 6, … read by antidiagonals downwards.
  • A081467 (program): a(n) = smallest (n+k) such that n divides the sum {(n+1)+ (n+2) + … (n+k)} or n divides kn + k(k+1)/2.
  • A081468 (program): a(n) is the smallest multiple of n of the type k*n + k*(k+1)/2, i.e., the smallest sum (n+1) to (n+k) which is divisible by n.
  • A081469 (program): a(n) = A081468(n)/n.
  • A081472 (program): a(n) = the smallest (n+k) such that the LCM of numbers from (n+1) to (n+k) is a multiple of n.
  • A081475 (program): Consider the mapping f(x/y) = (x+y)/(2xy) where x/y is a reduced fraction. Beginning with x_0 = 1 and y_0 = 2, repeated application of this mapping produces a sequence of fractions x_n/y_n; a(n) is the n-th numerator.
  • A081476 (program): Consider the mapping f(x/y) = (x+y)/(2xy) where x/y is a reduced fraction. Beginning with x_0 = 1 and y_0 = 2, repeated application of this mapping produces a sequence of fractions x_n/y_n; a(n) is the n-th denominator.
  • A081477 (program): Complement of A086377.
  • A081478 (program): Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,… Sequence contains the denominators.
  • A081489 (program): a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3).
  • A081490 (program): Leading term of n-th row of A081491.
  • A081491 (program): Triangle read by rows in which the n-th row contains n terms of an arithmetic progression with a common difference of (n-1) and the first term of (n+1)-th row is 1 more than the last term of the n-th row.
  • A081492 (program): Sum of terms in n-th row of A081491.
  • A081493 (program): Triangle read by rows in which the n-th row begins with n and contains n terms of an Arithmetic progression with a common difference of (n-1).
  • A081494 (program): Start with Pascal’s triangle; form a triangle by sliding down n steps from top on both sides and including the horizontal row, deleting the inner numbers; a(n) = sum of entries on perimeter of triangle.
  • A081495 (program): Start with Pascal’s triangle; form a rhombus by sliding down n steps from top on both sides then sliding down inwards to complete the rhombus and then deleting the inner numbers; a(n) = sum of entries on perimeter of rhombus.
  • A081496 (program): Start with Pascal’s triangle; a(n) is the sum of the numbers on the periphery of the n-th central rhombus containing exactly 4 numbers.
  • A081498 (program): Consider the triangle in which the n-th row starts with n, contains n terms and the difference of successive terms is 1,2,3,… up to n-1. Sequence gives row sums.
  • A081499 (program): Sum at 45 degrees to horizontal in triangle of A081498.
  • A081500 (program): In the following triangle the n-th row begins with n and contains n-1 smallest numbers coprime to n and greater than n. Sequence gives the leading diagonal.
  • A081501 (program): In the following triangle the n-th row begins with n and contains n-1 smallest numbers coprime to n and greater than n. Sequence gives the row sums.
  • A081502 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.
  • A081503 (program): Number of steps to reach a single digit when map in A081502 is iterated.
  • A081515 (program): Sum of terms in n-th row of A081517.
  • A081516 (program): Final term in n-th row of A081517.
  • A081518 (program): Final term in row n of A081520.
  • A081519 (program): Sum of terms in row n of A081520.
  • A081523 (program): Sum of terms in row n of A081521.
  • A081524 (program): a(n) = A081523(n)/n.
  • A081528 (program): a(n) = n*lcm{1,2,…,n}.
  • A081530 (program): a(n) = running sum of the first n harmonic numbers, multiplied by the LCM of 1..n.
  • A081534 (program): Numbers k such that 1 + 2 … + k divides lcm(1,2,…,k), or, in other words, lcm(1,2,…,k) is a multiple of the k-th triangular number.
  • A081539 (program): Triangle read by rows in which the n-th row contains the n numbers in increasing order formed by the concatenation of first n-1 numbers. (The digits of the numbers with 2 or more digits are taken as one entity.) First row is taken to be 0.
  • A081541 (program): Triangle read by rows: the n-th row contains n numbers sorted in decreasing value, each build by dropping a different number from the sequence [n,n-1,n-2,….,1] and concatenating the n-1 others. By definition the first row contains 0.
  • A081543 (program): G.f.: Sum_{k >= 1} x^k/(1-x^k)^(k+1).
  • A081551 (program): Triangle, read by rows, in which the n-th row contains n smallest n-digit numbers.
  • A081552 (program): Leading terms of rows in A081551.
  • A081553 (program): Sum of n-th row of A081551.
  • A081554 (program): a(n) = sqrt(2)*( (3+2*sqrt(2))^n - (3-2*sqrt(2))^n ).
  • A081555 (program): a(n) = 6*a(n-1) - a(n-2) - 4, a(0)=3, a(1)=7.
  • A081557 (program): Binomial transform of expansion of exp(cosh(x)), A005046.
  • A081559 (program): Expansion of e.g.f.: exp(cosh(2*x)-1), even powers only.
  • A081560 (program): Binomial transform of expansion of exp(cosh(2*x)).
  • A081561 (program): Second binomial transform of expansion of exp(cosh(2*x)).
  • A081562 (program): Binomial transform of expansion of exp(2cosh(x)), A000807.
  • A081565 (program): Binomial transform of expansion of exp(3cosh(x)).
  • A081567 (program): Second binomial transform of F(n+1).
  • A081568 (program): Third binomial transform of Fibonacci(n+1).
  • A081569 (program): Fourth binomial transform of F(n+1).
  • A081570 (program): Fifth binomial transform of F(n+1).
  • A081571 (program): Sixth binomial transform of F(n+1).
  • A081572 (program): Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.
  • A081574 (program): Fourth binomial transform of Fibonacci numbers F(n).
  • A081575 (program): Fifth binomial transform of Fibonacci numbers F(n).
  • A081576 (program): Square array of binomial transforms of Fibonacci numbers, read by antidiagonals.
  • A081577 (program): Pascal-(1,2,1) array read by antidiagonals.
  • A081578 (program): Pascal-(1,3,1) array.
  • A081579 (program): Pascal-(1,4,1) array.
  • A081580 (program): Pascal-(1,5,1) array.
  • A081581 (program): Pascal-(1,6,1) array.
  • A081582 (program): Pascal-(1,7,1) array.
  • A081583 (program): Third row of Pascal-(1,2,1) array A081577.
  • A081584 (program): Fourth row of Pascal-(1,2,1) array A081577.
  • A081585 (program): Third row of Pascal-(1,3,1) array A081578.
  • A081586 (program): Fourth row of Pascal-(1,3,1) array A081578.
  • A081587 (program): Third row of Pascal-(1,4,1) array A081579.
  • A081588 (program): Fourth row of the Pascal-(1,4,1) array A081579.
  • A081589 (program): Third row of Pascal-(1,5,1) array A081580.
  • A081590 (program): Fourth row of Pascal-(1,5,1) array A081580.
  • A081591 (program): Third row of Pascal-(1,6,1) array A081581.
  • A081592 (program): A self generating sequence: “there are n a(n)’s in the sequence”. Start with 1,2 and use the rule : “a(n)=k implies there are n following k’s (k is 1 or 2)”.
  • A081593 (program): Third row of Pascal-(1,7,1) array A081582.
  • A081594 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 2x+y.
  • A081595 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 4x+y.
  • A081596 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 5x+y.
  • A081597 (program): Let n = 10*x + y where 0 <= y <= 9, x >= 0. Then a(n) = 6*x + y.
  • A081598 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 7x+y.
  • A081599 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 8x+y.
  • A081600 (program): Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 9x+y.
  • A081601 (program): Numbers n such that 3 does not divide Sum_{k=0..n} binomial(2k,k) = A006134(n).
  • A081603 (program): Number of 2’s in ternary representation of n.
  • A081604 (program): Number of digits in ternary representation of n.
  • A081605 (program): Numbers having at least one 0 in their ternary representation.
  • A081606 (program): Numbers having at least one 1 in their ternary representation.
  • A081607 (program): Number of numbers <= n having at least one 0 in their ternary representation.
  • A081608 (program): Number of numbers <= n having no 0 in their ternary representation.
  • A081609 (program): Number of numbers <= n having at least one 1 in their ternary representation.
  • A081610 (program): Number of numbers <= n having at least one 2 in their ternary representation.
  • A081611 (program): Number of numbers <= n having no 2 in their ternary representation.
  • A081613 (program): Length of iteration list when Collatz-function is iterated with initial value n!=A000142[n].
  • A081614 (program): Subsequence of A005428 with state = 1.
  • A081615 (program): Subsequence of A005428 where state = 2.
  • A081617 (program): Smallest k such that (product of first n primes)*k+1 is divisible by the (n+1)-th prime. Also (A075306(n)-1)/A002110(n).
  • A081619 (program): Numbers whose divisors can be arranged as equilateral triangle.
  • A081620 (program): Least m with n*(n+1)/2 divisors.
  • A081622 (program): Number of 6-core partitions of n.
  • A081623 (program): Number of ways in which the points on an n X n square lattice can be equally occupied with spin “up” and spin “down” particles. If n is odd, we arbitrarily take the lattice to contain one more spin “up” particle than the number of spin “down” particles.
  • A081625 (program): a(n) = 2*5^n - 3^n.
  • A081626 (program): 2*6^n-4^n.
  • A081627 (program): 2*7^n-5^n.
  • A081628 (program): a(n) = 2*(-1)^n - (-5)^n.
  • A081630 (program): 2-(-3)^n.
  • A081631 (program): 2*2^n-(-2)^n.
  • A081632 (program): 2*3^n-(-1)^n.
  • A081652 (program): Greatest common divisor of n and sum of decimal digits of n-th prime.
  • A081653 (program): Greatest common divisor of sums of decimal digits of n and of n-th prime.
  • A081654 (program): a(n) = 2*4^n - 0^n.
  • A081655 (program): 2*5^n-1.
  • A081656 (program): 2*6^n-2^n.
  • A081657 (program): 2*7^n-3^n.
  • A081659 (program): a(n) = n + Fibonacci(n+1).
  • A081660 (program): n+A001045(n+1).
  • A081661 (program): Partial sums of A081660.
  • A081662 (program): Partial sums of n + Fibonacci(n+1).
  • A081663 (program): F(2n+1)+n*2^(n-1).
  • A081666 (program): n*3^(n-1)+A081567(n).
  • A081667 (program): a(n) = Fibonacci(binomial(n+2,2)).
  • A081668 (program): Expansion of 2sinh(x) + BesselI_0(2x).
  • A081669 (program): Expansion of exp(2x)+exp(x)BesselI_0(2x).
  • A081670 (program): 3^n-1+C(2n,n).
  • A081671 (program): Expansion of e.g.f. exp(4x) * I_0(2x).
  • A081672 (program): Expansion of exp(2x) - exp(0) + BesselI_0(2x).
  • A081673 (program): Expansion of exp(3*x) - exp(x)*(1-BesselI_0(2*x)).
  • A081674 (program): Generalized Poly-Bernoulli numbers.
  • A081675 (program): Generalized Poly-Bernoulli numbers.
  • A081676 (program): Largest perfect power <= n.
  • A081678 (program): a(n) = (4*6^n - 3*5^n - 3^n)/6.
  • A081679 (program): a(n)=(6^n-5^n-4^n-3^n+4*2^n)/2.
  • A081680 (program): A sum of decreasing powers.
  • A081681 (program): A sum of decreasing powers.
  • A081682 (program): (9^n-8^n-7^n-6^n+4*5^n)/2.
  • A081683 (program): (10^n-9^n-8^n-7^n+4*6^n)/2.
  • A081684 (program): 5^n-4^n-3^n-2^n+3.
  • A081685 (program): A sum of decreasing powers.
  • A081686 (program): A sum of decreasing powers.
  • A081687 (program): A sum of decreasing powers.
  • A081688 (program): 0 followed by A030124 - 1.
  • A081689 (program): A005228 - 1.
  • A081690 (program): From P-positions in a certain game.
  • A081691 (program): From P-positions in a certain game.
  • A081692 (program): Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n}, B_n = B_{n-1} + (A_n-A_{n-1})(A_n-A_{n-1}+1), where the mex of a set is the smallest nonnegative integer not in the set. Sequence gives A_n. B_n is in A081693.
  • A081693 (program): Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n}, B_n = B_{n-1} + (A_n-A_{n-1})(A_n-A_{n-1}+1), where the mex of a set is the smallest nonnegative integer not in the set. Sequence gives B_n. A_n is in A081692.
  • A081696 (program): Expansion of 1/(x + sqrt(1-4x)).
  • A081701 (program): a(n) = prime(n) * (prime(n) - 1)^(prime(n) - 1).
  • A081704 (program): Let f(0)=1, f(1)=t, f(n+1) = (f(n)^2+t^n)/f(n-1). f(t) is a polynomial with integer coefficients. Then a(n) = f(n) when t=3.
  • A081706 (program): Numbers n such that binary representation ends either in an odd number of ones followed by one zero or in an even number of ones.
  • A081707 (program): a(n) = tau(n) - bigomega(n) = A000005(n) - A001222(n).
  • A081708 (program): a(n) = a(n-1) + 64*a(n-2) starting with a(0) = 2 and a(1) = 1.
  • A081714 (program): a(n) = F(n)*L(n+1) where F=Fibonacci and L=Lucas numbers.
  • A081728 (program): Length of periods of Euler numbers modulo prime(n).
  • A081729 (program): Expansion of Sum(k>=0, x^(2^k)) + 1/(1+x). First differences of A007456 (gossip sequence) for n>1.
  • A081731 (program): a(n)=1/60*(A000364(n+2)-A000364(n)).
  • A081733 (program): Triangle read by rows, T(n,k) = 2^(n-k)*[x^k] Euler_polynomial(n, x), for n >= 0, k >= 0.
  • A081737 (program): a(n) = (n-1)*10 + n-th decimal digit of Pi=3.14159…
  • A081738 (program): a(n) = Sum_{2 <= p <= n, p prime} p^2.
  • A081742 (program): a(1)=1; then if n is a multiple of 3 a(n)=a(n/3)+1, if n is not a multiple of 3 but even a(n)=a(n/2)+1, a(n)=a(n-1)+1 otherwise.
  • A081743 (program): a(1)=1 then a(n)=a(n/2^k)+1 if n is even and 2^k is the largest power of 2 dividing n, a(n)=a(n-1) otherwise.
  • A081753 (program): a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.
  • A081754 (program): Numbers n such that the number of noncongruent solutions to x^(2^m) == 1 (mod n) is the same for any m>=1.
  • A081757 (program): Number of ways to write n as i*j+i-j, 0<i<j.
  • A081758 (program): Sum of prime factors (with repetition) of sum of prime factors (with repetition) of n.
  • A081759 (program): Numbers n such that 5n+6 is prime.
  • A081765 (program): Numbers k such that (k+2) divides 2^(k-1) - 1.
  • A081769 (program): a(n)-th term of the continued fraction for sum(k>=0,1/2^(2^k)) is 2.
  • A081770 (program): Numbers twice their squarefree kernel (A007947).
  • A081805 (program): a(n) = n minus (largest prime power in n factorization); a(1) = 0.
  • A081808 (program): Numbers n such that the largest prime power in the factorization of n equals phi(n).
  • A081810 (program): If n = p_1^e_1 * … * p_k^e_k, p_1 < … < p_k primes, then a(n) = Max{ p_i*e_i }.
  • A081812 (program): If n = p_1^e_1 * … * p_k^e_k, p_1 < … < p_k primes, then a(n) = Max{ Max(p_i, e_i) }.
  • A081814 (program): Deuteron-electron mass ratio.
  • A081821 (program): Rydberg constant, in inverse meters.
  • A081823 (program): Decimal expansion of the elementary charge e in coulomb (C).
  • A081832 (program): a(1)=a(2)=1, a(n) = a(n+1-2*a(n-1)) + a(n-2*a(n-2)).
  • A081834 (program): a(1)=1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.
  • A081835 (program): a(1)=1, a(n) = a(n-1) + 5 if n is already in the sequence, a(n) = a(n-1) + 4 otherwise.
  • A081839 (program): a(1)=0, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.
  • A081840 (program): a(1)=0, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A081841 (program): a(1)=0, a(n)=a(n-1)+3 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
  • A081842 (program): a(1)=0, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.
  • A081843 (program): a(1)=0, a(n)=a(n-1)+5 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.
  • A081848 (program): Number of numbers whose base-3/2 expansion (see A024629) has n digits.
  • A081853 (program): Consider recurrence b(0) = (2n+1)/2, b(n) = b(n-1)*ceiling(b(n-1)); sequence gives first integer reached.
  • A081854 (program): (8*n-3)*(4*n-1)*(8*n^2-5*n+1).
  • A081857 (program): Jacobsthal sequence (A001045) as represented in base 4.
  • A081858 (program): Numbers k such that 2*k+1 divides 2^k-1.
  • A081861 (program): (1/24)*(sigma_3(2n-1)-sigma_1(2n-1)).
  • A081862 (program): (1/168)*(sigma_7(2n-1)-sigma_1(2n-1)).
  • A081863 (program): Largest integer m such that m divides (sigma_(2n+1)(2k-1)-sigma(2k-1)) for all k>=1.
  • A081864 (program): Sum of 5th powers of the divisors of odd numbers: a(n) = sigma_5(2n-1).
  • A081865 (program): a(n) = sigma_7(2n-1).
  • A081866 (program): a(n)=sigma_9(2n-1).
  • A081867 (program): a(n)=sigma_11(2n-1).
  • A081869 (program): a(1)=2; for n>1, a(n)=2*a(n-1)-1 if that number is composite, a(n)=a(n-1)+1 otherwise.
  • A081870 (program): a(1)=1, a(n)=2*a(n-1)+1 if that number is not squarefree, a(n)=a(n-1)+1 otherwise.
  • A081871 (program): a(1)=1, a(n)=2*a(n-1)+1 if that number is composite, a(n)=a(n-1)+1 otherwise.
  • A081875 (program): a(n) = Sum_{d|n} phi(n/d)*C(2*d,d)/2.
  • A081890 (program): A sum of decreasing powers.
  • A081891 (program): A sum of decreasing powers.
  • A081892 (program): Second binomial transform of C(n+2,2).
  • A081893 (program): Third binomial transform of C(n+2,2).
  • A081894 (program): Fourth binomial transform of C(n+2,2).
  • A081895 (program): Second binomial transform of binomial(n+3, 3).
  • A081896 (program): A sequence related to binomial(n+3, 3).
  • A081897 (program): A sequence related to binomial(n+3, 3).
  • A081898 (program): A sequence related to binomial(n+4, 4).
  • A081899 (program): A sequence related to binomial(n+4, 4).
  • A081900 (program): A sequence related to binomial(n+4, 4).
  • A081901 (program): A sequence related to binomial(n+5, 5).
  • A081902 (program): A sequence related to binomial(n+5, 5).
  • A081903 (program): A sequence related to binomial(n+5, 5).
  • A081904 (program): A sequence related to binomial(n+6, 6).
  • A081905 (program): A sequence related to binomial(n+6, 6).
  • A081906 (program): A sequence related to binomial(n+6, 6).
  • A081907 (program): A sequence related to binomial(n+2, 2).
  • A081908 (program): a(n) = 2^n*(n^2 - n + 8)/8.
  • A081909 (program): a(n) = 3^n(n^2 - n + 18)/18.
  • A081910 (program): 4^n*(n^2-n+32)/32.
  • A081911 (program): a(n) = 5^n*(n^2 - n + 50)/50.
  • A081912 (program): a(n) = 6^n*(n^2 - n + 72)/72.
  • A081913 (program): a(n) = 2^n*(n^3 - 3n^2 + 2n + 48)/48.
  • A081914 (program): a(n) = 3^n*(n^3 - 3n^2 + 2n + 162)/162.
  • A081915 (program): a(n) = 4^n*(n^3 - 3n^2 + 2n + 384)/384.
  • A081916 (program): a(n) = 5^n*(n^3 - 3n^2 + 2n + 750)/750.
  • A081917 (program): a(0)=1, a(n)= n^(n-2)(7n^2-3n+2)/6 (n>0).
  • A081918 (program): a(0) = 1; a(n) = n^(n-1)(3n-1)/2 (n>0)
  • A081919 (program): E.g.f.: exp(x)/sqrt(1-x^2).
  • A081920 (program): Expansion of exp(2x)/sqrt(1-x^2).
  • A081921 (program): Expansion of exp(3x)/sqrt(1-x^2).
  • A081922 (program): Expansion of exp(4x)/sqrt(1-x^2).
  • A081923 (program): Expansion of exp(2x)/(1-x)^2.
  • A081924 (program): E.g.f.: exp(3*x)/(1-x)^2.
  • A081933 (program): a(1) = 1, a(n) is the smallest number coprime to n and beginning with a(n-1).
  • A081942 (program): a(1) = 1, a(n) = smallest number greater than a(n-1) such that a(n-1)*a(n) + 1 is prime.
  • A081946 (program): a(n) = Sum_{i=1..n} floor(r*floor(i/r)), where r=sqrt(2).
  • A081954 (program): Triangle read by rows: T(n, k) = 2^(n-k)*3^k, n >= 1, 0 <= k < n.
  • A081955 (program): a(n) = 2^r*3^s where r = n(n+1)/2 and s = n(n-1)/2.
  • A081971 (program): Consider the harmonic progression 1,1/2,1/3,1/4,1/5,…, group the terms such that the n-th group contains n members like this (1/1),(1/2,1/3),(1/4,1/5,1/6), (1/7,1/8,1/9,1/10),… a(n) = the numerator of the reduced rational sum of the terms of the n-th group.
  • A081972 (program): Consider the geometric progression 1,1/2,1/4,1/8,1/16,1/32,1/64,… Group the terms such that the n-th group contains n terms like this (1/1),(1/2,1/4),(1/8,1/16,1,32),(1/64,1/128,1/256,1/512),… a(n) = floor[1/s(n)] where s(n) is the sum of the members of the n-th group.
  • A081988 (program): Product of digits + 1 is a prime.
  • A081989 (program): Product of digits + 1 is a square greater than 1.
  • A082010 (program): a(n) = n/2 if n is even, otherwise floor(8*n/5)+1.
  • A082019 (program): Diagonal of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.
  • A082020 (program): Decimal expansion of 15/Pi^2.
  • A082022 (program): In the following square array a(i,j) = Least Common Multiple of i and j. Sequence contains the product of the terms of the n-th antidiagonal.
  • A082023 (program): Number of partitions of n into 2 parts which are not relatively prime.
  • A082028 (program): Expansion of exp(x)*(1+x)/(1-x)^2.
  • A082029 (program): Expansion of exp(2*x)*(1+x)/(1-x)^2.
  • A082030 (program): Expansion of e.g.f. exp(x)/(1-x)^3.
  • A082031 (program): Expansion of e.g.f. exp(2*x)/(1-x)^3.
  • A082032 (program): Expansion of e.g.f.: exp(2*x)/(1-2*x).
  • A082033 (program): a(n) = (3n+1)*n!.
  • A082034 (program): a(n) = (4*n + 1)*n!.
  • A082035 (program): a(n) = (4n^2+2n+1) * n!.
  • A082036 (program): a(n) = (9*n^2+3*n+1) * n!.
  • A082037 (program): A square array of linear-factorial numbers, read by antidiagonals.
  • A082038 (program): A square array of quadratic-factorial numbers, read by antidiagonals.
  • A082039 (program): Symmetric square array defined by T(n,k)=(k^2*n^2 + kn + 1), read by antidiagonals.
  • A082040 (program): a(n) = 9*n^2 + 3*n + 1.
  • A082041 (program): 16n^2+4n+1.
  • A082042 (program): (n^2+1)n!.
  • A082043 (program): A symmetric square array of numbers, read by antidiagonals.
  • A082044 (program): Main diagonal of A082043: a(n) = n^4 + 2n^2 + 1.
  • A082045 (program): Diagonal sums of number array A082043.
  • A082046 (program): A symmetric square array of numbers, read by antidiagonals.
  • A082047 (program): Diagonal sums of number array A082046.
  • A082048 (program): a(n) = least number greater than n having greater smallest prime factor than that of n.
  • A082050 (program): Sum of divisors of n that are not of the form 3k+1.
  • A082051 (program): Sum of divisors of n that are not of the form 3k+2.
  • A082052 (program): Sum of divisors of n that are not of the form 4k+1.
  • A082053 (program): Sum of divisors of n that are not of the form 4k+3.
  • A082054 (program): Sum of common prime divisors (without multiplicity) of sigma(n) and phi(n).
  • A082055 (program): Product of common prime-divisors (without multiplicity) of sigma(n) and phi(n).
  • A082061 (program): Greatest common prime-divisor of n and phi(n)=A000010(n); a(n)=1 if no common prime-divisor exists.
  • A082062 (program): Greatest common prime-divisor of n and sigma(n)=A000203(n); a(n)=1 if no common prime-divisor exists.
  • A082063 (program): Greatest common prime divisor of n and sigma_2(n) = A001157(n), or 1 if the two are relatively prime.
  • A082064 (program): Greatest common prime-divisor of phi(n) and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.
  • A082066 (program): Greatest common prime-divisor of sigma_1(n)=A000203(n) and sigma_2(n)=A001157(n); a(n)=1 if no common prime-divisor exists.
  • A082067 (program): Smallest prime that divides n and phi(n)=A000010(n), or 1 if n and phi(n) are relatively prime.
  • A082068 (program): Smallest common prime-divisor of n and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.
  • A082069 (program): Smallest common prime-divisor of n and Sigma_2(n) = A001157(n); a(n) = 1 if no common prime-divisor exists.
  • A082070 (program): Smallest prime that divides phi(n) and sigma(n) = A000203(n), or 1 if phi(n) and sigma(n) are relatively prime.
  • A082072 (program): Smallest prime that divides sigma(n) = A000203(n) and sigma_2(n) = A001157(n), or 1 if sigma(n) and sigma_2(n) are relatively prime.
  • A082073 (program): First difference set of primes with 4k+1 form: A002144.
  • A082074 (program): One quarter of first differences of primes of the form 4*k+1 (A002144).
  • A082075 (program): First differences of primes of the form 4*k+3 (A002145).
  • A082076 (program): First differences of primes of the form 4*k+3 (A002145), divided by 4.
  • A082081 (program): a(n) = fixed point when the pseudo-log function A008475[ ] is iterated.
  • A082083 (program): a(n)=A082081[n! ]=A082081[A000142[n]] Fixed points of iterated A008475 function initiated at factorials as initial values.
  • A082084 (program): a(n)=A029908[n! ]=A029908[A000142[n]] Fixed points of iterated A001414 function if started at factorials as initial values.
  • A082086 (program): Fixed points when A001414 is iterated and started at factorials of prime numbers.
  • A082087 (program): a(n) is the fixed point if function A008472 (= sum of prime factors with no repetition) is iterated when started at initial value n!.
  • A082088 (program): a(n) is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at initial value prime[n]!.
  • A082089 (program): a(n)-th prime is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at factorial of n-th prime.
  • A082090 (program): Length of iteration sequence if function A056239, a pseudo-logarithm is iterated and started at n. Fixed point equals zero for all initial values.
  • A082091 (program): a(n) = one more than the number of iterations of A005361 needed to reach 1 from the starting value n.
  • A082092 (program): Composite k such that d(k) < d(sigma(k)), i.e., A000005(k) < A000005(A000203(k)).
  • A082097 (program): a(n) = d(a(n-1)) + n = A000005(a(n-1)) + n, with a(1)=1.
  • A082100 (program): a(n) = A001157(A002110(n)), sum of squares of divisors of primorial numbers.
  • A082105 (program): A symmetric square array of numbers, read by antidiagonals.
  • A082106 (program): Main diagonal of number array A082105.
  • A082107 (program): Diagonal sums of number array A082105.
  • A082108 (program): a(n) = 4n^2 + 6n + 1.
  • A082109 (program): Third row of number array A082105.
  • A082110 (program): Array T(n,k) = k^2*n^2+5*k*n+1, read by antidiagonals.
  • A082111 (program): a(n) = n^2 + 5*n + 1.
  • A082112 (program): a(n) = 4n^2 + 10n + 1.
  • A082113 (program): n^4+5n^2+1.
  • A082114 (program): Diagonal sums of number array A082110.
  • A082115 (program): Fibonacci sequence (mod 3).
  • A082116 (program): Fibonacci sequence (mod 5).
  • A082117 (program): Fibonacci sequence (mod 6).
  • A082119 (program): Smallest positive difference between d and n/d for any divisor d of n.
  • A082120 (program): Smallest difference > 1 between d and n/d for any divisor d of n.
  • A082127 (program): Rounded base-3 logarithm of A082126(n).
  • A082129 (program): Rounded base-2 logarithm of A082128(n+4).
  • A082130 (program): Numbers k such that 2*k-1 and 2*k+1 are semiprimes.
  • A082133 (program): Expansion of e.g.f. x*exp(2*x)*cosh(x).
  • A082134 (program): Expansion of e.g.f. x*exp(3*x)*cosh(x).
  • A082135 (program): Expansion of e.g.f. x*exp(4*x)*cosh(x).
  • A082136 (program): Expansion of e.g.f. x*exp(5*x)*cosh(x).
  • A082137 (program): Square array of transforms of binomial coefficients, read by antidiagonals.
  • A082138 (program): A transform of C(n,3).
  • A082139 (program): A transform of binomial(n,5).
  • A082140 (program): A transform of binomial(n,6).
  • A082141 (program): A transform of C(n,7).
  • A082143 (program): First subdiagonal of number array A082137.
  • A082144 (program): A subdiagonal of number array A082137.
  • A082145 (program): A subdiagonal of number array A082137.
  • A082146 (program): G.f.: (1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).
  • A082147 (program): a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
  • A082148 (program): a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 10^k*N(n,k), where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
  • A082149 (program): A transform of C(n,2).
  • A082150 (program): A transform of C(n,2).
  • A082151 (program): A transform of C(n,2).
  • A082152 (program): Dispersion of the complement of the pentagonal numbers.
  • A082153 (program): Dispersion of the complement of row 1 of A082152.
  • A082154 (program): Dispersion of the complement of the hexagonal numbers.
  • A082155 (program): Dispersion of the complement of row 1 of A082154.
  • A082156 (program): Dispersion of the complement of row 1 of A056536.
  • A082173 (program): a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
  • A082179 (program): (c(p)-2)/p where p runs through the primes and where c(p) denotes the p-th Catalan number = 1/(p+1)*C(2p,p).
  • A082181 (program): a(0) = 1, for n>=1, a(n) = Sum_{k=0..n} 9^k*N(n,k), where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
  • A082183 (program): Smallest k > 0 such that T(n) + T(k) = T(m), for some m, T(i) being the triangular numbers, n > 1.
  • A082184 (program): The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.
  • A082186 (program): 1 + sum of first n terms of A001221.
  • A082204 (program): Begin with a 1, then place the smallest (as far as possible distinct) digits, such that, beginning from the n-th term, n terms form a palindrome.
  • A082206 (program): Digit sum of A082205(n).
  • A082233 (program): Square array T(n,k) = 2*n + k, read by antidiagonals in a zigzag fashion, n >= 0 and k >= 1.
  • A082234 (program): In the following square array numbers are entered like this a(1,1),a(1,2),a(2,1),a(3,1),a(2,2),a(1,3),a(1,4),a(2,3),a(3,2),a(4,1),a(5,1),a(4,2),… such that every entry is the geometric mean of the two diametrically opposite neighbors (wherever a pair exists). 1 2 4 8 16 32 64… 3 6 12 24 48 96 192… 9 18 36 72 144 288 576… 27 54 108 216 432 864 1728… … Sequence contains numbers as they are entered.
  • A082242 (program): Multiples of 4 that are the concatenation of 4 consecutive natural numbers.
  • A082243 (program): A082242(n)/4.
  • A082245 (program): Sum of (n-1)-th powers of divisors of n.
  • A082252 (program): Concatenation of (3n-2), (3n-1) and 3n divided by 3.
  • A082253 (program): Concatenation of (5n-4), (5n-3), (5n-2), (5n-1) and 5n divided by 5.
  • A082254 (program): Concatenation of (6n-5), (6n-4), (6n-3), (6n-2), (6n-1) and 6n divided by 6.
  • A082256 (program): (Concatenation of 9n-8, 9n-7, 9n-6, 9n-5, 9n-4, 9n-3, 9n-2, 9n-1 and 9n) divided by 9.
  • A082267 (program): Number of palindromes that use nonzero digits and have a digit sum of n.
  • A082274 (program): Palindromes k such that k + 2 is also a palindrome.
  • A082275 (program): Palindromes k such that k + 11 is also a palindrome.
  • A082285 (program): a(n) = 16n + 13.
  • A082286 (program): a(n) = 18*n + 10.
  • A082287 (program): a(1) = 1; for n > 1, n appears omega(n) times, where omega(n)=A001221(n) is the number of distinct prime factors of n, a(1)=1.
  • A082288 (program): n>1 appears bigomega(n) times, where bigomega(n)=A001222(n) is the number of prime factors of n (with repetition), a(1)=1.
  • A082289 (program): Expansion of x^4*(2+x)/((1+x)*(1-x)^5).
  • A082290 (program): Expansion of (1+x+x^2)/((1+x^2)*(1+x)^4*(1-x)^5).
  • A082291 (program): Expansion of x(2 + 5x - x^2)/((1 - x)(1 - 6x + x^2)).
  • A082292 (program): a(n) = (2n+1) * (2n)! / sqrt(4*(n+1)*Product_{k=1..2n+1} lcm(k, 2n+2-k)).
  • A082293 (program): Numbers having exactly one square divisor > 1.
  • A082294 (program): Numbers having exactly two square divisors > 1.
  • A082295 (program): Numbers having more than two square divisors > 1.
  • A082296 (program): Solutions to 13^x+17^x == 19 mod 23.
  • A082297 (program): Main diagonal of array A083861.
  • A082298 (program): G.f.: (1-3*x-sqrt(9*x^2-10*x+1))/(2*x).
  • A082299 (program): Greatest common divisor of n and its sum of prime factors (with repetition).
  • A082300 (program): Numbers relatively prime to the sum of their prime factors (with repetition).
  • A082301 (program): G.f.: (1 - 4*x - sqrt(16*x^2 - 12*x + 1))/(2*x).
  • A082302 (program): Expansion of g.f.: (1 - 5*x - sqrt(25*x^2 - 14*x + 1))/(2*x).
  • A082303 (program): McKay-Thompson series of class 32e for the Monster group.
  • A082304 (program): McKay-Thompson series of class 16d for the Monster group.
  • A082305 (program): G.f.: (1 - 6*x - sqrt(36*x^2 - 16*x + 1))/(2*x).
  • A082306 (program): Expansion of e.g.f. (1+x)*exp(2*x)*cosh(x).
  • A082307 (program): Expansion of e.g.f. (1+x)*exp(3*x)*cosh(x).
  • A082308 (program): Expansion of e.g.f. (1+x)*exp(4*x)*cosh(x).
  • A082309 (program): Expansion of e.g.f.: (1+x)*exp(5*x)*cosh(x).
  • A082311 (program): A Jacobsthal sequence trisection.
  • A082343 (program): Numerator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).
  • A082344 (program): Denominator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).
  • A082365 (program): A Jacobsthal number sequence.
  • A082366 (program): G.f.: (1 - 7*x - sqrt(49*x^2 - 18*x + 1))/(2*x).
  • A082367 (program): G.f.: (1-8*x-sqrt(64*x^2-20*x+1))/(2*x).
  • A082368 (program): a(n) = (4*n-1)! / (n! * n! * n! * (n-1)! * 3!).
  • A082369 (program): Numbers congruent to 13 mod 30.
  • A082375 (program): Irregular triangle read by rows: row n begins with n and decreases by 2 until 0 or 1 is reached, for n >= 0.
  • A082383 (program): a(0)=1, a(n)=2^n+n-2*a(n-1).
  • A082384 (program): a(0)=1, a(n)=2^n+n^2-2*a(n-1).
  • A082388 (program): a(1) = 1, a(2) = 2; further terms are defined by rules that for k >= 2, a(2^k-i) = a(2^k+i) for 1 <= i <= 2^k-1 and a(2^k) = a(2^(k-1)) + Sum_{i < 2^k} a(i).
  • A082389 (program): a(n) = floor((n+2)*phi) - floor((n+1)*phi) where phi=(1+sqrt(5))/2.
  • A082390 (program): Numbers on a computer numpad, read in a clockwise spiral.
  • A082391 (program): Start with the sequence a(1 to 4) = “1,3,2,3”. Then in step s, append “1”, “1,2”, or “1,2,3”, whichever ends with a(s+2).
  • A082392 (program): Expansion of (1/x) * sum(k>=0, x^2^k/(1-2x^2^(k+1))).
  • A082395 (program): Number of shifted Young tableaux with height <= 3.
  • A082397 (program): Number of directed aggregates of height <= 2 with n cells.
  • A082398 (program): Number of directed, diagonally convex polyominoes with n cells.
  • A082405 (program): a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6.
  • A082406 (program): Numbers k such that k divides Sum_{j=1..k} binomial(2j,j).
  • A082410 (program): a(1)=0. Thereafter, the sequence is constructed using the rule: for any k >= 0, if a(1), a(2), …, a(2^k+1) are known, the next 2^k terms are given as follows: a(2^k+1+i) = 1 - a(2^k+1-i) for 1 <= i <= 2^k.
  • A082412 (program): a(n) = (2*8^n + 2^n)/3.
  • A082413 (program): a(n) = (2*9^n + 3^n)/3.
  • A082414 (program): a(n) = (2*10^n + 4^n)/3.
  • A082415 (program): Numbers n such that in all partitions of n into distinct coprimes these coprimes are also mutually relatively prime.
  • A082416 (program): Parity of A073941(n).
  • A082417 (program): Numbers k such that P(k) < P(k+1) > P(k+2), where P(k) is the largest prime factor of k (A006530).
  • A082418 (program): Numbers n such that P(n) > P(n+1) < P(n+2), where P(n) = largest prime factor of n (A006530).
  • A082423 (program): a(1)=1, a(n)=ceiling(n/(n+1)*sum(k=1,n-1,a(k))).
  • A082425 (program): a(1)=1, a(n) = n*(a(n-1) + a(n-2) + … + a(2) + a(1)) - 1.
  • A082426 (program): a(1)=1, a(n)=n*(a(n-1)+a(n-2)+…+a(2)+a(1)) + 2.
  • A082427 (program): a(1)=1, a(n)=n*(a(n-1)+a(n-2)+…+a(2)+a(1)) - 2.
  • A082428 (program): a(1)=1, a(n)=n*(a(n-1)+a(n-2)+…+a(2)+a(1)) + 3.
  • A082429 (program): a(n) is the cardinality of the smallest subset S1 of S={1,2,3,…,n} such that every element of S is either in S1 or is the sum of two elements of S1.
  • A082430 (program): a(1)=1; for n > 1, a(n) = n*(a(n-1) + a(n-2) + … + a(2) + a(1)) + 4.
  • A082446 (program): a(0)=0, a(1)=1, a(2)=0; thereafter, if k>=0 and a block of the first 3*2^k terms is known, then a(3*2^k+i)=1-a(i) for 0<=i<3*2^k.
  • A082447 (program): a(n) = the number k such that s(k)=0 where s(0)=n and s(i)=s(i-1)-(s(i-1) modulo i).
  • A082448 (program): Add 1, multiply by 1, add 2, multiply by 2, etc.; start with 4.
  • A082450 (program): a(n) = 5*(n^2-n+2)/2.
  • A082451 (program): Sum over divisors d of n of Kronecker symbol (-60, d).
  • A082454 (program): a(n) = prime(n) + prime(n-1) - a(n-1) with a(0) = a(1) = 0.
  • A082455 (program): a(n) = prime(n) + prime(n-1) + a(n-1), a(0) = 0, a(1) = 4.
  • A082458 (program): Multiply by 1, add 1, multiply by 2, add 2, etc., starting with 0.
  • A082459 (program): Multiply by 1, add 1, multiply by 2, add 2, etc.
  • A082460 (program): a(n) = pi(n) - a(n - 1) = A000720(n) - a(n - 1).
  • A082462 (program): Let chi(k) = 1 if prime(k+1) - prime(k) = 2, = 0 otherwise; sequence gives a(n) = sum_{k <= n} chi(k).
  • A082465 (program): Least k>=1 such that n^2+kn+1 is prime.
  • A082471 (program): a(1)=1, a(n) = Sum_{k=1..n-1} Fibonacci(k)*a(k).
  • A082472 (program): a(1) = 1, a(n) = Sum_{k=1..n-1} a(k)*2^k.
  • A082476 (program): a(n) = Sum_{d|n} mu(d)^2*tau(d)^2.
  • A082477 (program): Number of divisors d of n such that d+1 is also a divisor of n+1.
  • A082480 (program): a(n) = Product_{k=1..n} (F(k)+1) where F(k) denotes the k-th Fibonacci number.
  • A082481 (program): Number of 1’s in binary representation of C(2n,n).
  • A082482 (program): Floor of (2^n-1)/n.
  • A082483 (program): Numbers n such that 1/(5-s(n)) is an integer where s(k)=sum(i=1,k,1/2^floor(sqrt(i))).
  • A082485 (program): Numbers n such that 1/(2-s(n)) is an integer where s(k)=sum(i=1,k,1/3^floor(sqrt(i))).
  • A082486 (program): Decimal expansion of the constant c satisfying Sum_{k>=1} 1/c^sqrtint(k) = 1 where sqrtint(k) = floor(sqrt(k)).
  • A082490 (program): Exponent of highest power of 3 dividing sum(0<=k<n, C(2n,n)).
  • A082491 (program): a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166).
  • A082493 (program): a(n) = n*ceiling(2^n/n) - 2^n.
  • A082494 (program): a(n) = n - (2^n (mod n)).
  • A082495 (program): a(n) = (2^n - 1) mod n.
  • A082496 (program): Numbers of the form 2p+1, where p and p+2 are a pair of twin primes.
  • A082498 (program): a(0)=0, a(1)=1, a(2n)=a(n), a(2n+1)=a(n)+a(n-1).
  • A082500 (program): a(n) = ceiling(n/2) if n is odd, or prime(n/2) otherwise.
  • A082505 (program): a(n) = sum of (n-1)-th row terms of triangle A134059.
  • A082506 (program): a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).
  • A082507 (program): Generated by a 3rd-order formal recursion with suitable initial values as follows: a(n) = n - a(n-1) - a(n-2) - a(n-3); a(0)=a(1)=a(2)=0.
  • A082508 (program): Differences between consecutive primes that are powers of 2 in order of their appearance. Differences not powers of 2 are deleted from A001223.
  • A082509 (program): Differences between consecutive primes being not powers of 2 in order of their appearance. Differences which are powers of 2 are omitted from A001223.
  • A082510 (program): Differences of consecutive primes being divisible by 6 in order of their appearance in A001223: terms not divisible by 6 are omitted from A001223.
  • A082511 (program): a(n) = 3^n mod 2n.
  • A082513 (program): a(n)=A000720(n)-A000005(n).
  • A082514 (program): a(n) = pi(n) + tau(n).
  • A082515 (program): a(n)=A000720(n)+A000010(n).
  • A082516 (program): Differences between consecutive Niven (or Harshad) numbers.
  • A082522 (program): Numbers of the form p^(2^k) with p prime and k>0.
  • A082523 (program): Number of times k^2 + (n-k)^2 is a square for 1 <= k <= n-1.
  • A082524 (program): a(1)=1, a(2)=2, then use the rule when a(n) is the end of a run, n appears a(n) times.
  • A082525 (program): Numerators of coefficients in (1+x)^(1+x) power series.
  • A082526 (program): Denominators of coefficients in (1+x)^(1+x) power series.
  • A082527 (program): Least k such that x(k)=0 where x(1)=n x(k)=k^2*floor(x(k-1)/k^2).
  • A082528 (program): Least k such that x(k)=0 where x(1)=n x(k)=k^3*floor(x(k-1)/k^3).
  • A082530 (program): a(1)=1, a(n)=5*a(n-1)+1 if a(n-1) is odd, a(n)=a(n-1)/2+1 otherwise.
  • A082531 (program): (-1)^n * coefficient of x^n in 1/x-1/(1-eta(x)) power series.
  • A082532 (program): a(n) = n^2 - 2*floor(n/sqrt(2))^2.
  • A082541 (program): a(n) = (7*3^n - 4*0^n)/3.
  • A082542 (program): a(n) = prime(n) + 2 - (prime(n) mod 4).
  • A082543 (program): Take a string of n x’s and insert n-1 ^’s and n-1 pairs of parentheses in all possible ways. Sequence gives number of distinct integer values when x=sqrt(2).
  • A082545 (program): a(n) = (2*n)! * Sum_{k=0..n} binomial(n,k)/(n+k)!.
  • A082548 (program): a(n) is the number of values of k such that k can be expressed as the sum of distinct primes with largest prime in the sum equal to prime(n).
  • A082550 (program): Number of sets of distinct positive integers whose arithmetic mean is an integer, the largest integer of the set being n.
  • A082551 (program): Denote sigma(n)-n by s(n); a(n)=1 if s(n)>n, a(n)=0 if s(n)=n, a(n)=-1 if s(n)<n.
  • A082554 (program): Primes whose base-2 representation is a block of 1’s, followed by a block of 0’s, followed by a block of 1’s.
  • A082555 (program): Primes whose base 3 representation does not contain a 0.
  • A082556 (program): G.f.: Product_{m>=1} 1/(1-x^m)^30.
  • A082557 (program): G.f.: Product_{m>=1} 1/(1-x^m)^32.
  • A082558 (program): Expansion of Product_{m>=1} 1/(1-x^m)^48.
  • A082559 (program): G.f.: Product_{m>=1} 1/(1-x^m)^64.
  • A082560 (program): a(1)=1, a(n)=2*a(n-1) if n is odd, or a(n)=a(n/2)+1 if n is even.
  • A082562 (program): a(n) = number of values of m such that m can be expressed as the sum of distinct odd numbers with largest odd number in the sum = 2n+1.
  • A082564 (program): Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.
  • A082568 (program): First nontrivial square root of unity mod A033949(n), i.e., smallest x > 1 such that x^2 == 1 mod A033949(n).
  • A082569 (program): a(1)=2; a(n)=ceiling(n*(a(n-1)-1/a(n-1))).
  • A082570 (program): a(1)=1, a(n)=ceiling(n*(a(n-1)+1/a(n-1))).
  • A082573 (program): a(1)=1, a(n)=ceiling(n*(a(n-1)+3/a(n-1))).
  • A082574 (program): a(1)=1, a(n) = ceiling(r(3)*a(n-1)) where r(3) = (1/2)*(3 + sqrt(13)) is the positive root of X^2 = 3*X + 1.
  • A082575 (program): Nonnegative numbers in (3*A005836) union (3*A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].
  • A082577 (program): a(n) = first digit to the right of decimal point of n*(sqrt(5)-1)/2.
  • A082578 (program): A binomial sum.
  • A082579 (program): Expansion of e.g.f.: exp( x/(1-x)^2 ).
  • A082580 (program): A sum of Lah numbers and binomial coefficients.
  • A082582 (program): Expansion of (1 + x^2 - sqrt( 1 - 4*x + 2*x^2 + x^4)) / (2*x) in powers of x.
  • A082585 (program): a(1)=1, a(n) = ceiling(r(5)*a(n-1)) where r(5) = (1/2)*(5 + sqrt(29)) is the positive root of X^2 = 5*X + 1.
  • A082587 (program): G.f.: (3+x+x^2+2*x^3)/(1-x^2-x^4).
  • A082588 (program): a(1) = 1, a(n) = Sum_{d | n and d < n} a(d)^2 for n > 1.
  • A082590 (program): Expansion of 1/((1 - 2*x)*sqrt(1 - 4*x)).
  • A082593 (program): Values in Pfennigs of German money before the introduction of the Euro.
  • A082594 (program): Constant term when a polynomial of degree n-1 is fitted to the first n primes.
  • A082601 (program): Tribonacci array: to get the next row, right-adjust the previous 3 rows and add them, then append a final 0.
  • A082605 (program): Using Euler’s 6-term sequence A014556, we define the partial recurrence relation a(0)=2, a(1)=3, a(2)=5; a(k) = 2*a(k-1) - 1 + (-1)^(k-1)*2^(k-2), 3 <= k <= 5.
  • A082630 (program): Limit of the sequence obtained from S(0) = (1,1) and, for n > 0, S(n) = I(S(n-1)), where I consists of inserting, for i = 1, 2, 3…, the term a(i) + a(i+1) between any two terms for which 7*a(i+1) <= 11*a(i).
  • A082639 (program): Numbers n such that 2*n*(n+2) is a square.
  • A082642 (program): Expansion of Molien series for 5-dimensional representation of dihedral group of order 10.
  • A082643 (program): a(n) = ceiling(n*(n+1/2)/2).
  • A082644 (program): a(n) = floor(n*(n-1/2)/2).
  • A082645 (program): a(n) = floor((2*n^2 + n - 4)/4).
  • A082647 (program): Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.
  • A082648 (program): Consider f(m) = Sum_{k=1..m} k! (A007489) when m is very large; a(n) = n-th digit from end.
  • A082649 (program): Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).
  • A082650 (program): Number of primes < n of form 1+k*spf(n), where spf(n) is the smallest prime factor of n (A020639).
  • A082651 (program): Positive integer values of n such that 5n^2+11 is a square.
  • A082652 (program): Triangle read by rows: T(n,k) is the number of squares that can be found in a k X n rectangular grid of little squares, for 1 <= k <= n.
  • A082654 (program): Order of 4 mod n-th prime: least k such that prime(n) divides 4^k-1, n >= 2.
  • A082655 (program): Number of distinct letters needed to spell English names of numbers 1 through n.
  • A082656 (program): Trajectory of 39 under map x -> x/2 if x even, 3x+9 if x odd.
  • A082657 (program): Integers expressible as the sum of a square and a triangular number in just one way.
  • A082658 (program): Integers expressible as the sum of a square and a triangular number in exactly two distinct ways.
  • A082659 (program): Integers expressible as the sum of a square and a triangular number in exactly three distinct ways.
  • A082660 (program): Number of ways n can be expressed as the sum of a square and a triangular number.
  • A082662 (program): Numbers k such that the odd part of k is less than sqrt(2k).
  • A082663 (program): Odd semiprimes pq with p < q < 2p.
  • A082664 (program): Numbers k such that A082647(k) = A000005(k) - 1.
  • A082667 (program): a(n) = floor(2n/3) * ceiling(2n/3) / 2.
  • A082669 (program): Let p(n) = upper member of n-th pair of twin primes; sequence gives a(n) = p(n)*(p(n)-1)/2.
  • A082679 (program): Number of LEGO towers, one piece per floor, where every floor is perpendicular to the one below it (so we have a kind of 3-dimensional zigzag pattern).
  • A082681 (program): Denominator of Sum_{i=n(n-1)/2+1..n(n+1)/2} 1/i.
  • A082682 (program): Algebraic degree of R(e^(-n * Pi)), where R(q) is the Rogers-Ramanujan continued fraction.
  • A082683 (program): Smaller of the two prime numbers whose product is A082663(n).
  • A082684 (program): Larger of the two prime numbers whose product is A082663(n).
  • A082685 (program): (2*5^n + 2^n)/3.
  • A082686 (program): Odd nonprime integers n which have an odd number of proper divisors.
  • A082687 (program): Numerator of Sum_{k=1..n} 1/(n+k).
  • A082688 (program): Denominator of Sum_{k=1..n} 1/(n+k).
  • A082689 (program): Numerator of n*sum(k=1,(-1)^(k+1)/(n+k)).
  • A082690 (program): Denominator of n*sum(k=1,(-1)^(k+1)/(n+k)).
  • A082691 (program): a(1)=1, a(2)=2, then if the first 3*2^k-1 terms are a(1), a(2), …, a(3*2^k - 1), the first 3*2^(k+1)-1 terms are a(1), a(2), …, a(3*2^k - 1), a(1), a(2), …, a(3*2^k - 1), a(3*2^k-1) + 1.
  • A082692 (program): Partial sums of A082691.
  • A082693 (program): Pyramidal sequence built with powers of 2.
  • A082694 (program): Partial sums of A082693.
  • A082724 (program): a(n) = (3*11^n + 3^n)/4.
  • A082725 (program): a(n) = n/A100762(n).
  • A082727 (program): a(0)=1, a(n)=abs(n-2*a(n-1)).
  • A082729 (program): Least positive number that can be written using all divisors of n and the operations add and subtract.
  • A082732 (program): a(1) = 1, a(2) = 3, a(n) = LCM of all the previous terms + 1.
  • A082735 (program): Product of n-th group of terms in A074147.
  • A082741 (program): Numbers that have digits consisting only of line segments or both line segments and curves (base 10 digits are 1, 2, 4, 5, 7).
  • A082742 (program): Indices of occurrences of 2 in A004738.
  • A082743 (program): a(0)=1, a(1)=2; for n >= 2, a(n) is smallest palindrome greater than 1 which is congruent to 1 (mod n).
  • A082754 (program): Triangle read by rows: T(n, k) = abs(n^k-k^n), 1<=k<=n.
  • A082758 (program): Sum of the squares of the trinomial coefficients (A027907).
  • A082759 (program): a(n) = Sum_{k = 0..n} binomial(n,k)*trinomial(n,k), where trinomial(n,k) = trinomial coefficients.
  • A082761 (program): Trinomial transform of the Fibonacci numbers (A000045).
  • A082762 (program): Trinomial transform of Lucas numbers (A000032).
  • A082763 (program): Roman numeral contains an asymmetric symbol (L).
  • A082764 (program): Trinomial transform of the Pell numbers (A000129).
  • A082765 (program): Trinomial transform of the factorial numbers (A000142).
  • A082766 (program): Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).
  • A082767 (program): Number of edges in the prime graph.
  • A082768 (program): Numbers that begin with 1, 3, 7 or 9.
  • A082772 (program): Integers n such that there exists at least one divisor d of n, 1 < d < n, such that d divides n and d+1 divides n+1.
  • A082775 (program): Convolution of natural numbers >= 2 and the partition numbers (A000041).
  • A082784 (program): Characteristic function of multiples of 7.
  • A082787 (program): (2/3)*(2*n-1)!*binomial(3*n,2*n).
  • A082791 (program): Smallest k such that k*n begins with 2: a(n) = A082811(n)/n.
  • A082792 (program): Smallest multiple of n beginning with 3.
  • A082794 (program): Smallest multiple of n beginning with 4.
  • A082795 (program): Smallest multiple of n beginning with 5.
  • A082796 (program): Smallest multiple of n beginning with 6.
  • A082797 (program): Smallest multiple of n beginning with 7.
  • A082798 (program): Smallest multiple of n beginning with 8.
  • A082799 (program): Smallest multiple of 3 beginning with n.
  • A082800 (program): Smallest multiple of 4 beginning with n.
  • A082801 (program): Smallest multiple of 6 beginning with n.
  • A082802 (program): Smallest multiple of 7 beginning with n.
  • A082803 (program): Smallest multiple of 8 beginning with n.
  • A082804 (program): Smallest multiple of 9 beginning with n.
  • A082811 (program): Smallest multiple of n beginning with 2.
  • A082840 (program): a(n) = 4*a(n-1) - a(n-2) + 3, with a(0) = -1, a(1) = 1.
  • A082841 (program): a(n) = 4*a(n-1) - a(n-2) for n>1, a(0)=3, a(1)=9.
  • A082844 (program): Start with 3,2 and apply the rule a(a(1)+a(2)+…+a(n)) = a(n), fill in any undefined terms with a(t) = 2 if a(t-1) = 3 and a(t) = 3 if a(t-1) = 2.
  • A082845 (program): Partial sums of A082844.
  • A082849 (program): Product of cototient values of consecutive integers.
  • A082850 (program): Let S(0) = {}, S(n) = {S(n-1), S(n-1), n}; sequence gives S(infinity).
  • A082851 (program): Partial sums of A082850.
  • A082852 (program): a(0)=0, a(n) = A014137(A072643(n)-1).
  • A082853 (program): Integers 0 to Catalan(n)-1 followed by integers 0 to Catalan(n+1)-1 etc.
  • A082854 (program): Integers 1 to Catalan(n) followed by integers 1 to Catalan(n+1) etc.
  • A082855 (program): a(0)=0, a(1)=1, a(n) = A014137(A081288(n-1)-1).
  • A082863 (program): Number of distinct prime factors of n^2-1.
  • A082865 (program): a(n) is the sum of the preceding terms that are coprime to n.
  • A082870 (program): Tribonacci array.
  • A082873 (program): Independence number of king KG_2 on triangle board B_n.
  • A082874 (program): Independence number of king KG_4 on triangle board B_n.
  • A082877 (program): a(n) = A002884(n) / A070731(n).
  • A082887 (program): a(n)=gcd[mod[n!,2^n],mod[(n+1)!,2^(n+1)]].
  • A082893 (program): a(n) is the closest number to n-th prime which is divisible by n.
  • A082894 (program): a(n) is the closest number to 2^n which is divisible by n.
  • A082895 (program): Closest number to sigma(n) = A000203(n) which is divisible by n.
  • A082896 (program): a(n) = A082893(n)/n.
  • A082898 (program): a(n) = A082895(n)/n, where A082895(n) is the closest number to sigma(n) which is divisible by n.
  • A082899 (program): a(n) = A082893(n)-A000040(n), that is difference of n-th prime and number closest to it and divisible by n.
  • A082900 (program): a(n) = A082894(n)-A000079(n), that is difference of 2^n and the number closest to it and divisible by n.
  • A082901 (program): a(n) = A082895(n)-A000203(n); the distance from sigma(n) to that multiple of n which is closest to sigma(n), positive terms for cases where the closest multiple is after sigma(n), and negative terms where it is before sigma(n). In case of ties, a positive term is selected.
  • A082902 (program): a(n) = gcd(2^n, sigma(2,n)) = gcd(A000079(n), A001157(n)).
  • A082903 (program): Highest power of two that divides the sum of divisors of n.
  • A082906 (program): Sum of terms in n-th row of modified Pascal’s triangle displayed in A082905.
  • A082907 (program): A modified Pascal’s triangle, read by rows, and modified as follows: binomial(n,j) is replaced by gcd(2^n, binomial(n,j)), i.e., the largest power of 2 dividing binomial(n,j).
  • A082908 (program): Largest value of gcd(2^n, binomial(n,j)) with j=0..n-1; maximal value of largest power of 2 dividing binomial(n,j) in the n-th row of Pascal’s triangle.
  • A082909 (program): a(n) = Sum_{d|n} (d mod 3).
  • A082910 (program): a(n) = prime(prime(n+1)-prime(n)).
  • A082911 (program): a(n) = prime(n+pi(n)) - prime(n) = A000040(n+A000720(n)) - A000040(n).
  • A082916 (program): Numbers k such that k and binomial(2*k, k) are relatively prime.
  • A082928 (program): If n is prime, a(n) = n+1; if n is even, a(n) = n/2; otherwise a(n) = n.
  • A082934 (program): A082928(1) + A082928(2) + … + A082928(n).
  • A082936 (program): a(n) = (1/(3*n))*Sum_{d|n, d even} phi(2*n/d)*binomial(3d/2,d).
  • A082942 (program): (n^2+1)*(4*n^2+1)*(4*n^2+3).
  • A082953 (program): a(n) = A000252(n) / A070732(n).
  • A082962 (program): Numerators of continued fraction convergents to (sqrt(37)-4)/3.
  • A082964 (program): a(n) = m given by arctan(tan(n)) = n - m*Pi.
  • A082969 (program): Numbers n such that (n/4)^2-n/8=sum(k=1,n, k modulo(sum(i=0,k-1,1-t(i))) where t(i)=A010060(i) is the Thue-Morse sequence.
  • A082974 (program): a(n) = (a(n-1) + p(n)) mod p(n+1).
  • A082975 (program): Denominators of continued fraction convergents to (sqrt(37)-4)/3.
  • A082977 (program): Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.
  • A082981 (program): Start with the sequence S(0)={1,1} and for k>0 define S(k) to be I(S(k-1)) where I denotes the operation of inserting, for i=1,2,3…, the term a(i)+a(i+1) between any two terms for which 4a(i+1)<=5a(i). The listed terms are the initial terms of the limit of this process as k goes to infinity.
  • A082984 (program): Numbers k for which the 3x+1 problem takes at least k halving and tripling steps to reach 1.
  • A082985 (program): Coefficient table for Chebyshev’s U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  • A082986 (program): Largest x such that 1/x + 1/y + 1/z = 1/n.
  • A082987 (program): a(n)=sum(k=0,n,3^k*F(k)) where F(k) is the k-th Fibonacci number.
  • A082988 (program): a(n)=sum(k=0,n,4^k*F(k)) where F(k) is the k-th Fibonacci number.
  • A082989 (program): Number of ordered trees with n edges and having no root-to-leaf branches.
  • A082995 (program): Distance from n!+1 to next larger square.
  • A082996 (program): a(n) = card{ x <= n : bigomega(x) = 4 }.
  • A082997 (program): a(n) = card{ x <= n : omega(x) = 2 }.
  • A082999 (program): a(n) = A046195(n) mod 5.
  • A083000 (program): Values of x for which 9y^2 = x^2 + 2xy - 2x has integer solutions with positive y.
  • A083005 (program): a(1)=1, a(n)=2a(n-1)+1 if a(n-1) is prime, a(n)=a(n-1)+1 otherwise.
  • A083019 (program): Number of common divisors of n and F(n) where F(n) denotes the n-th Fibonacci number.
  • A083022 (program): Numbers n such that 4*n^2 - 3 is prime.
  • A083023 (program): a(n) = number of partitions of n into a pair of parts n=p+q, p>=q>=0, with p-q equal to a square >= 0.
  • A083025 (program): Number of primes congruent to 1 modulo 4 dividing n (with multiplicity).
  • A083026 (program): Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.
  • A083028 (program): Numbers that are congruent to {0, 2, 3, 5, 7, 8, 11} mod 12.
  • A083030 (program): Numbers that are congruent to {0, 4, 7} mod 12.
  • A083031 (program): Numbers that are congruent to {0, 3, 7} mod 12.
  • A083032 (program): Numbers that are congruent to {0, 4, 7, 10} mod 12.
  • A083033 (program): Numbers that are congruent to {0, 2, 3, 5, 7, 9, 10} mod 12.
  • A083034 (program): Numbers that are congruent to {0, 1, 3, 5, 7, 8, 10} mod 12.
  • A083035 (program): a(n) = floor(sqrt(2)*n)-2*floor(n/sqrt(2)).
  • A083036 (program): Partial sums of A083035.
  • A083037 (program): a(n)=2*A083036(n)-n. Also -A123737(n).
  • A083038 (program): A fractal sequence.
  • A083039 (program): Number of divisors of n that are <= 3.
  • A083040 (program): Number of divisors of n that are <= 4
  • A083043 (program): Integers y such that 11*x^2 - 9*y^2 = 2 for some integer x.
  • A083044 (program): Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), where x=3/2, n >= 0, k >= 0.
  • A083045 (program): Main diagonal of table A083044.
  • A083046 (program): Antidiagonal sums of table A083044.
  • A083049 (program): Antidiagonal sums of table A083047.
  • A083051 (program): First column of table A083050.
  • A083054 (program): a(n) = floor(sqrt(3)*n) - 3*floor(n/sqrt(3)).
  • A083055 (program): a(n) = cardinality{ k<=n / A083054(k)=1}.
  • A083056 (program): a(n) = 3*A083055(n)-n.
  • A083057 (program): A fractal sequence.
  • A083058 (program): Number of eigenvalues equal to 1 of n X n matrix A(i,j)=1 if j=1 or i divides j.
  • A083062 (program): a(n) = (n+1)^n/(n+2) - (-1)^n/(n+2).
  • A083063 (program): a(n) = (n+1)^(n-1)/(n+2) + (-1)^n/(n+2).
  • A083064 (program): Square number array T(n,k)=(k(k+2)^n+1)/(k+1) read by antidiagonals.
  • A083065 (program): 4th row of number array A083064.
  • A083066 (program): 5th row of number array A083064.
  • A083067 (program): 6th row of number array A083064.
  • A083068 (program): 7th row of number array A083064.
  • A083069 (program): Main diagonal of number array A083064.
  • A083070 (program): First super-diagonal of number array A083064.
  • A083071 (program): First subdiagonal of number array A083064.
  • A083072 (program): A subdiagonal of number array A083064.
  • A083073 (program): A subdiagonal of number array A083064.
  • A083074 (program): n^3 - n^2 - n - 1.
  • A083075 (program): Square array read by antidiagonals: T(n,k) = (k*(2*k+3)^n + 1)/(k+1).
  • A083076 (program): Third row of number array A083075.
  • A083077 (program): Fifth row of number array A083075.
  • A083078 (program): 6th row of number array A083075.
  • A083079 (program): 4th column of number array A083075.
  • A083080 (program): Main diagonal of number array A083075.
  • A083081 (program): First super-diagonal of number array A083075.
  • A083082 (program): First subdiagonal of number array A083075.
  • A083083 (program): A diagonal of number array A083075.
  • A083084 (program): A diagonal of number array A083075.
  • A083085 (program): (2+(-5)^n)/3.
  • A083086 (program): a(n) (2*2^n + (-4)^n)/3.
  • A083088 (program): First column of table A083087.
  • A083089 (program): Numbers that are congruent to {0, 2, 4, 6, 7, 9, 11} mod 12.
  • A083091 (program): Antidiagonal sums of table A083087.
  • A083093 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 3.
  • A083094 (program): Numbers k such that Sum_{j=0..k} (binomial(k,j) mod 3) is odd.
  • A083095 (program): a(n) = A083094(n)/4.
  • A083096 (program): Numbers k such that 3 divides Sum_{j=1..k} binomial(2*j,j).
  • A083097 (program): a(n) = A083096(n)/6.
  • A083098 (program): a(n) = 2*a(n-1) + 6*a(n-2).
  • A083099 (program): a(n) = 2*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.
  • A083100 (program): a(n) = 2*a(n-1) + 7*a(n-2).
  • A083101 (program): a(n) = 2*a(n-1) + 10*a(n-2).
  • A083102 (program): a(n) = 2*a(n-1) + 10*a(n-2).
  • A083120 (program): Numbers that are congruent to {0, 2, 4, 5, 7, 9, 10} mod 12.
  • A083127 (program): a(n) = 3*n^3 + n^2 - 4*n.
  • A083131 (program): Number the letters of English alphabet from 1 to 26; sequence specifies capital letters with left-right symmetry.
  • A083141 (program): Main diagonal of array in A083140.
  • A083158 (program): Palindromes which are the arithmetic mean of two distinct palindromes.
  • A083173 (program): Triangle read by rows: the n-th row contains the first n-1 multiples of prime(n) followed by the next multiple that will make the row sum a multiple of n.
  • A083174 (program): Leading diagonal of A083173.
  • A083175 (program): Row sums in A083175.
  • A083176 (program): Arithmetic means of rows of A083173.
  • A083177 (program): Let P(k) = floor(k/2). Start with n, apply P repeatedly until reach 1. a(n) = concatenation of numbers obtained.
  • A083178 (program): Numbers with a digit sum of n and a maximum product of digits. In case of two identical products choose the largest number.
  • A083186 (program): Sum of first n primes whose indices are primes.
  • A083196 (program): a(n) = 8*n^4 + 9*n^2 + 2.
  • A083199 (program): Exponent of largest power of 2 dividing A061419(n).
  • A083200 (program): Polynexus numbers of order 7.
  • A083210 (program): Numbers with no subset of their divisors such that the complement has the same sum.
  • A083215 (program): a(n) = 1 + Sum(prime(i)*(2*i-1): 1<=i<=n).
  • A083217 (program): a(n) = (2*5^n+(-1)^n)/3.
  • A083218 (program): a(n) = n mod (spf(n+1)+1), where spf(n) is the smallest prime factor of n (A020639).
  • A083219 (program): a(n) = n - 2*floor(n/4).
  • A083220 (program): a(n) = n + (n mod 4).
  • A083222 (program): a(n) = (4*5^n + (-5)^n)/5.
  • A083223 (program): a(n) = (5*6^n+(-6)^n)/6.
  • A083224 (program): a(n) = (6*7^n + (-7)^n)/7.
  • A083225 (program): a(n) = (7*8^n + (-8)^n)/8.
  • A083226 (program): a(n) = (8*9^n + (-9)^n)/9.
  • A083227 (program): a(n) = (9*10^n + (-10)^n)/10.
  • A083228 (program): A Jacobsthal related sequence.
  • A083229 (program): a(n) = (3*3^n + (-5)^n)/4.
  • A083230 (program): Number of repunit divisors of n.
  • A083231 (program): a(n) = (3*5^n + (-3)^n)/4.
  • A083232 (program): a(n) = (3*7^n+(-1)^n)/4.
  • A083233 (program): a(n) = (3*8^n + 0^n)/4.
  • A083234 (program): a(n) = (3*10^n + 2^n)/4.
  • A083236 (program): First order recursion: a(0)=2; a(n) = prime(n) - a(n-1).
  • A083237 (program): First order recursion: a(0)=5; a(n)=prime(n)-a(n-1).
  • A083238 (program): First order recursion: a(0)=1; a(n) = sigma(1,n) - a(n-1).
  • A083239 (program): First order recursion: a(0)=1; a(n)=phi(n)-a(n-1)=A000010(n)-a(n-1).
  • A083241 (program): a(n) + a(n-1) + a(n-2) + a(n-3) = prime(n), n>2, a(0)=a(1)=a(2)=0;.
  • A083244 (program): k is in the sequence iff the number of numbers unrelated to k is larger than that of related ones[=divisors and coprimes] to k: A045763(k) > A073757(k) or A045763(k) > k/2 or A073757(k) < k/2.
  • A083245 (program): Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2*A045763(n) = 2*A073757(n)-n.
  • A083249 (program): Numbers n with A045763(n) = n + 1 - d(n) - phi(n) < d(n) < phi(n).
  • A083254 (program): a(n) = 2*phi(n) - n.
  • A083256 (program): a(n) = A046523(n-th nonprime number) = A046523(A018252(n)).
  • A083257 (program): a(n) = A071364(n-th nonprime number) = A071364(A018252(n)).
  • A083258 (program): a(n) = gcd(A046523(n), n).
  • A083259 (program): a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.
  • A083261 (program): a(n) = gcd(A046523(n+1), A046523(n)).
  • A083262 (program): a(n) = sigma(A046523(n)), sum of divisors of the least number with the same prime signature as n.
  • A083266 (program): Sum of related numbers (counted in A073757) belonging to n: a(n) = A000203(n) + A023896(n) - 1; related = {divisor-set, RRS}.
  • A083267 (program): Product of related numbers (counted in A073757) belonging to n; related = {divisor-set, RRS}: a(n) = A007955(n)*A001783(n).
  • A083268 (program): a(n) is the lcm of related numbers to n (counted in A073757): related = {divisor-set, RRS}.
  • A083271 (program): a(n) = n*tau(n) + 1.
  • A083272 (program): a(n) = n*tau(a(n-1)) + 1 = n*A000005(a(n-1)) + 1, a(0) = 1.
  • A083277 (program): k appears 3k-2 times.
  • A083282 (program): a(n) = n^(3*n).
  • A083292 (program): a(n) = n*floor(n/10) + (n mod 10).
  • A083294 (program): a(n) = (4 + (-9)^n)/5.
  • A083295 (program): a(n) = (4*2^n + (-8)^n)/5.
  • A083296 (program): a(n) = (4*3^n + (-7)^n)/5.
  • A083297 (program): a(n) = (4*4^n + (-6)^n)/5.
  • A083299 (program): a(n) = (4*6^n + (-4)^n)/5.
  • A083300 (program): a(n) = (4*7^n + (-3)^n)/5.
  • A083301 (program): a(n) = (4*8^n + (-2)^n)/5.
  • A083302 (program): a(n) = (4*9^n + (-1)^n)/5.
  • A083304 (program): a(n) = (4*(n+5)^n + n^n)/5.
  • A083305 (program): (4*(n+10)^n+n^n)/5.
  • A083306 (program): a(n) = (4*(n+15)^n + n^n)/5.
  • A083307 (program): a(n) = (4*n^n + (n-10)^n)/5.
  • A083308 (program): a(n) = (4*(n+1)^n + (n-9)^n)/5.
  • A083312 (program): Largest integer m such that 1+2+…+m divides n.
  • A083313 (program): a(0)=1; a(n) = 3^n - 2^(n-1) for n >= 1.
  • A083314 (program): (2*4^n-(3^n-1))/2.
  • A083315 (program): (2*5^n-(4^n-2^n))/2.
  • A083316 (program): a(n) = (2*6^n - (5^n - 3^n))/2.
  • A083318 (program): a(0) = 1; for n>0, a(n) = 2^n + 1.
  • A083319 (program): 4^n+3^n-2^n.
  • A083320 (program): a(n) = 5^n + 4^n - 3^n.
  • A083321 (program): a(n) = (-1)^n + (-2)^n - (-3)^n.
  • A083322 (program): a(n) = 2^n - A081374(n).
  • A083323 (program): a(n) = 3^n - 2^n + 1.
  • A083324 (program): An alternating sum of decreasing powers.
  • A083325 (program): a(n) = 5^n - 4^n + 3^n.
  • A083326 (program): a(n) = 6^n - 5^n + 4^n.
  • A083327 (program): a(n) = (5^n - 4^n + 3^n - 2^n)/2.
  • A083328 (program): a(n) = (6^n - 5^n + 4^n - 3^n)/2.
  • A083329 (program): a(0) = 1; for n > 0, a(n) = 3*2^(n-1) - 1.
  • A083330 (program): a(n) = (3*4^n - 2*3^n + 2^n)/2.
  • A083331 (program): a(n) = (3*5^n - 2*4^n + 3^n)/2.
  • A083332 (program): a(n) = 10*a(n-2) - 16*a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.
  • A083333 (program): a(n) = 10*a(n-2) - 16*a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.
  • A083334 (program): a(n) = 12*a(n-2) - 25*a(n-4).
  • A083335 (program): a(n)=12a(n-2)-25a(n-4).
  • A083336 (program): a(n)=4a(n-2)-a(n-4).
  • A083337 (program): a(n) = 2*a(n-1) + 2*a(n-2); a(0)=0, a(1)=3.
  • A083339 (program): a(n) is the number of distinct prime factors of n that occur in partitions into two primes when n is even and into three primes when n is odd.
  • A083345 (program): Numerator of r(n) = Sum(e/p: n=Product(p^e)).
  • A083346 (program): Denominator of r(n) = Sum(e/p: n=Product(p^e)).
  • A083347 (program): Numbers n such that Sum(e/p: n=Product(p^e)) < 1.
  • A083356 (program): Total area of all incongruent integer-sided rectangles of area <= n.
  • A083363 (program): Diagonal of table A083362.
  • A083364 (program): Antidiagonal sums of table A083362.
  • A083365 (program): Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
  • A083368 (program): A Fibbinary system represents a number as a sum of distinct Fibonacci numbers (instead of distinct powers of two). Using representations without adjacent zeros, a(n) = the highest bit-position which changes going from n-1 to n.
  • A083370 (program): Primes satisfying f(2p)=p when f(1)=5 (see comment).
  • A083371 (program): Primes p such that q-p >= 8, where q is the next prime after p.
  • A083374 (program): a(n) = n^2 * (n^2 - 1)/2.
  • A083375 (program): n appears prime(n) times.
  • A083377 (program): a(n) = the largest integer whose square has n digits and first digit 1.
  • A083378 (program): a(n) is the largest integer whose cube has n digits and first digit 1, except that a(2)=2.
  • A083381 (program): Square array giving the number of trellis edges T(i,j) (i >= 0, j >= 0), read by antidiagonals.
  • A083384 (program): a(n) = n*Sum(((k-1)/2)*k!*Stirling_2(n,k),k=1..n).
  • A083385 (program): Total height of all elements in all preferential arrangements of n elements, where elements at the bottom level have height 1.
  • A083390 (program): m such that 2m + 1 divides lcm(1,3,5,…,2m - 1).
  • A083392 (program): Alternating partial sums of A000217.
  • A083398 (program): Number of diagonals needed to produce a list {1,..n} in the Wythoff array.
  • A083399 (program): Number of divisors of n that are not divisors of other divisors of n.
  • A083402 (program): Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the main diagonal and 1 elsewhere; a(n) is the size of the conjugacy class of this matrix in GL(n,2).
  • A083403 (program): Write the numbers 1, 2, … in a triangle with n terms in the n-th row; a(n) = number of squarefree integers in n-th row.
  • A083410 (program): a(n) = A083385(n)/n.
  • A083411 (program): a(n) = Sum(((k-1)/2)*k!*Stirling_2(n,k),k=1..n).
  • A083412 (program): Wythoff array read by antidiagonals.
  • A083413 (program): a(n) = Sum_{d|n} d*2^(d-1) for n > 0.
  • A083416 (program): Add 1, double, add 1, double, etc.
  • A083420 (program): a(n) = 2*4^n - 1.
  • A083421 (program): a(n)=2*5^n-2^n.
  • A083423 (program): a(n) = (5*3^n + (-3)^n)/6.
  • A083424 (program): a(n) = (5*4^n + (-2)^n)/6.
  • A083425 (program): a(n) = (5*5^n + (-1)^n)/6.
  • A083426 (program): (4*7^n+2^n)/5.
  • A083445 (program): Largest n-digit number minus the product of its digits; i.e., a(n) = 99999… (n 9’s) - 9^n.
  • A083446 (program): a(n) = ((10^n - 1) - 9^n)/9.
  • A083447 (program): a(n) = floor( n*R(n)/(n+R(n))), where R(n) is the digit reversal of n (A004086).
  • A083449 (program): a(n) = A019566(n)/9, where A019566(n) = concat(n,…,1) - concat(1,…,n).
  • A083451 (program): (n concatenated n times) - n^n.
  • A083452 (program): a(n) = (n times concatenation of n - n^n) divided by n, or a(n) = A083451(n) /n.
  • A083453 (program): a(n) = (concatenation of numbers from n to 1) - n^n.
  • A083456 (program): Smallest nontrivial k such that k^n + 1 is a palindrome (k>1 for n>1).
  • A083457 (program): Smallest nontrivial k such that k^n - 1 is a palindrome (k >1 for n>1).
  • A083460 (program): Palindromes arising in A083456.
  • A083461 (program): Palindromes arising in A083457.
  • A083474 (program): T(n)^2-n!, where T(n) is the n-th triangular number.
  • A083475 (program): Consider the set of all the numbers n*k where 1 <= k <= n. Then a(n) = number of palindromic members of this set.
  • A083476 (program): Indices of terms of A083475 with a zero entry.
  • A083477 (program): Smallest palindrome > 1 and == 1 mod n-th palindrome.
  • A083479 (program): The natural numbers with all terms of A033638 inserted.
  • A083481 (program): Smallest k such that n(n+1)*k is a square.
  • A083482 (program): Square root of smallest square of the type n(n+1)*k.
  • A083483 (program): Number of forests with two connected components in the complete graph K_{n}.
  • A083487 (program): Triangle read by rows: T(r,c) = 2*r*c + r + c (1 <= c <= r).
  • A083502 (program): Smallest k such that n*(n+k) + 1 is an n-th power.
  • A083503 (program): n-th powers arising in A083502.
  • A083504 (program): Triangle read by rows: for 1 <= k <= n, T(n, k) is the total perimeter of all squares contained in a square grid with n rows and k columns.
  • A083510 (program): Members of A000124 which are the arithmetic mean of two other members.
  • A083511 (program): Members of A000124 which are multiples of 11.
  • A083514 (program): Number of steps for iteration of map x -> (4/3)*ceiling(x) to reach an integer > 3n+1 when started at 3n+1, or -1 if no such integer is ever reached.
  • A083523 (program): Smallest Fibonacci number divisible by 2^n.
  • A083528 (program): a(n) = 5^n mod 2*n.
  • A083529 (program): a(n) = 5^n mod 3*n.
  • A083530 (program): a(n) = 7^n mod (2*n).
  • A083531 (program): First difference sequence of A002191. Differences between possible values for sum of divisors of n.
  • A083532 (program): First difference sequence of A007369. Differences between impossible values for sum of divisors of n.
  • A083537 (program): a(0) = a(1) = 0, a(2n) = a(n)+1, a(2n+1) = a(n-1).
  • A083538 (program): a(n) = sigma(n)*sigma(n+1)/gcd(sigma(n+1), sigma(n))^2.
  • A083539 (program): a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.
  • A083542 (program): a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.
  • A083548 (program): Least common multiple of cototient values of consecutive integers.
  • A083549 (program): Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.
  • A083550 (program): Product of 2 consecutive prime differences of two successive terms of A001223.
  • A083551 (program): Least common multiple of 2 consecutive prime differences, of two successive terms of A001223.
  • A083553 (program): Product of prime(n+1)-1 and prime(n)-1.
  • A083554 (program): Least common multiple of prime(n+1)-1 and prime(n)-1.
  • A083558 (program): p(p^2-p+1) as p runs through the primes.
  • A083559 (program): Nearest integer to 1/(Sum_{k>=n} 1/k^4).
  • A083564 (program): a(n) = L(n)*L(2n), where L(n) are the Lucas numbers (A000204).
  • A083570 (program): A de Bruijn sequence of length 9 over {0, 1, 2}, repeated.
  • A083572 (program): Smaller of two consecutive star numbers (A000567) whose sum is a square.
  • A083575 (program): a(0) = 6; for n>0, a(n) = 2*a(n-1) - 1.
  • A083577 (program): Prime star numbers.
  • A083578 (program): a(n) = (6^n + (-4)^n)/2.
  • A083579 (program): Generalized Jacobsthal numbers.
  • A083580 (program): Binomial transform of A083579.
  • A083581 (program): 8/3-5(-2)^n/3.
  • A083582 (program): a(n) = (8*2^n-5*(-1)^n)/3.
  • A083583 (program): a(n) = (8*3^n - 5*0^n)/3.
  • A083584 (program): a(n) = (8*4^n - 5)/3.
  • A083585 (program): (8*5^n - 5*2^n)/3.
  • A083586 (program): Binomial transform of A083580.
  • A083587 (program): a(n) = 4*3^n/3 - 5*0^n/6 - (n-1)2^(n-1).
  • A083588 (program): Binomial transform of A083587.
  • A083589 (program): Expansion of 1/((1-4*x)*(1-x^4)).
  • A083590 (program): Expansion of 1/((1-5*x)*(1-x^5)).
  • A083591 (program): Inverse binomial transform of A083589.
  • A083592 (program): Inverse binomial transform of A083590.
  • A083593 (program): Expansion of 1/((1-2*x)*(1-x^4)).
  • A083594 (program): a(n) = (7 - 4*(-2)^n)/3.
  • A083595 (program): a(n) = (7*2^n - 4(-1)^n)/3.
  • A083597 (program): a(n) = (7*4^n - 4)/3.
  • A083650 (program): Expansion of f(-x, x^3) * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.
  • A083651 (program): Triangular array, read by rows: T(n,k) = k-th bit in binary representation of n (0<=k<=n).
  • A083652 (program): Sum of lengths of binary expansions of 0 through n.
  • A083656 (program): a(n) = Sum_{i=1..n} floor(r*floor(r*i)), where r=sqrt(2).
  • A083657 (program): a(n) = Sum_{i=1..n} floor(r*floor(r*i)) where r=sqrt(3).
  • A083658 (program): a(n) = a(n-1) + a(n-2) + gcd(a(n-1), a(n-2)) for n > 1; a(0)=1, a(1)=1.
  • A083659 (program): Denominator of fraction equal to the continued fraction [p(n); p(n-1),…,5,3,2].
  • A083661 (program): G.f.: 1/(1-x) * sum(k>=0, x^2^(k+2)/(1+x^2^k)).
  • A083662 (program): a(n) = a(floor(n/2)) + a(floor(n/4)), n > 0; a(0)=1.
  • A083667 (program): Number of antisymmetric binary relations on a set of n labeled points.
  • A083669 (program): Number of ordered quintuples (a,b,c,d,e), -n <= a,b,c,d,e <= n, such that a+b+c+d+e = 0.
  • A083672 (program): Binomial transform of 1,8,48,256,1280,6144,… (cf. A002697).
  • A083679 (program): Decimal expansion of log(4/3).
  • A083680 (program): Decimal expansion of (3/2)*log(3/2).
  • A083681 (program): Sum of divisors of semiprimes.
  • A083682 (program): Sum of aliquot divisors of semiprimes.
  • A083683 (program): a(n) = 11*2^n + 1.
  • A083686 (program): a(0) = 8; for n>0, a(n) = 2*a(n-1) - 1.
  • A083690 (program): a(n) = number of partitions of n wherein the sum of the 1’s is no more than the sum of the other parts.
  • A083694 (program): a(n) = 2*A002532(n).
  • A083695 (program): a(n) = 2*a(n-1) + 5*a(n-2).
  • A083704 (program): a(n)=sum(k=1,n,floor(r*floor(k/r))) where r=sqrt(3).
  • A083705 (program): a(n) = 2*a(n-1) - 1 with a(0) = 10.
  • A083706 (program): a(n) = 2^(n+1)+n-1.
  • A083707 (program): G.f.: (x+4*x^3+x^5)/((1-x)^2*(1-x^2)^2*(1-x^3)).
  • A083708 (program): G.f.: (x+4*x^3+x^5)/((1-x)^2*(1-x^2)^2*(1-x^3)^2).
  • A083709 (program): G.f.: (1-x+2*x^2+2*x^3+2*x^4-x^5+x^6)/((1-x)*(1-x^2)^2*(1-x^3)).
  • A083710 (program): Number of integer partitions of n with a part dividing all the other parts.
  • A083711 (program): A083710(n) - A000041(n-1).
  • A083713 (program): a(n) = (8^n - 1)*3/7.
  • A083714 (program): (greatest prime <= n) - (greatest prime factor of n).
  • A083715 (program): (greatest prime <= n) mod (greatest prime factor of n).
  • A083716 (program): a(n) = integer part of (greatest prime <= n)/(greatest prime factor of n); a(1) = 1.
  • A083717 (program): (Greatest prime <= n) * (greatest prime factor of n).
  • A083718 (program): (greatest prime <= n) + (greatest prime factor of n).
  • A083719 (program): a(n) = n * [1 + sum(k=1 to n-1) prime(k)].
  • A083720 (program): Product of primes less than greatest prime factor of n but not dividing n.
  • A083721 (program): Number of primes greater than the greatest prime factor of n but not greater than n.
  • A083722 (program): Product of primes greater than the greatest prime factor of n but not greater than n.
  • A083723 (program): a(n) = (prime(n)+1)*n - 1.
  • A083725 (program): a(n) = n * [1 + sum(k=1 to n) prime(k)].
  • A083726 (program): a(n) = (prime(n)+1)*n.
  • A083727 (program): a(n) = n * (2^n - 8).
  • A083728 (program): a(0) = 1, a(n) = 480*sigma(n).
  • A083729 (program): Decimal expansion of sqrt(2)/(sqrt(2)-1)^2 = 3*sqrt(2)+4.
  • A083730 (program): Greatest prime^2 factor of n, or a(n)=1 for squarefree n.
  • A083741 (program): a(n) = L(n) + a(L(n)), where L(n) = n - 2^floor(log_2(n)) (A053645).
  • A083742 (program): First differences of A006282.
  • A083743 (program): a(1) = 1; if a(n-1) + n is prime then a(n) = a(n-1) + n, else a(n) = a(n-1).
  • A083744 (program): a(1) = 1; if a(n-1) + n is composite then a(n) = a(n-1) + n, else a(n) = a(n-1).
  • A083746 (program): a(1) = 1, a(2) = 2; for n>2, a(n) = 3*(n-2)*(n-2)!.
  • A083751 (program): Number of partitions of n into >= 2 parts and with minimum part >= 2.
  • A083782 (program): n-th row of the following triangle contains n distinct natural numbers such that every sum of n-1 of them +1 is a prime,n >1, with a(1) = 1 by convention. Sequence contains the triangle by rows.
  • A083786 (program): Composite numbers mod 10.
  • A083794 (program): Numbers n such that tau(n) is different from tau(n-1), where tau(m) = number of divisors of m.
  • A083795 (program): Numbers n such that n and n-1 have the same number of divisors. Numbers not included in A083794.
  • A083811 (program): Numbers n such that 2n+1 is the digit reversal of n+1.
  • A083812 (program): 4n-1 is the digit reversal of n-1.
  • A083813 (program): a(n) = 3*(10^n-1).
  • A083817 (program): Interleaving of (1,2,3,4,5,…), (2,2,2,2,…) and (0,1,3,6,10,…).
  • A083818 (program): Numbers n such that 2n-1 is the digit reversal of n.
  • A083822 (program): a(n) = digit reversal of 3*n, divided by 3.
  • A083823 (program): a(1) = 15; then numbers obtained at every stage of division by 3 in the following process: multiply by 3, reverse the digits, divide by 3, reverse the digits, multiply by 3, reverse the digit, divide by 3.
  • A083824 (program): a(n) = digit reversal of 9*n, divided by 9.
  • A083852 (program): Decimal palindromes that are multiples of 11.
  • A083854 (program): Numbers that are squares, twice squares, three times squares, or six times squares, i.e., numbers whose squarefree part divides 6.
  • A083856 (program): Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).
  • A083857 (program): Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.
  • A083858 (program): Expansion of x/(1 - 3*x - 6*x^2).
  • A083859 (program): Main diagonal of generalized Fibonacci array A083856.
  • A083860 (program): First subdiagonal of generalized Fibonacci array A083856.
  • A083861 (program): Square array T(n,k) of second binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.
  • A083862 (program): Main diagonal of array A083857.
  • A083866 (program): Positions of zeros in Per Nørgård’s infinity sequence (A004718).
  • A083867 (program): a(n) is the number of divisors of the n-th decimal palindrome that are palindromes.
  • A083877 (program): Absolute value of determinant of n X n matrix where the element a(i,j) = if i + j > n then 2*(i + j -n) - 1, else 2*(n + 1 - i - j).
  • A083878 (program): a(0)=1, a(1)=3, a(n)=6a(n-1)-7a(n-2), n>=2.
  • A083879 (program): a(0)=1, a(1)=4, a(n) = 8*a(n-1) - 14*a(n-2), n >= 2.
  • A083880 (program): a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 23*a(n-2), n >= 2.
  • A083881 (program): a(n) = 6*a(n-1) - 6*a(n-2), with a(0)=1, a(1)=3.
  • A083882 (program): a(0)=1, a(1)=4, a(n)=8a(n-1)-13a(n-2), n>=2.
  • A083884 (program): a(n) = (3^(2*n) + 1) / 2.
  • A083885 (program): (4^n+2^n+0^n+(-2)^n)/4
  • A083886 (program): Expansion of e.g.f. exp(3*x)*exp(x^2).
  • A083888 (program): Number of divisors of n with largest digit = 1 (base 10).
  • A083889 (program): Number of divisors of n with largest digit = 2 (base 10).
  • A083890 (program): Number of divisors of n with largest digit = 3 (base 10).
  • A083891 (program): Number of divisors of n with largest digit = 4 (base 10).
  • A083892 (program): Number of divisors of n with largest digit = 5 (base 10).
  • A083893 (program): Number of divisors of n with largest digit = 6 (base 10).
  • A083894 (program): Number of divisors of n with largest digit = 7 (base 10).
  • A083895 (program): Number of divisors of n with largest digit = 8 (base 10).
  • A083896 (program): Number of divisors of n with largest digit = 9 (base 10).
  • A083897 (program): Number of divisors of n with largest digit <= 2 (base 10).
  • A083898 (program): Number of divisors of n with largest digit <= 3 (base 10).
  • A083899 (program): Number of divisors of n with largest digit <= 4 (base 10).
  • A083900 (program): Number of divisors of n with largest digit <= 5 (base 10).
  • A083901 (program): Number of divisors of n with largest digit <= 6 (base 10).
  • A083902 (program): Number of divisors of n with the largest digit of the divisor <= 7 (base 10).
  • A083903 (program): Number of divisors of n with largest digit <= 8 (base 10).
  • A083904 (program): G.f. 1/(1-x) * Sum_{k>=0} 3^k * x^2^(k+1)/(1+x^2^k).
  • A083905 (program): G.f.: 1/(1-x) * sum(k>=0, (-1)^k*x^2^(k+1)/(1+x^2^k)).
  • A083907 (program): a(1) = 1; for n>1, a(n) = n*a(n-1) if GCD(n,a(n-1)) = 1 else a(n) = a(n-1).
  • A083909 (program): Numbers of the form 10^(m-k)*(10^(m+k+1)-10^k), m, k >= 0.
  • A083910 (program): Number of divisors of n that are congruent to 0 modulo 10.
  • A083911 (program): Number of divisors of n that are congruent to 1 modulo 10.
  • A083912 (program): Number of divisors of n that are congruent to 2 modulo 10.
  • A083913 (program): Number of divisors of n that are congruent to 3 modulo 10.
  • A083914 (program): Number of divisors of n that are congruent to 4 modulo 10.
  • A083915 (program): Number of divisors of n that are congruent to 5 modulo 10.
  • A083916 (program): Number of divisors of n that are congruent to 6 modulo 10.
  • A083917 (program): Number of divisors of n that are congruent to 7 modulo 10.
  • A083918 (program): Number of divisors of n that are congruent to 8 modulo 10.
  • A083919 (program): Number of divisors of n that are congruent to 9 modulo 10.
  • A083920 (program): Number of nontriangular numbers <= n.
  • A083921 (program): Start with (1,2) and then concatenate 2^n+1 previous terms, n>=0 and add 2 if a(2^n+1)=1 or add 1 if a(2^n+1)=2.
  • A083922 (program): Partial sums of A083921.
  • A083924 (program): Characteristic function for A072795.
  • A083931 (program): a(n) = A000695(A014486(n)).
  • A083932 (program): A014486-encoding of the Catalan mountain ranges with only even-length slopes allowed.
  • A083933 (program): A063171-encoding of A083923-trees.
  • A083937 (program): A014486-encodings of the plane binary trees and plane general trees whose left(most) subtree is just a “stick”: .
  • A083943 (program): A generalized Jacobsthal sequence.
  • A083944 (program): A generalized Jacobsthal sequence.
  • A083952 (program): Integer coefficients a(n) of A(x), where a(n) = 1 or 2 for all n, such that A(x)^(1/2) has only integer coefficients.
  • A084008 (program): a(n) = T(n) concatenated with reverse(T(n)) divided by 11, where T(n) is the n-th triangular number.
  • A084009 (program): a(n) = n^2 concatenated with reverse(n^2) divided by 11.
  • A084010 (program): a(n) = (2^n concatenated with Reverse(2^n))) divided by 11.
  • A084011 (program): Digit reversal of 11*n, divided by 11.
  • A084021 (program): Product of n and its 9’s complement.
  • A084052 (program): 2*n digit-reversed mod 2.
  • A084053 (program): 4*n digit-reversed mod 4.
  • A084054 (program): 5*n digit-reversed mod 5.
  • A084055 (program): 6*n digit-reversed mod 6.
  • A084056 (program): a(n) = -a(n-1) + a(n-2) + a(n-3), with a(0)=0, a(1)=1, a(2)=-3.
  • A084057 (program): a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=1.
  • A084058 (program): a(n) = 2*a(n-1) + 7*a(n-2) for n>1, a(0)=1, a(1)=1.
  • A084059 (program): a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=2.
  • A084060 (program): a(n) = 1/2 + (1-6*n)*(-1)^n/2.
  • A084062 (program): Main diagonal of number array A084061.
  • A084063 (program): First subdiagonal of number array A084061.
  • A084064 (program): Third row of number array A084061.
  • A084065 (program): Fourth row of number array A084061.
  • A084068 (program): a(1) = 1, a(2) = 2; a(2*k) = 2*a(2*k-1) - a(2*k-2), a(2*k+1) = 4*a(2*k) - a(2*k-1).
  • A084069 (program): Numbers k such that 7*k^2 = floor(k*sqrt(7)*ceiling(k*sqrt(7))).
  • A084070 (program): a(n) = 38*a(n-1) - a(n-2), with a(0)=0, a(1)=6.
  • A084075 (program): Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}.
  • A084076 (program): Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.
  • A084077 (program): Length of list created by n substitutions k -> Range(-abs(k+1), abs(k-1)) starting with {1}.
  • A084078 (program): Length of list created by n substitutions k -> Range[-abs(k+1), abs(k-1), 2] starting with {0}.
  • A084080 (program): Length of lists created by n substitutions k -> Range[k+1,1,-3] starting with {1}, counting down from k+1 to 1 step -3.
  • A084081 (program): Sum of lists created by n substitutions k -> Range[k+1,0,-2] starting with {0}, counting down from k+1 to 0 step -2.
  • A084083 (program): Length of lists created by n substitutions k -> Range[k+1,-Abs[k],-2] starting with {1}.
  • A084084 (program): Length of lists created by n substitutions k -> Range[0,1+Mod[k+1,3]] starting with {0}.
  • A084085 (program): Length of lists created by n substitutions k -> Range[0,Mod[k+1,4]] starting with {0}.
  • A084086 (program): a(n) = Fibonacci(2*n+1) + 2*Fibonacci(2*n-1) - 2^n - [n = 0], where [b] is the Iverson bracket of b.
  • A084087 (program): Numbers k not divisible by 3 such that the exponent of the highest power of 2 dividing k is even.
  • A084088 (program): Numbers k such that k == 2 (mod 3) and the exponent of the highest power of 2 dividing k is even.
  • A084089 (program): Numbers k such that k == 1 (mod 3) and the exponent of the highest power of 2 dividing k is even.
  • A084090 (program): Numbers k such that k is divisible by 3 or the exponent of the highest power of 2 dividing k is odd. Complement of A084087.
  • A084091 (program): Expansion of sum(k>=0, x^2^k/(1+x^2^k+x^2^(k+1))).
  • A084095 (program): First super-diagonal of number array A084061.
  • A084096 (program): Fifth row of number array A084061.
  • A084097 (program): Square array whose rows have e.g.f. exp(x)*cosh(sqrt(k)*x), k>=0, read by ascending antidiagonals.
  • A084098 (program): Expansion of e.g.f. exp(x)*tan(2*x)/2.
  • A084099 (program): Expansion of (1+x)^2/(1+x^2).
  • A084100 (program): Expansion of (1+x-x^2-x^3)/(1+x^2).
  • A084101 (program): Expansion of (1+x)^2/((1-x)*(1+x^2)).
  • A084102 (program): Inverse binomial transform of A084101.
  • A084103 (program): Expansion of (1+x)^3/(1+x^3).
  • A084104 (program): A period 6 sequence.
  • A084109 (program): n is congruent to 1 (mod 4) and is not the sum of two squares.
  • A084113 (program): Number of multiplications when calculating A084110(n).
  • A084114 (program): Number of divisions when calculating A084110(n).
  • A084115 (program): A084113(n) minus A084114(n).
  • A084116 (program): Numbers m such that A084115(m) = 1.
  • A084120 (program): a(n) = 6*a(n-1) - 3*a(n-2), a(0)=1, a(1)=3.
  • A084126 (program): Prime factor <= other prime factor of n-th semiprime.
  • A084127 (program): Prime factor >= other prime factor of n-th semiprime.
  • A084128 (program): a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
  • A084130 (program): a(n) = 8*a(n-1) - 8*a(n-2), a(0)=1, a(1)=4.
  • A084131 (program): a(n) = 10*a(n-1) - 17*a(n-2), a(0) = 1, a(1) = 5.
  • A084132 (program): a(n) = 4*a(n-1) + 6*a(n-2), a(0)=1, a(1)=2.
  • A084134 (program): a(n) = 8*a(n-1) - 6*a(n-2), a(0) = 1, a(1) = 4.
  • A084135 (program): a(n) = 10*a(n-1) - 15*a(n-2), a(0)=1, a(1)=5.
  • A084136 (program): Binomial transform of cosh(sqrt(2)*x)^2.
  • A084137 (program): Binomial transform of A084136.
  • A084150 (program): A Pell related sequence.
  • A084151 (program): Binomial transform of a Pell convolution.
  • A084152 (program): Exponential self-convolution of Jacobsthal numbers (divided by 2).
  • A084153 (program): Binomial transform of a Jacobsthal convolution.
  • A084154 (program): Binomial transform of sinh(x)*cosh(sqrt(2)*x).
  • A084155 (program): A Pell-related fourth-order recurrence.
  • A084156 (program): Binomial transform of sinh(x)*cosh(sqrt(3)*x).
  • A084157 (program): a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.
  • A084158 (program): a(n) = A000129(n)*A000129(n+1)/2.
  • A084159 (program): Pell oblongs.
  • A084169 (program): A Pell Jacobsthal product.
  • A084170 (program): a(n) = (5*2^n + (-1)^n - 3)/3.
  • A084171 (program): Binomial transform of generalized Jacobsthal numbers A084170.
  • A084172 (program): a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).
  • A084173 (program): a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).
  • A084174 (program): a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).
  • A084175 (program): Jacobsthal oblong numbers.
  • A084177 (program): Binomial transform of Jacobsthal oblongs.
  • A084178 (program): Inverse binomial transform of Fibonacci oblongs.
  • A084179 (program): Expansion of the g.f. x/((1+2x)(1-x-x^2)).
  • A084181 (program): 2^n+(-2)^n-(-1)^n.
  • A084182 (program): 3^n+(-1)^n-[1/(n+1)], where [] represents the floor function.
  • A084183 (program): Jacobsthal reverse-pair sequence.
  • A084184 (program): Partial sums of a Jacobsthal related sequence.
  • A084188 (program): a(0)=1, a(n+1) = 2*a(n) + b(n+2), where b(n)=A004539(n) is the n-th bit in the binary expansion of sqrt(2).
  • A084196 (program): Number of primes q<prime(n) such that q+1 divides prime(n)+1.
  • A084213 (program): Binomial transform of A081250.
  • A084214 (program): Inverse binomial transform of a math magic problem.
  • A084215 (program): Expansion of g.f.: (1+x^2)/(1-2*x).
  • A084218 (program): a(n) = sigma_4(n^2)/sigma_2(n^2).
  • A084219 (program): Inverse binomial transform of A053088.
  • A084220 (program): a(n) = sigma_6(n^2)/sigma_3(n^2).
  • A084221 (program): a(n+2) = 4*a(n), with a(0)=1, a(1)=3.
  • A084222 (program): a(n) = -2*a(n-1) + 3*a(n-2), with a(0)=1, a(1)=2.
  • A084228 (program): a(1)=1, a(2)=2; thereafter a(n) = sum of digits of (a(1)+a(2)+a(3)+…+a(n-1)).
  • A084231 (program): Numbers k such that the root-mean-square value of 1, 2, …, k, i.e., sqrt((1/k)*Sum_{j=1..k} j^2), is an integer.
  • A084232 (program): RMS values associated with A084231.
  • A084240 (program): a(n) = -5*a(n-1) - 4*a(n-2), a(0)=1, a(1)=0.
  • A084241 (program): a(n) = -5*a(n-1)-4*a(n-2) with n>1, a(0)=0, a(1)=1.
  • A084244 (program): a(0)=1, a(1)=5, a(n) = -3*a(n-1), n>1.
  • A084245 (program): a(n) = Mod[n+(Mod[Prime[n],7]-3),10]
  • A084247 (program): a(n) = -a(n-1) + 2a(n-2), a(0)=1, a(1)=2.
  • A084253 (program): a(n) is the denominator of the coefficient of z^(2n-1) in the Maclaurin expansion of Sqrt[Pi]Erfi[z].
  • A084258 (program): Decimal expansion of Sum_{k>=1} coth(Pi*k)/k^3.
  • A084259 (program): a(n) = n! * (n+1)! * (n+2)! * (n+3).
  • A084261 (program): A binomial transform of factorial numbers.
  • A084262 (program): Binomial transform of double factorials.
  • A084263 (program): a(n) = (-1)^n/2+(n^2+n+1)/2.
  • A084264 (program): Binomial transform of A084263.
  • A084265 (program): a(n) = (n^2 + 3*n + 1 + (-1)^n) / 2.
  • A084266 (program): Binomial transform of A084265.
  • A084267 (program): Partial sums of a binomial quotient.
  • A084290 (program): Difference between consecutive primes arising before difference (d=2) between twin primes. In A001223, terms before those ones which equal to two.
  • A084291 (program): Difference between consecutive primes arising after difference (d=2) between twin primes. In A001223, terms following those ones which equal to two.
  • A084294 (program): Number of primes in the interval [prime(n),n+prime(n)].
  • A084300 (program): a(n) = phi(n) mod 6.
  • A084301 (program): a(n) = sigma(n) mod 6.
  • A084302 (program): Remainder of tau(n) modulo 6.
  • A084309 (program): a(n) = gcd(prime(n)-1, n).
  • A084310 (program): a(n) = gcd(prime(n)+1, n).
  • A084311 (program): a(n) = gcd(prime(n)-1,n-1).
  • A084320 (program): Number of powers of two between 2 consecutive factorials (2! including).
  • A084326 (program): a(0)=0, a(1)=1; for n>1, a(n) = 6*a(n-1)-4*a(n-2).
  • A084328 (program): a(0)=0, a(1)=1; a(n) = 13*a(n-1) - 11*a(n-2).
  • A084329 (program): a(0)=0, a(1)=1, a(n)=20a(n-1)-20a(n-2).
  • A084330 (program): a(0)=0, a(1)=1, a(n) = 31*a(n-1) - 29*a(n-2).
  • A084338 (program): a(1) = 1, a(2) = 2, a(3) = 3, a(n+3) = a(n) + a(n+1).
  • A084339 (program): 7*n digit-reversed mod 7.
  • A084340 (program): 8*n digit-reversed mod 8.
  • A084341 (program): 13*n digit-reversed mod 13.
  • A084345 (program): Numbers with a nonprime number of 1’s in their binary expansion (complement of A052294).
  • A084346 (program): Triangle read by rows in which row n gives decomposition of Fib(n)*Fib(n+1) into non-adjacent Fibonacci numbers (given by their indices).
  • A084348 (program): Triangle in which row n gives periodic part of a certain map.
  • A084349 (program): Squarefree numbers that are not the sum of two squares.
  • A084351 (program): Length of period of sequences r(k,n)=floor(sin(1)*k!)-n*floor(sin(1)*k!/n) when n is fixed.
  • A084357 (program): Number of sets of sets of lists.
  • A084359 (program): a(n) = number of partitions of n into pair of parts n=p+q, p>=q>=1, with p-q equal to a square >= 0.
  • A084360 (program): Number of partitions of n into pair of parts whose difference is a prime.
  • A084363 (program): a(n) = n^(n+1) - (n-1)^n.
  • A084364 (program): Define the operations M: multiply by 11, D: divide by 11, R: reverse digits. Sequence gives trajectory of 19 under action of M,R,D,R.
  • A084365 (program): Squarefree kernel of numbers that are not prime powers.
  • A084367 (program): a(n) = n*(2*n+1)^2.
  • A084368 (program): Prime(n) does not contain the digit 1.
  • A084369 (program): Numbers n such that the n-th prime number doesn’t contain any even digits.
  • A084371 (program): Squarefree kernels of powerful numbers (A001694).
  • A084372 (program): Least composite k such that nearest integer to average of smallest and largest prime factors of k equals n.
  • A084376 (program): G.f.: (1+x)/Product_{m>0} (1 - x^m).
  • A084377 (program): a(n) = n^3 + 7.
  • A084378 (program): a(n) = n^3 + 3.
  • A084379 (program): a(n) = n^3 + 17.
  • A084380 (program): a(n) = n^3 + 2.
  • A084381 (program): a(n) = n^3 + 5.
  • A084382 (program): a(n) = n^3 + 6.
  • A084383 (program): a(0)=0; for n>0, a(n) = smallest number that is not a concatenation of any number of distinct earlier terms in increasing order.
  • A084386 (program): Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n-1) + 3*a(n-3), with a(0) = a(1) = a(2) = 1.
  • A084400 (program): a(1) = 1; for n>1, a(n) = smallest number that does not divide the product of all previous terms.
  • A084421 (program): A005187(A000040(n)).
  • A084431 (program): Expansion of (1 + 6*x + 5*x^2)/((1-2*x)*(1+2*x)).
  • A084432 (program): G.f.: 2/(1-x) + sum(k>=0, t^2(3-t)/(1+t)/(1-t)^2, t=x^2^k).
  • A084439 (program): Number of triangular partitions of n of order 3.
  • A084468 (program): Odd numbers with exactly 3 ones in binary expansion.
  • A084471 (program): Change 0 to 00 in binary representation of n.
  • A084472 (program): Write n in binary and replace 0 with 00.
  • A084473 (program): Replace 0 with 0000 in binary representation of n.
  • A084474 (program): Write n in binary and replace 0 with 0000.
  • A084477 (program): Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.
  • A084480 (program): Number of tilings of a 4 X 2n rectangle with L tetrominoes.
  • A084481 (program): Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.
  • A084487 (program): Prime(n)+q, where q is the next prime such that q mod 4 = prime(n) mod 4.
  • A084488 (program): Prime(n)+q, where q is the next prime such that q mod 4 = (prime(n)+2) mod 4.
  • A084500 (program): a(0)=0, after which each n occurs A084506(n) times.
  • A084505 (program): Partial sums of A084506.
  • A084506 (program): The length of each successively larger 3-ball ground-state site swap given in A084501, i.e., the number of digits in each term of A084502.
  • A084508 (program): Partial sums of A084509. Positions of ones in the first differences of A084506.
  • A084509 (program): Number of ground-state 3-ball juggling sequences of period n.
  • A084515 (program): Partial sums of A084516.
  • A084516 (program): The length of each successively larger, indecomposable 3-ball ground-state site swap given in A084511, i.e., the number of digits in each term of A084512.
  • A084518 (program): Partial sums of A084519. Positions of ones in the first differences of A084516.
  • A084519 (program): Number of indecomposable ground-state 3-ball juggling sequences of period n.
  • A084520 (program): a(0)=0, after which each n occurs A084526(n) times.
  • A084525 (program): Partial sums of A084526.
  • A084526 (program): The length of each successively larger, indecomposable, ‘prime’ 3-ball ground-state site swap given in A084521, i.e., the number of digits in each term of A084522.
  • A084534 (program): Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).
  • A084535 (program): a(n) = floor(n^2 - n^(3/2)).
  • A084536 (program): Triangular array related to Motzkin triangle A026300.
  • A084543 (program): a(2,n) as defined in A003148.
  • A084544 (program): Alternate number system in base 4.
  • A084545 (program): Alternate number system in base 5.
  • A084546 (program): Triangle read by rows: T(n,k) = C( C(n,2), k) for n >= 1, 0 <= k <= C(n,2).
  • A084555 (program): Partial sums of A084556.
  • A084556 (program): n occurs n! times.
  • A084557 (program): a(0)=0, after which each n occurs A084556(n) times.
  • A084558 (program): a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.
  • A084561 (program): Numbers with a square number of 1’s in their binary expansion.
  • A084566 (program): a(0)=1, a(2n) = 2a(2n-1)+a(n), a(2n+1) = 2a(2n)+2a(n).
  • A084567 (program): Binomial transform of (1,-1,4,-16,64,-256,1024,….)=(3*0^n-(-4)^n)/4.
  • A084568 (program): a(0)=1, a(1)=5, a(n+2)=4a(n), n>0.
  • A084569 (program): Partial sums of A084570.
  • A084570 (program): Partial sums of A084263.
  • A084571 (program): Let a(1)=1; for n>1, a(n)=nextprime((3/2)*a(n-1)).
  • A084600 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x+2x^2)^n for n >= 0.
  • A084601 (program): Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2x^2)^n.
  • A084602 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 3x^2)^n.
  • A084603 (program): Coefficients of 1/sqrt(1 - 2*x - 11*x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.
  • A084604 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 4x^2)^n.
  • A084605 (program): G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.
  • A084606 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+2x^2)^n.
  • A084608 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.
  • A084609 (program): Coefficients of 1/(1-4x-8x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+3x^2)^n.
  • A084610 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-x^2)^n.
  • A084612 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+x-2x^2)^n.
  • A084614 (program): Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 3x^2)^n.
  • A084623 (program): Numerator of 2^(n-1)/n.
  • A084624 (program): floor(C(n+5,5)/C(n+2,2)).
  • A084625 (program): Binomial transform of A084624.
  • A084626 (program): Floor(C(n+6,6)/C(n+2,2)).
  • A084627 (program): Floor(C(n+6,6)/C(n+3,3)).
  • A084628 (program): a(n) = floor(binomial(n+7,7)/binomial(n+3,3)).
  • A084630 (program): Floor(C(n+7,7)/C(n+5,5)).
  • A084631 (program): Floor(C(n+8,8)/C(n+2,2)).
  • A084633 (program): Inverse binomial transform of repeated odd numbers.
  • A084634 (program): Binomial transform of 1,1,1,2,2,2,2,…
  • A084635 (program): Binomial transform of 1,0,1,0,1,1,1,…
  • A084636 (program): Binomial transform of (1,0,1,0,1,0,2,0,2,0,2,0….).
  • A084637 (program): Binomial transform of (1,0,1,0,1,0,1,1,1,1,1,….).
  • A084638 (program): Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,….).
  • A084639 (program): Expansion of x*(1+2*x)/((1+x)*(1-x)*(1-2*x)).
  • A084640 (program): Generalized Jacobsthal numbers.
  • A084641 (program): Binomial transform of n^7.
  • A084642 (program): A Jacobsthal ratio.
  • A084643 (program): a(n) = 3^(n-1)*(2*n-3) + 2^(n+1).
  • A084645 (program): Hypotenuses for which there exists a unique integer-sided triangle.
  • A084646 (program): Hypotenuses for which there exist exactly 2 distinct integer triangles.
  • A084648 (program): Hypotenuses for which there exist exactly 4 distinct integer triangles.
  • A084658 (program): Number of unlabeled 3-connected claw-free cubic graphs on 2n vertices.
  • A084660 (program): Decimal expansion of solution of area bisectors problem.
  • A084662 (program): a(1) = 4; a(n) = a(n-1) + gcd(a(n-1), n).
  • A084663 (program): a(1) = 8; a(n) = a(n-1) + gcd(a(n-1), n).
  • A084672 (program): G.f.: (1+x^2+2*x^4)/((1-x^3)*(1-x)^2).
  • A084675 (program): Product of the first n digits of the Golden Ratio phi = 1.6180339… (treating 0’s as if they were 1’s).
  • A084678 (program): a(n)=b(n,n) with b(n,1)=n and b(n,k)=binomial(b(n,k-1),d(n,n-k+1)) for 1<k<=n, where d(n,i) are the divisors of n, d(i)<d(j), 1<=i<j<=A000005(n).
  • A084680 (program): Order of 10 modulo n [i.e., least m such that 10^m = 1 (mod n)] or 0 when no such number exists.
  • A084681 (program): Order of 10 modulo 9n [i.e., least m such that 10^m = 1 (mod 9n)] or 0 when no such number exists.
  • A084683 (program): G.f.: (1+2*x^3+2*x^6)/((1-x)*(1-x-x^2+x^3-x^4-x^5+x^6)).
  • A084684 (program): Degrees of certain maps (see Comments and Formulas for more precise definitions).
  • A084686 (program): Take n-th prime p(n), rewrite it with digits in decreasing order to get b(n), then a(n)=(b(n)-p(n))/9.
  • A084688 (program): Nonnegative integers n such that 2^n uses only distinct decimal digits.
  • A084694 (program): Squarefree numbers which are products of three consecutive numbers. I.e., squarefree numbers of the form k^3 - k.
  • A084697 (program): a(1) = 2; for n >= 1, k>=1, a(n+1) = a(n) + k*n is the smallest such prime.
  • A084703 (program): Squares k such that 2*k+1 is also a square.
  • A084707 (program): a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 27; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4) for n > 3.
  • A084737 (program): Beginning with 1, numbers such that {a(n+2)-a(n+1)}/{a(n+1)-a(n)} =prime(n).
  • A084744 (program): Product of all composite numbers from 1 to the n-th nonprime number divided by product of all the prime divisors of each of those composite numbers which exceed the previously stated value.
  • A084747 (program): Leading diagonal of triangle shown below in which the n-th row contains the n smallest numbers > 0 such that when they are incremented by n yield a prime.
  • A084756 (program): For n, k > 0, let T(n, k) be given by T(n, 1) = n and T(n, k+1) = k*T(n, k)+1. Then a(n) = T(n, n).
  • A084757 (program): For n, k > 0, let T(n, k) be given by T(n, 1) = n and T(n, k+1) = k*T(n, k)+1. a(n) is the sum of the n-th antidiagonal.
  • A084768 (program): P_n(7), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 7*x + 12*x^2)^n.
  • A084769 (program): P_n(9), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 9*x + 20*x^2)^n.
  • A084770 (program): Coefficients of 1/(1-4x-16x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+5x^2)^n.
  • A084771 (program): Coefficients of expansion of 1/sqrt(1 - 10*x + 9*x^2); also, a(n) is the central coefficient of (1 + 5*x + 4*x^2)^n.
  • A084772 (program): Coefficients of 1/sqrt(1 - 12*x + 16*x^2); also, a(n) is the central coefficient of (1 + 6*x + 5*x^2)^n.
  • A084773 (program): Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6x+8x^2)^n.
  • A084774 (program): Coefficients of 1/sqrt(1-14*x+9*x^2); also, a(n) is the central coefficient of (1+7x+10x^2)^n.
  • A084783 (program): Triangle, read by rows, such that the diagonal (A084785) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.
  • A084784 (program): Binomial transform = self-convolution: first column of the triangle (A084783).
  • A084785 (program): Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).
  • A084792 (program): Primes other than prime(prime(n)+n-1).
  • A084840 (program): Write the numbers 1, 2, … in a triangle with n terms in the n-th row; a(n) = number of abundant integers in n-th row.
  • A084841 (program): Write the numbers 1, 2, … in a triangle with n terms in the n-th row; a(n) = number of deficient integers in n-th row.
  • A084844 (program): Denominators of the continued fraction n + 1/(n + 1/…) [n times].
  • A084845 (program): Numerators of the continued fraction n+1/(n+1/…) [n times].
  • A084847 (program): 2*3^n+2^(2n-1)*(n-2).
  • A084849 (program): a(n) = 1 + n + 2*n^2.
  • A084850 (program): 2^(n-1)*(n^2+2n+2).
  • A084851 (program): Binomial transform of binomial(n+2,2).
  • A084854 (program): Triangular array, read by rows: T(n,k) = concatenated decimal representations of n and k, 1<=k<=n.
  • A084856 (program): Prime(n+2)^2-prime(n)^2.
  • A084857 (program): Inverse binomial transform of n^2*3^(n-1).
  • A084858 (program): Binomial transform of A001651.
  • A084859 (program): Binomial transform of Cullen numbers A002064.
  • A084860 (program): Expansion of (1 - 2x + 2x^2 - x^3)/(1 - 2x)^2.
  • A084861 (program): Expansion of (1-3x+4x^2-3x^3+x^4)/(1-2x)^2.
  • A084863 (program): Number of solutions to n = 2*u^2 + 3*v^2, u*v>0.
  • A084864 (program): Numbers that can be written in the form 2*u^2 + 3*v^2, u*v>0.
  • A084865 (program): Primes of the form 2x^2 + 3y^2.
  • A084868 (program): Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.
  • A084869 (program): Number of 2-multiantichains of an n-set.
  • A084874 (program): Number of (k,m,n)-antichains of multisets with k=3 and m=2.
  • A084879 (program): Number of (k,m,n)-multiantichains of multisets with k=3 and m=2.
  • A084888 (program): Number of partitions of n^3 into two squares>0.
  • A084890 (program): Triangular array, read by rows: T(n,k) = arithmetic derivative of n*k, 1<=k<=n.
  • A084891 (program): Multiples of 2, 3, 5, or 7, but not 7-smooth.
  • A084899 (program): Binomial transform of heptagonal numbers A000566.
  • A084900 (program): 3^(n-2)n(5n+1)/2.
  • A084901 (program): a(n) = 4^(n-2)*n*(5*n+3)/2.
  • A084902 (program): a(n) = 5^(n-1)*n*(n+1)/2.
  • A084903 (program): Binomial transform of positive cubes.
  • A084911 (program): Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + …
  • A084915 (program): a(n) = (n!)^2*n.
  • A084918 (program): Numbers n >= 1000, such that if prime P divides n, then so does each smaller prime.
  • A084919 (program): First differences of A048093.
  • A084920 (program): a(n) = (prime(n)-1)*(prime(n)+1).
  • A084921 (program): a(n) = lcm(p-1, p+1) where p is the n-th prime.
  • A084922 (program): a(n) = (prime(n)-1)*(prime(n)+1)/6.
  • A084930 (program): Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x).
  • A084934 (program): Rectangular array T(m,n) (m>=1, n>=1) read by antidiagonals: row m consists of the numbers ( i + mj : i >= 0, j >= 0 ), sorted in increasing order, with repetitions allowed.
  • A084935 (program): Diagonal sums of the array T in A084934.
  • A084936 (program): Nonsquarefree numbers divided by their squarefree kernels.
  • A084939 (program): Pentagorials: n-th polygorial for k=5.
  • A084940 (program): Heptagorials: n-th polygorial for k=7.
  • A084941 (program): Octagorials: n-th polygorial for k=8.
  • A084942 (program): Enneagorials: n-th polygorial for k=9.
  • A084943 (program): Decagorials: n-th polygorial for k=10.
  • A084944 (program): Hendecagorials: n-th polygorial for k=11.
  • A084947 (program): a(n) = Product_{i=0..n-1} (7*i+2).
  • A084948 (program): a(n) = Product_{i=0..n-1} (8*i+2).
  • A084949 (program): a(n) = Product_{i=0..n-1} (9*i+2).
  • A084964 (program): Follow n+2 by n. Also solution of a(n+2)=a(n)+1, a(0)=2, a(1)=0.
  • A084967 (program): Multiples of 5 whose GCD with 6 is 1.
  • A084968 (program): Multiples of 7 coprime to 30.
  • A084969 (program): Numbers whose smallest prime factor is 11.
  • A084970 (program): Numbers whose smallest prime factor is 13.
  • A084978 (program): Number of ways to represent n as a+b*(c+d*(e+f*(…x+y*(z)…))) in positive integers.
  • A084980 (program): Triangle of (multi)factorials: n-th row is (n+1)!!… {n “!”s}, (n+1)!… {n-1 “!”s}, …, (n+1)!.
  • A084984 (program): Numbers containing no prime digits.
  • A084990 (program): a(n) = n*(n^2+3*n-1)/3.
  • A084996 (program): Numbers which can be written as the product of two distinct primes and containing only prime decimal digits.
  • A085001 (program): a(n) = (3*n+1)*(3*n+4).
  • A085002 (program): a(n) = floor(phi*n) - 2*floor(phi*n/2) where phi is the golden ratio.
  • A085003 (program): Partial sums of A085002.
  • A085004 (program): a(n)=2*A085003(n)-n.
  • A085005 (program): A Von Koch curve related to the Golden ratio.
  • A085006 (program): Let S(0)={1,1,2} S(n)={S(n-1), S(n-1)-{x},{3-x}} where x is the last element of S(n-1), then sequence is S(infinity).
  • A085007 (program): Partial sums of A085006.
  • A085020 (program): a(n) = Sum_{d|n, (d+1) prime} (d + 1).
  • A085025 (program): a(n) = (5*n+1)*(5*n+6).
  • A085026 (program): a(n) = (6*n+1)*(6*n+7).
  • A085027 (program): a(n) = (4*n+3)*(4*n+7).
  • A085036 (program): a(n) = (5*n+2)*(5*n+7).
  • A085037 (program): Smallest square divisible by the n-th triangular number (n(n+1)/2).
  • A085046 (program): a(n) = n^2 - (1 + (-1)^n)/2.
  • A085053 (program): Number of primes of the form nk+1, where k=1 to n; 0 if no such number exists.
  • A085056 (program): (Product of all composite numbers from 1 to n)/ ( product of the prime divisors of all composite numbers up to n). More precisely, denominator = product of the largest squarefree divisors of composite numbers up to n.
  • A085057 (program): a(n) is the smallest integer of the form a*b*c…/p*q*r…, where the numerator and the denominator contain n numbers each and a,b,c,…p,q,r… are all the integers from 1 to 2n.
  • A085058 (program): a(n) = A001511(n+1) + 1.
  • A085059 (program): a(1) = 1, a(n+1) = a(n)-n if a(n) > n else a(n+1) = a(n) + n.
  • A085060 (program): Integer reached in A085058.
  • A085062 (program): A085060(n)/9 - 1/3.
  • A085068 (program): Number of steps >= 1 for iteration of map x -> (4/3)*ceiling(x) to reach an integer when started at n, or -1 if no such integer is ever reached.
  • A085071 (program): Integers reached in A085068.
  • A085078 (program): The largest number with the prime signature of n! using primes <= n.
  • A085086 (program): a(1) = 1; if n is composite then a(n) = Sum_{i < n, i not prime} a(i), else if n is prime then a(n) = sum_{ j < n, j is a noncomposite} a(j).
  • A085087 (program): a(1) = 1; for n>1, a(n) = a(n-1)*n if n is prime, a(n) = a(n-1)/n if n is composite dividing a(n-1) else a(n) = a(n-1).
  • A085089 (program): Number of distinct prime signatures arising up to n.
  • A085090 (program): If 2n-1 is prime then a(n) = 2n-1, otherwise a(n) = 0.
  • A085097 (program): Ramanujan sum c_n(3).
  • A085099 (program): Least natural number k such that k^2 + n is prime.
  • A085110 (program): a(1)=1, then add 1 multiply by 2 to get a(2), subtract 1 and multiply by 3 to get a(3), add 1 and multiply by 4 to get a(4) and so on.
  • A085117 (program): Decimal expansion of largest Stoneham number S(3,2).
  • A085118 (program): Primes together with twice the odd primes.
  • A085121 (program): Number of ways of writing n as the sum of three odd squares.
  • A085125 (program): Multiples of 2 which are members of A002473. Or multiples of 2 with the largest prime divisor < 10.
  • A085126 (program): Multiples of 3 which are members of A002473. Or multiples of 3 with the largest prime divisor < 10.
  • A085127 (program): Multiples of 4 which are members of A002473. Or multiples of 4 with the largest prime divisor < 10.
  • A085128 (program): Multiples of 5 which are members of A002473. Or multiples of 5 with the largest prime divisor <= 7.
  • A085129 (program): Multiples of 6 which are members of A002473. Or multiples of 6 with the largest prime divisor < 10.
  • A085131 (program): Multiples of 8 which are members of A002473. Or multiples of 8 with the largest prime divisor < 10.
  • A085132 (program): Multiples of 9 which are members of A002473. Or multiples of 9 with the largest prime divisor < 10.
  • A085139 (program): a(n) = Sum_{i=0..n-1} (1 + (-1)^(n-1-i))/2 * Sum_{j=0..i} a(j)*a(i-j) for n > 0, with a(0) = 1.
  • A085140 (program): Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.
  • A085141 (program): Greatest nonnegative integer k such that k(3k+1)/2 <= n.
  • A085144 (program): a(0)=0, a(2n) = a(n)+1, a(2n+1) = -a(n).
  • A085145 (program): Positions of 0 in A085144.
  • A085151 (program): Numbers generated by the Fibonacci polynomial x^4 + 3x^2 + 1.
  • A085152 (program): All prime factors of n and n+1 are <= 5. (Related to the abc conjecture.)
  • A085157 (program): Quintuple factorials, 5-factorials, n!!!!!, n!5.
  • A085158 (program): Sextuple factorials, 6-factorials, n!!!!!!, n!6.
  • A085182 (program): n occurs A076050(n) (= A007001(n)+1) times.
  • A085183 (program): a(n) = A053645(A057520(n)), i.e., the terms of A014486 without their most significant bit (1) and the least significant bit (0).
  • A085184 (program): Sequence A085183 shown in base 4. Quaternary code for binary trees.
  • A085185 (program): Sequence A014486 shown in base 4.
  • A085186 (program): Sequence A085195 shown in base 4.
  • A085192 (program): First differences of A014486.
  • A085193 (program): Repeating part of A085192.
  • A085194 (program): Terms of A085193 halved. The repeating part in the first differences of A057520.
  • A085195 (program): Partial sums of A085194.
  • A085197 (program): Positions of ones in A007001. Repeating part in each sub-permutation A082315[A014137(n-1)..A014138(n-1)] normalized to begin from 1.
  • A085207 (program): Array A(x,y): concatenation of binary expansions of x & y, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), … Zero is expanded as an empty string.
  • A085208 (program): Transpose of A085207.
  • A085209 (program): Array A085207 in binary.
  • A085210 (program): Array A085208 in binary.
  • A085223 (program): Row 1 of A085201.
  • A085224 (program): A014486-encodings of the plane general trees whose rightmost subtree (branching from the root) is just a stick: /.
  • A085225 (program): Row 1 of A085203.
  • A085234 (program): (Greatest power of smallest prime factor of n) < square root(n).
  • A085238 (program): Sort the numbers 2^i and 3^j. Then a(n) is the exponent of the n-th term.
  • A085239 (program): Sort the numbers 2^i and 3^j. Then a(n) is the base of the n-th term. Set a(1)=1.
  • A085241 (program): a(n) = A085239(n+1) - A085239(n).
  • A085242 (program): a(n) = A085239(n) - 2.
  • A085243 (program): a(n) = A085238(n+1) - A085238(n).
  • A085246 (program): Satisfies a(1)=1, a(A026835(n+1)) = a(n)+1, with a(m)=0 for all m not found in A026835, where A026835(n+2)=A026835(n+1)+a(n)+1.
  • A085250 (program): 4 times hexagonal numbers: a(n) = 4*n*(2*n-1).
  • A085252 (program): Number of ways to write n as sum of two powerful numbers (A001694).
  • A085253 (program): Numbers having no representation as sum of two powerful numbers (A001694).
  • A085254 (program): Numbers having a unique representation as sum of two powerful numbers (A001694).
  • A085259 (program): Integer part of the conversion from Centigrade to Fahrenheit.
  • A085260 (program): Ratio-determined insertion sequence I(0.0833344) (see the link below).
  • A085262 (program): Indices of nonzero terms of A085246, where a(n+2)=a(n+1)+A085246(n)+1 and a(2^(n-1)+1)=2^n.
  • A085263 (program): Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).
  • A085264 (program): Smallest number with exactly n representations as sum of a squarefree number (A005117) and a square (A000290).
  • A085265 (program): Numbers that can be written as sum of a positive squarefree number and a positive square.
  • A085267 (program): Numbers having at least two representations as sum of a squarefree number and a nonzero square.
  • A085268 (program): Integer part of the conversion from Fahrenheit to Centigrade.
  • A085269 (program): Integer part of the conversion from centimeters to inches.
  • A085270 (program): Integer part of the conversion from miles to kilometers.
  • A085271 (program): Difference between n-th composite number and its smallest prime divisor.
  • A085273 (program): Difference between n-th composite number and its largest prime divisor.
  • A085275 (program): Sum of n-th composite number and its largest prime divisor.
  • A085277 (program): Expansion of (1+x)^2/((1-2x)(1-3x)).
  • A085278 (program): Expansion of (1+2x)^2/((1-x^2)(1-2x)).
  • A085279 (program): Expansion of (1 - 2*x - 2*x^2)/((1 - 2*x)*(1 - 3*x)).
  • A085280 (program): Expansion of (1-4x+x^2)/((1-x)(1-3x)(1-4x)).
  • A085281 (program): Expansion of (1 - 3*x + x^2)/((1-2*x)*(1-3*x)).
  • A085282 (program): Expansion of (1 - 5*x + 5*x^2)/((1-x)*(1-3*x)*(1-4*x)).
  • A085283 (program): a(n) = n*n^n - (n-1)(n-1)^n.
  • A085284 (program): C(n+3,3)n^3/4.
  • A085287 (program): Expansion of (1+4x)/((1-x^2)(1-3x)).
  • A085292 (program): Product of Lucas (A000204) and a Pell companion series (A001333).
  • A085293 (program): Product of Lucas (A000204) and a Pell Companion series (A002203).
  • A085296 (program): Runs of zeros in Catalan sequence modulo 3: consecutive occurrences of binomial(2*k,k)/(k+1) == 0 (mod 3).
  • A085297 (program): Nonzero residues of Catalan sequence modulo 3; related to the Thue-Morse sequence (A001285).
  • A085301 (program): Number of factorials between two primorials.
  • A085302 (program): a(n) is the partial sum of A085301(j) from j=1 to n; a(n)-1 shows the number of factorials below n-th primorial.
  • A085303 (program): Positions of 2 in A085301.
  • A085314 (program): Number of distinct 11th powers modulo n.
  • A085333 (program): a(n) is the least n-th power of a prime that is the sum of two positive cubes, or 0 if no solution exists (for n=3k).
  • A085339 (program): Modulo 91 remainders of 6th powers.
  • A085340 (program): a(n) is the value of determinant of the following special matrix: diagonal values equal to n-2; upper triangular entries equal to -1; lower triangular values are +1.
  • A085341 (program): Number of primes between sigma(n) and n.
  • A085342 (program): Number of primes between phi(n) and n, where n is included in the count if it is a prime, while phi(n) is never included in the count even if it is a prime.
  • A085343 (program): Number of primes between sigma(n) and phi(n).
  • A085348 (program): Ratio-determined insertion sequence I(0.264) (see the link below).
  • A085349 (program): Ratio-determined insertion sequence I(0.26688) (see the link below).
  • A085350 (program): Binomial transform of poly-Bernoulli numbers A027649.
  • A085351 (program): Expansion of (1-3*x)/((1-4*x)*(1-5*x)).
  • A085352 (program): Expansion of (1-4x)/((1-5x)(1-6x)).
  • A085353 (program): Expansion of (1-5x)/((1-6x)(1-7x)).
  • A085354 (program): a(n) = 3*4^n - (n+4)*2^(n-1).
  • A085356 (program): a(n) = polygorial(n,3)/polygorial(3,n), n >= 3.
  • A085357 (program): Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).
  • A085358 (program): Runs of zeros in binomial(3k,k)/(2k+1) (Mod 2): relates ternary trees (A001764) to the infinite Fibonacci word (A003849).
  • A085360 (program): Partial sums of A026905; the convolution of the natural numbers with the partition function.
  • A085362 (program): a(0)=1; for n>0, a(n) = 2*5^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
  • A085363 (program): a(0)=1, for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
  • A085364 (program): a(0)=1, for n>0: a(n) = 6*13^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
  • A085368 (program): Sum of numerators and denominators of convergents to 1/e.
  • A085369 (program): Cutting sequence for 1/e.
  • A085370 (program): Niven (or Harshad) numbers that are not divisible by 3.
  • A085371 (program): Non-Niven (or non-Harshad) numbers that are divisible by 3.
  • A085373 (program): a(n) = binomial(2n+1, n+1)*binomial(n+2, 2).
  • A085374 (program): a(n) = binomial(2n+1, n+1)*binomial(n+3, 3).
  • A085375 (program): a(n) = binomial(2*n+1, n+1)*binomial(n+4, 4).
  • A085376 (program): Ratio-dependent insertion sequence I(0.36704) (see the link below).
  • A085377 (program): a(n) = 15n^2 + 13n^3.
  • A085378 (program): Difference between primes p and the largest prime divisor of p-1.
  • A085382 (program): Sum of prime p and largest prime divisor of p-1.
  • A085383 (program): Triangle read by rows, 1 <= k <= n: T(n,k) = floor(n/k)*ceiling(n/k).
  • A085384 (program): Ramanujan sum c_n(4).
  • A085385 (program): Binomial transform of hexagonal pyramidal numbers A002412.
  • A085386 (program): E.g.f. cosh(x+x^2/2).
  • A085387 (program): E.g.f exp(x)*cosh(x+x^2/2).
  • A085388 (program): First differences of k^n.
  • A085389 (program): a(n) = (n*(n+1)^(n-1) + 0^n)/(n+1).
  • A085390 (program): a(n) = (n(n+1)^(n-2)+0^(n-2))/(n+1).
  • A085391 (program): Square array of centered numbers, read by antidiagonals.
  • A085392 (program): a(n) = largest prime divisor of n, or 1 if n is 1 or a prime.
  • A085403 (program): Expansion of (1-x+sqrt(1-6x+x^2))/2 in powers of x.
  • A085405 (program): Common residues of binomial(3n+2,n+1)/(3n+2) modulo 2.
  • A085407 (program): Runs of zeros in binomial(3k+2,k+1)/(3k+2) modulo 2 (A085405).
  • A085409 (program): Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.
  • A085415 (program): Take prime[n] and continue adding 1, 2, …, a(n) until one reaches a prime.
  • A085416 (program): Take prime[n] and continue adding 1,2,…, A085415(n) until one reaches a prime a(n).
  • A085417 (program): Take prime[n] and continue adding n,n+1,…, n+a(n)-1 until one reaches a prime.
  • A085418 (program): Primes reached in A085417.
  • A085423 (program): a(n) = floor(log_2(3n)).
  • A085424 (program): Number of ones in the symmetric signed digit expansion of n with q=2 (i.e., the representation of n in the (-1,0,1)_2 number system).
  • A085425 (program): Number of minus ones in the symmetric signed digit expansion of n with q=2 (i.e., the representation of n in the (-1,0,1)_2 number system).
  • A085428 (program): Sum of the smallest and largest prime divisors of the n-th composite number.
  • A085431 (program): a(n) = (2^(n-1) + prime(n+1)-prime(n))/2.
  • A085432 (program): a(n) = number of triangles ABC with side lengths (n, m, p) such that n, m, p are integers, 1 <= n <= m <= p, gcd(n, m, p) = 1, the Nagel point N of ABC lies on the incircle C(I, r).
  • A085433 (program): Resultant of the polynomial x^3-1 and the Chebyshev polynomial of the second kind U_n(x).
  • A085434 (program): n-th even number not a power of 2 whose largest and smallest factors do not add or subtract to a twin prime.
  • A085435 (program): Resultant of the polynomial x^n-1 and the Chebyshev polynomial of the second kind U_2(x).
  • A085438 (program): a(n) = Sum_{i=1..n} binomial(i+1,2)^3.
  • A085439 (program): a(n) = Sum_{i=1..n} binomial(i+1,2)^4.
  • A085440 (program): a(n) = Sum_{i=1..n} binomial(i+1,2)^5.
  • A085441 (program): a(n) = Sum_{i=1..n} binomial(i+1,2)^6.
  • A085442 (program): a(n) = Sum_{i=1..n} binomial(i+1,2)^7.
  • A085443 (program): Where records in A070172 occur.
  • A085444 (program): Numbers divisible by twice the sum of the products of each of their digits.
  • A085447 (program): a(n) = 6*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=6.
  • A085449 (program): Horadam sequence (0,1,4,2).
  • A085454 (program): Array defined by T(i,1)=i, T(1,j)=2j, T(i,j)=T(i-1,j)+T(i-1,j-1) read by antidiagonals
  • A085455 (program): Sum(Sum a(j)a(i-j),(j=0,..,i)),(i=0,..,n)=(-3)^n.
  • A085456 (program): Sum(Sum a(j)a(i-j),(j=0,..,i)),(i=0,..,n)=(-7)^n.
  • A085457 (program): Sum(Sum a(j)a(i-j),(j=0,..,i)),(i=0,..,n)=(-11)^n.
  • A085458 (program): a(n) = 4*Sum_{i=0..n-1} C(2*i+1, i)*C(n-1, n-1-i)*(-1)^(n-1-i)*2^i for n > 0, a(0) = 1.
  • A085461 (program): Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.
  • A085462 (program): Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4 and v3<=v4.
  • A085463 (program): Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1<=v4, v1<=v5, v2<=v4, v2<=v5 and v3<=v4.
  • A085464 (program): Number of monotone n-weightings of complete bipartite digraph K(4,2).
  • A085465 (program): Number of monotone n-weightings of complete bipartite digraph K(3,3).
  • A085466 (program): a(n) is the denominator of the polynomial in e^2 giving the (2n)th du Bois Reymond constant.
  • A085473 (program): a(n) = 6*n^2 + 3*n + 1.
  • A085474 (program): C(2*n+4,4)-C(2*n,4).
  • A085475 (program): Square array of binomial related numbers, read by antidiagonals.
  • A085476 (program): Periodic Pascal array, read by antidiagonals.
  • A085478 (program): Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
  • A085479 (program): Product of three solutions of the Diophantine equation x^3 - y^3 = z^2.
  • A085480 (program): Expansion of 3*x*(1+2*x)/(1-3*x-3*x^2).
  • A085482 (program): Product of three solutions of the Diophantine equation x^2 - y^2 = z^3.
  • A085487 (program): a(n) = p^n + q^n, p = (1 + sqrt(21))/2, q = (1 - sqrt(21))/2.
  • A085490 (program): Number of pairs with two different elements which can be obtained by selecting unique elements from two sets with n+1 and n^2 elements respectively and n common elements.
  • A085492 (program): Numbers n having no partition into distinct divisors of n+1.
  • A085501 (program): Number of prime powers p^k <= n that are not prime (k = 0 or k > 1).
  • A085503 (program): Sub-triangle of A008292: take every second term of every second row.
  • A085504 (program): Horadam sequence (0,1,9,3).
  • A085521 (program): a(n) = Product_{k=0..n} (2^(2k+1)+1).
  • A085522 (program): Product_{k=0..n} (3^(2k+1)+1).
  • A085523 (program): Product_{k=0..n} (4^(2k+1)+1).
  • A085524 (program): a(0) = 0; a(n) = n^(2*n-1) for n > 0.
  • A085525 (program): a(n) = n^(2*n + 2).
  • A085526 (program): a(n) = n^(2n+1).
  • A085527 (program): a(n) = (2n+1)^n.
  • A085528 (program): a(n) = (2*n+1)^(n+1).
  • A085529 (program): a(n) = (2n+1)^(2n+1).
  • A085530 (program): a(n) = (2n+1)^(2n).
  • A085531 (program): a(n) = (2*n+1)^(2*n-1).
  • A085532 (program): (2n)^(n+1).
  • A085533 (program): (2n)^(2n+1).
  • A085534 (program): a(n) = (2n)^(2n).
  • A085535 (program): a(n) = (2n)^(2n-1).
  • A085536 (program): a(n) = (3n)^(3n).
  • A085537 (program): a(n) = n^4 - n^3.
  • A085538 (program): a(n) = n^5 - n^4.
  • A085539 (program): a(n) = n^6 - n^5.
  • A085540 (program): a(n) = n*(n+1)^3.
  • A085542 (program): Determinant of the n X n matrix M_(i,j)=i/gcd(i,j)=lcm(i,j)/j.
  • A085543 (program): Number of divisors of the partition numbers (A000041).
  • A085550 (program): Decimal expansion of (sqrt(13)-3)/2.
  • A085551 (program): Decimal expansion of (sqrt(29)-5)/2.
  • A085552 (program): n-th digit after decimal point of sqrt(n^2+4)-n)/2.
  • A085558 (program): Numbers with the property that the number of prime digits < number of nonprime digits.
  • A085561 (program): Number of prime divisors of the partition numbers (counted with multiplicity).
  • A085562 (program): Sum of the nonprime digits of n.
  • A085563 (program): Sum of the prime digits of n.
  • A085565 (program): Decimal expansion of lemniscate constant A.
  • A085570 (program): If n mod 2 = 0 then 2*Sum(floor(C(n,w)/(2*w+1)),w=0..n/2-1)+floor(C(n,n/2)/(n+1)) otherwise 2*Sum(floor(C(n,w)/(2*w+1)),w=0..(n-1)/2).
  • A085573 (program): 2*Sum(floor(C(n,w)/w),w=1..n/2-1)+floor(C(n,n/2)/(n/2)) if n is even, otherwise 2*Sum(floor(C(n,w)/w),w=1..(n-1)/2).
  • A085577 (program): Size of maximal subset of the n^2 cells in an n X n grid such that there are at least 3 edges between any pair of chosen cells.
  • A085579 (program): See comments lines for definition.
  • A085580 (program): a(n) = (n+1)-st digit after decimal point of d, where d = (sqrt((n+1)^2 + 4(n+1)) - (n+1))/2.
  • A085583 (program): Number of (3412,1234)-avoiding involutions in S_n.
  • A085584 (program): Number of (3412,2341)-, (3412,4123)- and (3412,52341)-avoiding involutions in S_n.
  • A085585 (program): Squares with all but one even digits.
  • A085598 (program): Primes p with same final decimal digit as k, p = prime(k).
  • A085599 (program): Number of pairs of coprimes (n-i,n+i), 1<i<n.
  • A085600 (program): Number of simple graphs with 3 edges on n vertices.
  • A085601 (program): a(n) = 2 * (4^n + 2^n) + 1.
  • A085602 (program): Numbers of the form (2n+1)^(2n+1) + 1.
  • A085603 (program): (2n)^(2n) + 1.
  • A085606 (program): a(n) = (n-1)^n - 1.
  • A085613 (program): a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2).
  • A085614 (program): Number of elementary arches of size n.
  • A085621 (program): Mean prime gaps that do not occur in A049036.
  • A085622 (program): Maximal number of segments (equivalently, corners) in a rook circuit of a 2n X 2n board.
  • A085624 (program): Sum of the entries in the character table of the dihedral group D_{2n} of order 2n.
  • A085625 (program): Numbers that are the sum of 2 squares in exactly 2 ways.
  • A085626 (program): Partial sums of A051935.
  • A085639 (program): Ramanujan sum c_n(5).
  • A085640 (program): Resultant of the polynomial x^3-1 and the Chebyshev polynomial of the first kind T_n(x).
  • A085641 (program): Smallest prime == 1 mod(p*q*…k) where p, q, …k are all the distinct prime divisors of n. Or, smallest prime == 1 (mod the largest squarefree divisor of n).
  • A085642 (program): Number of columns in the character table of the symmetric group S_n that have zero sum.
  • A085644 (program): a(1) = 1; a(n+1) = a(n)*2n+2n+1.
  • A085652 (program): Fibonacci sequence in base 2 of the alternate number system.
  • A085680 (program): Size of largest code of length n and constant weight 2 that can correct a single adjacent transposition.
  • A085681 (program): Integers of the form 2^n*p where p is a prime > 2^n.
  • A085683 (program): a(n) = Sum_{k = 1..N-1} floor(N/k) where N is the n-th prime.
  • A085687 (program): G.f.: 8/(1+sqrt(1-8*x))^3.
  • A085688 (program): a(1) = 11; a(n) = if n == 2 mod 3 then a(n-1)-3, if n == 0 mod 3 then a(n-1)-2, if n == 1 mod 3 then a(n-1)*2.
  • A085689 (program): a(1) = 4; a(n) = if n == 2 mod 3 then a(n-1)/2, if n == 0 mod 3 then a(n-1)*2, if n == 1 mod 3 then a(n-1)*3.
  • A085691 (program): Triangle read by rows: T(n,k) is the number of triangles of side k in triangular matchstick arrangement of side n; n>=1 and k>=1.
  • A085695 (program): a(n) = Fibonacci(n)*Fibonacci(3n)/2.
  • A085696 (program): a(n) = L(n) * L(n+1) * L(n+2) / 2, where L(n) = Lucas number (A000032).
  • A085697 (program): a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number.
  • A085708 (program): Arithmetic derivative of 10^n.
  • A085710 (program): Smallest k such that kn+1 is a semiprime.
  • A085711 (program): Numbers k such that sopfr(k) is a semiprime.
  • A085712 (program): Semiprimes n such that lpf(n)^spf(n)+1 is also semiprime, where lpf(n) is larger prime factor of n and spf(n) is smaller prime factor of n.
  • A085717 (program): Consider the square lattice L and the sublattice K of index 5 spanned by (2,-1), (1,2); a(n) = number of points (x,y) in M with x >= 0, y >= 0, x+y <= n.
  • A085722 (program): Numbers n such that n^2 + 1 is a semiprime.
  • A085727 (program): Number of semiprimes between n and 2n (inclusive).
  • A085729 (program): Sum of prime factors of prime powers.
  • A085730 (program): Euler’s totient function applied to the sequence of prime powers.
  • A085731 (program): Greatest common divisor of n and its arithmetic derivative.
  • A085736 (program): Numbers n such that all groups of order n are solvable.
  • A085739 (program): Partial sums of A034953(n).
  • A085740 (program): a(n) = T(n)^2 - n^2, where T(n) is a triangular number.
  • A085741 (program): a(n) = T(n)^n, where T() are the triangular numbers (A000217).
  • A085742 (program): a(n) = T(n^3) - T(n), where T() are the triangular numbers (A000217).
  • A085743 (program): a(n) = T(n^3) - T(n^2), where T() are the triangular numbers (A000217).
  • A085744 (program): a(n) = A000217(n^3) - n^3.
  • A085746 (program): Numbers n such that n^2 + n + 1 is a semiprime.
  • A085750 (program): Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.
  • A085759 (program): Prime powers of the form 4n+1.
  • A085760 (program): Prime powers of the form 4n+3.
  • A085761 (program): Number of triangular numbers between n and 2n (inclusive).
  • A085763 (program): Number of palindromes between n and 2n (inclusive).
  • A085765 (program): Partial sums and bisection of A086450.
  • A085766 (program): Smallest m such that n divides the tetrahedral number A000292(m+1).
  • A085767 (program): Smallest m such that n divides the pentagonal number A000326(m).
  • A085776 (program): Numbers n such that n concatenated with n+1 is a semiprime.
  • A085779 (program): Smallest m such that the triangular number A000217(n) divides m!.
  • A085781 (program): a(n) = 2*binomial(2*n+1,n+1) - 2^n.
  • A085786 (program): a(n) = n*(2*n^2 + n + 1)/2.
  • A085787 (program): Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, …
  • A085788 (program): Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2)=t(3)+t(6)=6+21=27.
  • A085789 (program): Partial sums of n 3-spaced triangular numbers beginning with t(2), e.g., a(2) = t(2) + t(5) = 3 + 15 = 18.
  • A085799 (program): Determinant of the symmetric n X n matrix A defined by A[i,j] = abs(i^2 - j^2) for 1 <= i,j <= n.
  • A085801 (program): Maximum number of nonattacking queens on an n X n toroidal board.
  • A085802 (program): Sum of digits of n is a semiprime.
  • A085810 (program): Number of three-choice paths along a corridor of height 5, starting from the lower side.
  • A085811 (program): Number of partitions of n including 3, but not 1.
  • A085812 (program): Sum(sum(binomial(i,j),i=n..2*n),j=0..n).
  • A085814 (program): Even entries (A048967) minus the odd entries (A001316) in row n of Pascal’s triangle (A007318).
  • A085820 (program): Possible two-digit endings of primes (with leading zeros).
  • A085837 (program): Denominators of unit fractions having non-terminating decimal expansions.
  • A085840 (program): Triangle read by rows: T(n,m) = 4^m * (2*n+1)! / ( (2*n - 2*m + 1)! * (2*m)! ), row n has n+1 terms.
  • A085841 (program): Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).
  • A085842 (program): Numbers k whose divisors (apart from 1 and k) sum to a prime.
  • A085855 (program): Number of 1’s in decimal expansion of Fibonacci(n).
  • A085856 (program): Number of 2’s in decimal expansion of Fibonacci(n).
  • A085857 (program): Number of 3’s in decimal expansion of Fibonacci(n).
  • A085858 (program): Number of 4’s in decimal expansion of Fibonacci(n).
  • A085859 (program): Number of 5’s in decimal expansion of Fibonacci(n).
  • A085860 (program): Number of 6’s in decimal expansion of Fibonacci(n).
  • A085862 (program): Number of 8’s in decimal expansion of Fibonacci(n).
  • A085863 (program): Number of 9’s in decimal expansion of Fibonacci(n).
  • A085864 (program): a(1) = 2, a(n+1) = a(n)*{tau(a(n))}.
  • A085866 (program): a(1) = 3, a(n+1) = a(n)*phi(a(n)), where phi(n) is Euler’s totient function.
  • A085879 (program): Smallest n-th power k^n == 1 (mod 10), where k>1.
  • A085880 (program): Triangle T(n,k) read by rows: multiply row n of Pascal’s triangle (A007318) by the n-th Catalan number (A000108).
  • A085881 (program): Triangle T(n,k) read by rows: multiply row n of Pascal’s triangle (A007318) by A001147(n).
  • A085891 (program): Maximal product of three numbers with sum n: a(n) = max(r*s*t), n = r+s+t.
  • A085899 (program): a(n) = floor( 2*(1 + n + 2*n^2 + 4*n^3)/(1 + 2*n + n^2)).
  • A085903 (program): G.f.: (1 + 2*x^2)/((1 + x)*(1 - 2*x)*(1 - 2*x^2)).
  • A085904 (program): Numbers n such that n, n+1 and n+2 are highly composite numbers (2), i.e., all prime divisors <= 7 (A002473).
  • A085906 (program): Ramanujan sum c_n(6).
  • A085912 (program): Group the natural numbers such that the product of the terms of the n-th group is divisible by n!. (1),(2),(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),… Sequence contains the product pertaining to groups.
  • A085913 (program): Group the natural numbers such that the product of the terms of the n-th group is divisible by n!. (1),(2),(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),… Sequence contains the first term of every group.
  • A085915 (program): Group the natural numbers such that the product of the terms of the n-th group is divisible by n!: (1), (2), (3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16, 17, 18), (19, 20, 21, 22, 23, 24), … Sequence contains the product of the terms of the n-th group divided by n!. a(n) = A085912(n)/(n!).
  • A085918 (program): Primes p such that for some k the number of terms > 0 and < 1 in the Farey sequence of order k is p.
  • A085923 (program): a(0) = 1, a(n+1) = (n+1)*(a(n) + n).
  • A085931 (program): Leading diagonal of A085930.
  • A085938 (program): a(n) is the (n+1)-digit number in which the first digit is 1 and the subsequent digits increase by steps of n (mod 10).
  • A085939 (program): Horadam sequence (0,1,6,4).
  • A085945 (program): Number of subsets of {1,2,…,n} with relatively prime elements.
  • A085952 (program): First n-digit number that occurs in the sequence A085951.
  • A085959 (program): Multiples of 37.
  • A085960 (program): Size of the largest code of length 4 and minimum distance 3 over an alphabet of size n. This is usually denoted by A_{n}(4,3).
  • A085970 (program): Number of integers ranging from 2 to n that are not prime-powers.
  • A085971 (program): Union of primes and numbers that are not prime powers (A000040, A024619).
  • A085972 (program): Number of numbers <= n that are primes or not prime powers.
  • A085975 (program): Number of 1’s in decimal expansion of prime(n).
  • A085976 (program): Number of 2’s in decimal expansion of prime(n).
  • A085977 (program): Number of 3’s in decimal expansion of prime(n).
  • A085979 (program): Number of 5’s in decimal expansion of prime(n).
  • A085980 (program): Number of 6’s in decimal expansion of prime(n).
  • A085981 (program): Number of 7’s in decimal expansion of prime(n).
  • A085983 (program): Number of 9’s in decimal expansion of prime(n).
  • A085986 (program): Squares of the squarefree semiprimes (p^2*q^2).
  • A085987 (program): Product of exactly four primes, three of which are distinct (p^2*q*r).
  • A085989 (program): Numbers that can be expressed as a sum of two squares, each >=2.
  • A085990 (program): Number of topological types of polygons with 2n different sides.
  • A086008 (program): Number of 0’s in decimal expansion of n^2.
  • A086009 (program): Number of 1’s in decimal expansion of n^2.
  • A086010 (program): Number of 2’s in decimal expansion of n^2.
  • A086011 (program): Number of 3’s in decimal expansion of n^2.
  • A086012 (program): Number of 4’s in decimal expansion of n^2.
  • A086013 (program): Number of 5’s in decimal expansion of n^2.
  • A086014 (program): Number of 6’s in decimal expansion of n^2.
  • A086016 (program): Number of 8’s in decimal expansion of n^2.
  • A086017 (program): Number of 9’s in decimal expansion of n^2.
  • A086020 (program): a(n) = Sum_(i=1..n) binomial(i+2,3)^2 [ Sequential sums of the tetragonal numbers or “tetras” (pyramidal, square) raised to power 2 (drawn from the 4th diagonal - left or right - of Pascal’s Triangle) ].
  • A086021 (program): a(n) = Sum_{i=1..n} C(i+2,3)^3.
  • A086022 (program): a(n) = Sum_{i=1..n} C(i+2,3)^4.
  • A086023 (program): a(n) = Sum_{i=1..n} C(i+3,4)^2.
  • A086024 (program): a(n) = Sum_{i=1..n} C(i+3,4)^3.
  • A086025 (program): a(n) = Sum_{i=1..n} C(i+4,5)^2.
  • A086026 (program): a(n) = Sum_{i=1..n} C(i+4,5)^3.
  • A086027 (program): a(n) = Sum_{i=1..n} binomial(i+5,6)^2.
  • A086028 (program): a(n) = Sum_{i=1..n} C(i+5,6)^3.
  • A086029 (program): a(n) = Sum_{i=1..n} C(i+6,7)^2.
  • A086030 (program): a(n) = Sum_{i=1..n} C(i+6,7)^3.
  • A086046 (program): Sum of first n 4-almost primes.
  • A086047 (program): Sum of first n 5-almost primes.
  • A086052 (program): Sum of first n 6-almost primes.
  • A086059 (program): Sum of first n 7-almost primes.
  • A086061 (program): Sum of first n 8-almost primes.
  • A086062 (program): Sum of first n 3-almost primes.
  • A086064 (program): In decimal representation: smallest k>1 such that n is a substring of n*k.
  • A086070 (program): Where records in A086068 occur.
  • A086071 (program): Number of 0’s in decimal expansion of triangular number n(n+1)/2.
  • A086072 (program): Number of 1’s in decimal expansion of triangular number n(n+1)/2.
  • A086073 (program): Number of 2’s in decimal expansion of triangular number n(n+1)/2.
  • A086074 (program): Number of 3’s in decimal expansion of triangular number n(n+1)/2.
  • A086075 (program): Number of 4’s in decimal expansion of triangular number n(n+1)/2.
  • A086076 (program): Number of 5’s in decimal expansion of triangular number n(n+1)/2.
  • A086077 (program): Number of 6’s in decimal expansion of triangular number n(n+1)/2.
  • A086078 (program): Number of 7’s in decimal expansion of triangular number n(n+1)/2.
  • A086079 (program): Number of 8’s in decimal expansion of triangular number n(n+1)/2.
  • A086080 (program): Number of 9’s in decimal expansion of triangular number n(n+1)/2.
  • A086084 (program): A086070 in binary.
  • A086086 (program): Primes p such that p - floor(sqrt(p)) is prime.
  • A086088 (program): Decimal expansion of the limit of the ratio of consecutive terms in the tetranacci sequence A000078.
  • A086089 (program): Decimal expansion of 3*sqrt(3)/(2*Pi).
  • A086090 (program): 2^n+n3^n.
  • A086091 (program): 3^n+3n4^(n-1).
  • A086092 (program): 4^n+3n5^(n-1).
  • A086093 (program): a(n) = 3^n + 2*n*4^(n-1).
  • A086099 (program): a(n) = OR(k AND (n-k): 0<=k<=n), AND and OR bitwise.
  • A086100 (program): A086099 in binary.
  • A086106 (program): Decimal expansion of positive root of x^4 - x^3 - 1 = 0.
  • A086111 (program): Numerator of the mean deviation of a discrete uniform distribution on n elements.
  • A086112 (program): Denominator of the mean deviation of a discrete uniform distribution on n elements.
  • A086113 (program): Number of 3 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.
  • A086114 (program): Number of 4 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.
  • A086115 (program): Number of 5 X n (0,1) matrices such that each row and each column is nondecreasing or nonincreasing.
  • A086116 (program): Numerator of mean deviation of a symmetrical binomial distribution on n elements.
  • A086117 (program): Denominator of mean deviation of a symmetrical binomial distribution on n elements.
  • A086130 (program): a(n) = lcm(n, A003415(n)).
  • A086141 (program): Permutation of A025487 (least prime signatures) which, when values are factored, exhibit self-similarity (cf. A008687).
  • A086142 (program): a(n) = floor( log( prime(n) )^2 ).
  • A086144 (program): a(n) = 2*A071640(n) - n.
  • A086148 (program): Sum of the orders of the elements in the dihedral group D_2n.
  • A086156 (program): a(n) = sigma(n^2) - n*sigma(n).
  • A086159 (program): Number of partitions of n into the first three triangular numbers, 1, 3 and 6.
  • A086161 (program): Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^2.
  • A086162 (program): Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^3.
  • A086167 (program): a(n) = sum of the first n lower twin primes.
  • A086168 (program): a(n) = sum of the first n upper twin primes.
  • A086169 (program): Sum of the first n twin prime pairs.
  • A086170 (program): a(1)=1; a(n)=a(n-1)+1 if n is in the sequence; a(n)=a(n-1)+2 if n and (n-1) are not in the sequence; a(n)=a(n-1)+3 if n is not in the sequence but (n-1) is in the sequence.
  • A086178 (program): Decimal expansion of 1 + 2*sqrt(2).
  • A086180 (program): Decimal expansion of 1 + sqrt(6).
  • A086192 (program): Tribonacci numbers that start with first three squares.
  • A086197 (program): Numerators of running averages of A051903.
  • A086198 (program): Denominators of running averages of A051903.
  • A086201 (program): Decimal expansion of 1/(2*Pi).
  • A086205 (program): Determinant of n X n matrix M_(i,j)=binomial(i^2, j).
  • A086206 (program): Number of n X n matrices with entries in {0,1} with no zero row and with zero main diagonal.
  • A086213 (program): Tribonacci numbers that start with first three cubes.
  • A086221 (program): Bisection of A086652.
  • A086223 (program): Every integer can be represented uniquely as m = k*2^(j+1)+2^j-1. Sequence gives values of k for m = repunit(n).
  • A086224 (program): a(n) = 7*2^n-1.
  • A086225 (program): a(n) = 11*2^n - 1.
  • A086227 (program): a(n) = Sum_{1<=k<=4*n, gcd(k,n)=1} (i^k*tan(k*Pi/(4*n)))/(4*i), where i is the imaginary unit.
  • A086228 (program): Determinant of n X n matrix M(i,j)=binomial(2i+1, j).
  • A086229 (program): Determinant of n X n matrix M(i,j) = binomial(2i-1,j), (i,j) ranging from 1 to n.
  • A086246 (program): Expansion of (1 + x - sqrt(1 - 2*x - 3*x^2)) / 2 in powers of x.
  • A086247 (program): Differences of successive 7-smooth numbers.
  • A086253 (program): Decimal expansion of Feller’s alpha coin-tossing constant.
  • A086254 (program): Decimal expansion of Feller’s beta coin-tossing constant.
  • A086260 (program): Number of symmetric n X n conference matrices.
  • A086261 (program): Number of antisymmetric n X n conference matrices.
  • A086270 (program): Rectangular array T(k,n) of polygonal numbers, by antidiagonals.
  • A086271 (program): Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.
  • A086272 (program): Rectangular array T(n,k) of central polygonal numbers, by antidiagonals.
  • A086273 (program): Rectangular array T(n,k) of central polygonal numbers, by antidiagonals.
  • A086274 (program): Antidiagonal sums of A086272 (and of A086273).
  • A086275 (program): Number of distinct Gaussian primes in the factorization of n.
  • A086285 (program): Numbers k such that 1 + 2k + 3k^2 is prime.
  • A086286 (program): Smallest prime factor of 7-smooth numbers.
  • A086287 (program): Greatest prime factor of 7-smooth numbers.
  • A086288 (program): Number of distinct prime factors of 7-smooth numbers.
  • A086289 (program): Total number of prime factors of 7-smooth numbers.
  • A086291 (program): Maximal exponent in prime factorization of 7-smooth numbers.
  • A086292 (program): Number of divisors of 7-smooth numbers.
  • A086293 (program): Sum of divisors of 7-smooth numbers.
  • A086294 (program): Sum of distinct prime factors of 7-smooth numbers.
  • A086295 (program): Sum of all prime factors of 7-smooth numbers.
  • A086296 (program): Euler’s totient of 7-smooth numbers.
  • A086297 (program): Squarefree kernels of 7-smooth numbers.
  • A086298 (program): Numbers n such that 1-2n+3n^2 is prime.
  • A086299 (program): a(n) = if n is 7-smooth then 1 else 0: characteristic function of 7-smooth numbers.
  • A086300 (program): Arithmetic derivative of 7-smooth numbers.
  • A086302 (program): a(n) = 4*n^4 + 24*n^3 + 48*n^2 + 36*n + 8.
  • A086303 (program): Numbers n such that n+15 is prime.
  • A086304 (program): Numbers n such that n+6 is prime.
  • A086325 (program): Let u(1)=0, u(2)=1, u(k)=u(k-1)+u(k-2)/(k-2); then a(n)=n!*u(n).
  • A086327 (program): Number of factors over Q in the factorization of the Chebyshev polynomial of the second kind U_n(x).
  • A086330 (program): a(n) = Sum_{m >= 2} m! mod n.
  • A086331 (program): Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).
  • A086341 (program): a(n) = 2*2^floor(n/2) - (-1)^n.
  • A086343 (program): a(n) starts new run of consecutive values in A055938.
  • A086344 (program): a(n) = -2*a(n-1) + 4*a(n-2), a(0) = 1, a(1) = 0.
  • A086346 (program): On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square.
  • A086347 (program): On a 3 X 3 board, number of n-move routes of chess king ending in a given side square.
  • A086348 (program): On a 3 X 3 board, number of n-move routes of chess king ending in the central square.
  • A086349 (program): On a 3 X 3 board, the number of n-move paths for a chess king.
  • A086350 (program): Square array of Pell related numbers, read by antidiagonals.
  • A086351 (program): T(n,3) of A086350.
  • A086352 (program): Main diagonal of square array A086350.
  • A086353 (program): Fixed point if nonzero-digit product of n! is iterated.
  • A086354 (program): Fixed point if (nonzero-digit product)-function at initial value 2^n is iterated.
  • A086355 (program): Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.
  • A086358 (program): Digital root of n!.
  • A086359 (program): Fixed point if [decimal-digit-sum]-function at initial-value=A000984(n)=C[2n,n] is iterated.
  • A086360 (program): Fixed point if (decimal-digit-sum)-function at initial value = n-th primorial = A002110(n) is iterated.
  • A086363 (program): Array T(m,n) read by antidiagonals: if X and Y are two (possibly empty) finite sets with m and n elements respectively and Z is the disjoint union of X and Y, then T(m,n) is the number of self-inverse partial functions f:Z ->Z which do not fix any element of Y.
  • A086369 (program): Number of factors over Q in the factorization of T_n(x) - 1 where T_n(x) is the Chebyshev polynomial of the first kind.
  • A086374 (program): Number of factors over Q in the factorization of T_n(x) + 1 where T_n(x) is the Chebyshev polynomial of the first kind.
  • A086375 (program): Number of factors over Q in the factorization of U_n(x) + 1 where U_n(x) is the Chebyshev polynomial of the second kind.
  • A086377 (program): a(1)=1; a(n)=a(n-1)+2 if n is in the sequence; a(n)=a(n-1)+2 if n and (n-1) are not in the sequence; a(n)=a(n-1)+3 if n is not in the sequence but (n-1) is in the sequence.
  • A086384 (program): Odd digits of e.
  • A086385 (program): Odd digits of Pi.
  • A086389 (program): Number of factors over Q in the factorization of U_n(x) - 1 where U_n(x) is the Chebyshev polynomial of the second kind.
  • A086396 (program): Even digits of e.
  • A086398 (program): a(1)=1; a(n)=a(n-1)+2 if n is in the sequence; a(n)=a(n-1)+2 if n and (n-1) are not in the sequence; a(n)=a(n-1)+4 if n is not in the sequence but (n-1) is in the sequence.
  • A086399 (program): Even digits of Pi.
  • A086403 (program): Numerators in continued fraction representation of (e-1)/(e+1).
  • A086404 (program): Square array of numbers T(n,k) = ((1+sqrt(3))*(k+sqrt(3))^n-(1-sqrt(3))*(k-sqrt(3))^n)/(2*sqrt(3)), read by antidiagonals.
  • A086405 (program): Row T(n,3) of number array A086404.
  • A086406 (program): Main diagonal of number array A086404.
  • A086410 (program): Smallest prime factor of 3-smooth numbers, with a(1)=1.
  • A086411 (program): Greatest prime factor of 3-smooth numbers.
  • A086412 (program): Number of distinct prime factors of 3-smooth numbers.
  • A086414 (program): Minimal exponent in prime factorization of 3-smooth numbers.
  • A086415 (program): Maximal exponent in prime factorization of 3-smooth numbers.
  • A086416 (program): Number of divisors of 3-smooth numbers.
  • A086417 (program): Sum of divisors of 3-smooth numbers.
  • A086418 (program): Sum of distinct prime factors of 3-smooth numbers.
  • A086419 (program): Sum of all prime factors of 3-smooth numbers.
  • A086420 (program): Euler’s totient of 3-smooth numbers: a(n) = A000010(A003586(n)).
  • A086435 (program): Maximum number of parts possible in a factorization of n into a product of distinct numbers > 1.
  • A086436 (program): Maximum number of parts possible in a factorization of n; a(1) = 1, and for n > 1, a(n) = A001222(n) = bigomega(n).
  • A086443 (program): Expansion of x^2/((1-4*x)*(1-3*x)^2).
  • A086444 (program): Sequence associated with palindromic structures.
  • A086445 (program): Partial sums of A005578.
  • A086449 (program): a(0) = 1, a(2n+1) = a(n), a(2n) = a(n) + a(n-1) + … + a(n-2^m) + … where a(n) = 0 for n < 0.
  • A086450 (program): a(0) = 1, a(2n+1) = a(n), a(2n) = a(n) + a(n-1) + … + a(n-m) + … where a(n<0) = 0.
  • A086452 (program): Number of maximal triangulations (using all 2(n+2) points) of a convex polygon having (n+2) sides and an interior point in the middle of each side.
  • A086453 (program): Least difference between 5^n and a power of 2.
  • A086454 (program): Number of divisors of prime powers: tau(p^e).
  • A086455 (program): Sum of divisors of prime powers: sigma(p^e).
  • A086456 (program): Expansion of (1 + x + sqrt(1 - 6*x + x^2))/2 in powers of x.
  • A086459 (program): Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, …, 2^(n-1)) right.
  • A086460 (program): Square array read by antidiagonals: T(n,k)=nk+0^n.
  • A086461 (program): Symmetric version of square array A086460.
  • A086462 (program): Expansion of (1+x)(1+4x)/((1-x)(1-4x)).
  • A086463 (program): Decimal expansion of Pi^2/18.
  • A086464 (program): Decimal expansion of 17/36*Zeta(4).
  • A086465 (program): Decimal expansion of (5 + 4*sqrt(5)*arcsch(2))/25.
  • A086466 (program): Decimal expansion of 2*sqrt(5)/5 arccsch(2).
  • A086467 (program): Decimal expansion of 2*arccsch(2)^2.
  • A086468 (program): Decimal expansion of 2*zeta(3)/5.
  • A086482 (program): Beginning with 1, the smallest number not included earlier such that the n-th partial product is an n-th power; or the geometric mean of the first n terms is an integer.
  • A086483 (program): Bit that is two places to left of least significant 1-bit in the binary expansion of n.
  • A086486 (program): Numbers n such that the sum of the distinct prime divisors divides rad(n)=A007947(n).
  • A086500 (program): Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), … Sequence contains the group sum.
  • A086507 (program): If n is even, a(n) = smallest prime == 1 (mod n), If n is odd, a(n) = smallest prime == -1 (mod n).
  • A086508 (program): If n is even, a(n) = smallest prime == -1 (mod n), If n is odd, a(n) = smallest prime == 1 (mod n).
  • A086514 (program): Difference between the arithmetic mean of the neighbors of the terms and the term itself follows the pattern 0,1,2,3,4,5,…
  • A086520 (program): Number of integers strictly greater than (n-sqrt(n))/2 and strictly less than (n+sqrt(n))/2.
  • A086525 (program): a(n) = a(( a(n-2))*(1-mod(n,2))+a(n-1)*(mod(n,2))) + a((n - a(n-2))*(1-mod(n,2))+(n-a(n-1))*(mod(n,2))).
  • A086529 (program): Beginning with 2, distinct even numbers such that the arithmetic mean of successive pairs of terms gives odd primes in their natural order. {a(n) + a(n+1)}/2 = prime(n+1).
  • A086543 (program): Number of partitions of n with at least one odd part.
  • A086552 (program): Numbers x such that tau(x)/tau(x-1) is an integer, where tau() is the number of divisors function.
  • A086568 (program): Smallest number having as many distinct prime factors as n has prime factors, when counted with multiplicity.
  • A086570 (program): Expansion of (1 + 3x + 5x^2 + 7x^3 + …) / (1 - 2x + 3x^2 - 4x^3 + …).
  • A086573 (program): a(n) = 2*(10^n - 1).
  • A086574 (program): a(n) = 3*(10^n-1).
  • A086575 (program): a(n) = 4*(10^n - 1).
  • A086576 (program): a(n) = 5*(10^n - 1).
  • A086577 (program): a(n) = 6*(10^n - 1).
  • A086578 (program): a(n) = 7*(10^n - 1).
  • A086579 (program): a(n) = 8*(10^n - 1).
  • A086580 (program): a(n) = 9*(10^n - 1).
  • A086581 (program): Number of Dyck paths of semilength n with no DDUU.
  • A086592 (program): Denominators in left-hand half of Kepler’s tree of fractions.
  • A086593 (program): Bisection of A086592, denominators of the left-hand half of Kepler’s tree of fractions.
  • A086594 (program): a(n) = 8*a(n-1) + a(n-2), starting with a(0)=2 and a(1)=8.
  • A086601 (program): Triangular numbers + 1 squared.
  • A086602 (program): a(n) = A000217(A000217(n))-n^2.
  • A086603 (program): a(n) = n^3*3^(n-1).
  • A086604 (program): 2^(n-3)n(9n^2-9n+4).
  • A086605 (program): a(n) = 9*n^3 - 18*n^2 + 10*n.
  • A086614 (program): Triangle read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.
  • A086615 (program): Antidiagonal sums of triangle A086614.
  • A086616 (program): Partial sums of the large Schroeder numbers (A006318).
  • A086618 (program): a(n) = Sum{k=0..n} binomial(n,k)^2*C(k), where C() = A000108() are the Catalan numbers.
  • A086619 (program): Product of first n terms of the binomial transform of the Catalan numbers (A007317).
  • A086621 (program): Main diagonal of square table A086620 of coefficients, T(n,k), of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.
  • A086622 (program): G.f. A(x) satisfies: A(x) = 1/(1-2*x) + x^2*A(x)^2.
  • A086630 (program): Main diagonal of square table A086629; coefficients of x^n*y^n in f(x,y) that satisfies f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.
  • A086631 (program): Antidiagonal sums of square table A086629.
  • A086640 (program): Arrange n^2 octagons that each have area 7 so that they leave (n-1)^2 square gaps each with area 2; a(n) is the total area of these polygons.
  • A086642 (program): Maximal number of zeros in a column of the character table of the symmetric group S_n.
  • A086645 (program): Triangle read by rows: T(n, k) = binomial(2n, 2k).
  • A086646 (program): Triangle, read by rows, of numbers T(n,k), 0 <= k <= n, given by T(n,k) = A000364(n-k)*binomial(2*n, 2*k).
  • A086650 (program): n-th composite number raised to the n-th prime number.
  • A086652 (program): a(n) = A000225(n+3)-A052955(n).
  • A086653 (program): 2^n + 3*n.
  • A086655 (program): (C(2p,p)-2)/(2p) where p runs through the primes.
  • A086659 (program): T(n,k) counts the set partitions of n containing k-1 blocks of length 1.
  • A086660 (program): Stirling transform of Hermite numbers: Sum_{k=0..n} Stirling2(n,k) * HermiteH(k,0).
  • A086663 (program): Number of non-attacking knights on an n X n board with all non-perimeteral squares removed.
  • A086664 (program): n - {sum of prime power components of n}.
  • A086665 (program): Difference between sum of divisors of n and integer log of n, i.e., A000203(n) - A001414(n).
  • A086666 (program): a(n) = sigma_2(n) - sigma_1(n).
  • A086668 (program): Number of divisors d of n such that 2d+1 is a prime.
  • A086669 (program): a(n) = number of divisors of n that are fundamental discriminants.
  • A086670 (program): Sum of floor(d/2) where d is a divisor of n.
  • A086671 (program): Sum of floor(sqrt(d)) where d runs through the divisors of n.
  • A086674 (program): Sum of signed indices from Euler’s Pentagonal Theorem (see A000041).
  • A086685 (program): Number of 1 <= i < n such that i*n+1 is prime.
  • A086686 (program): Number of 1<=i<n such that i*n-1 is prime.
  • A086687 (program): a(n) = binomial(n!,n).
  • A086689 (program): a(n) = Sum_{i=1..n} i^2*t(i), where t = A000217.
  • A086694 (program): A run of 2^n 1’s followed by a run of 2^n 0’s, for n=0, 1, 2, …
  • A086695 (program): a(n) = 100^n - 10^n - 1.
  • A086699 (program): Number of n X n matrices over GF(2) with rank n-1.
  • A086700 (program): Euler phi function applied to the triangular numbers.
  • A086701 (program): n-th prime number raised to the n-th composite number.
  • A086706 (program): Number of Niven numbers less than or equal to n.
  • A086707 (program): Smallest mode of the sequences n/(n-k)*binomial(n,n-k) (see link).
  • A086716 (program): Convolution of triangular numbers with partition numbers.
  • A086718 (program): Convolution of sequence of primes with sequence sigma(n).
  • A086726 (program): Decimal expansion of sum(1/(6*m)^2,m=1..infinity).
  • A086729 (program): Decimal expansion of Pi^2/72.
  • A086732 (program): Convolution of A000203 with partition function (A000041) of positive integers.
  • A086733 (program): Convolution of sigma(n) with phi(n).
  • A086736 (program): Sum(j=1,n,floor(A000041(j)/j)))).
  • A086737 (program): a(n) = A000217(A000041(n)).
  • A086738 (program): A000041(n) - A000203(n).
  • A086739 (program): A000041(n)-A000010(n).
  • A086740 (program): Floor(A000041(n)/n).
  • A086743 (program): Numbers n such that the coefficient of x^n equals 0 in Product_{k>=1} (1 - x^(3^k)).
  • A086746 (program): Multiples of 3018.
  • A086747 (program): Baum-Sweet sequence: a(n) = 1 if binary representation of n contains no block of consecutive zeros of odd length; otherwise a(n) = 0.
  • A086748 (program): Numbers m such that when C(2k, k) == 1 (mod m) then k is necessarily even.
  • A086749 (program): Partial sums of A038580.
  • A086755 (program): Sum_{k=1..n} (k(k+1))^2/2.
  • A086756 (program): a(n) = n^n mod 10^n.
  • A086760 (program): a(n) = 8n^2 + 88n + 43.
  • A086761 (program): Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.
  • A086764 (program): Triangle T(n, k), read by row, related to Euler’s difference table A068106 (divide column k of A068106 by k!).
  • A086767 (program): Last coefficient of the last term in the numerator of the simplified expansion of the solutions of FLT for n=2 for FLT n=1,2,3,..
  • A086769 (program): a(n) = sum{2^(b(i)-1): 1<=i<=n}, where b(n) is the differences between consecutive primes.
  • A086775 (program): Decimal expansion of the number defined by the continued fraction shown below.
  • A086779 (program): Numbers k such that k-th cyclotomic polynomial has exactly 7 nonzero terms.
  • A086783 (program): Discriminant of the polynomial x^n - 1.
  • A086784 (program): Number of non-trailing zeros in binary representation of n.
  • A086787 (program): a(n) = Sum_{i=1..n} ( Sum_{j=1..n} i^j ).
  • A086790 (program): a(n) = floor((1+n+2*n^2+4*n^3)/(1+2*n+n^2))
  • A086794 (program): Numbers n such that n-th cyclotomic polynomial has exactly 9 nonzero terms.
  • A086797 (program): Discriminant of the polynomial x^n - x - 1.
  • A086799 (program): Replace all trailing 0’s with 1’s in binary representation of n.
  • A086800 (program): Triangle read by rows in which row n lists differences between prime(n) and prime(k) for 1 <= k <= n.
  • A086801 (program): a(n) = prime(n) - 3.
  • A086802 (program): Triangle read by rows in which row n lists (prime(n)-prime(k))/2 for 2 <= k <= n.
  • A086803 (program): a(n) = Product_{i=2..n} (prime(n+1)-prime(i)).
  • A086804 (program): a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).
  • A086810 (program): Triangle obtained by adding a leading diagonal 1,0,0,0,… to A033282.
  • A086812 (program): Number of symmetric invertible n X n matrices over GF(2).
  • A086813 (program): a(1)=1 then a(n)= (1/2) *(5*a(n-1)+1) if a(n-1) is odd, a(n)=3/2*a(n-1) otherwise.
  • A086814 (program): a(n) = ceiling( (1 + n + 2*n^2 + 4*n^3)/(1 + 2*n + n^2) ).
  • A086815 (program): a(n)=(n-1)*n^(2*n)
  • A086816 (program): Smaller member of a twin prime pair with a triangular sum.
  • A086822 (program): a(n) = floor((6*n^0+5*n^1+4*n^2+3*n^3) / (1*n^0+1*n^1+1*n^2)).
  • A086828 (program): a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 7; thereafter, a(n) = a(n-1) + (n-1)*a(n-2).
  • A086831 (program): Ramanujan sum c_n(2).
  • A086843 (program): Odd numbers m such that the sequence defined by b(1) = m; for k>1, b(k) = floor(phi*b(k-1)), where phi = (1 + sqrt(5))/2, contains only odd numbers.
  • A086844 (program): Odd numbers m such that the sequence defined by b(1) = m; for k>1, b(k) = floor((1+sqrt(3))*b(k-1)) contains only odd numbers.
  • A086845 (program): a(1) = 0, a(n) = a(floor(n/2)) + 2*a(ceiling(n/2)) + floor(n/2).
  • A086847 (program): a(n) = gcd(A001608(n), n), where A001608 = Perrin sequence.
  • A086849 (program): Sum of first n nonsquares.
  • A086851 (program): a(0) = 1, a(n+1) = a(n)^2 - n.
  • A086858 (program): Let f(n) be the inverse of the function g(x) = x^x. Then a(n) = floor(f(n)).
  • A086862 (program): Differences between successive palindromes.
  • A086864 (program): a(n) = (n-1)*(n-2)*(n-3)*(3*n-10)*3^(n-5)/4.
  • A086866 (program): Third column of A059450.
  • A086871 (program): Row sums of A059450.
  • A086874 (program): Seventh power of odd primes.
  • A086876 (program): Run lengths in A071542.
  • A086879 (program): Number of symmetric singular n X n matrices over GF(2).
  • A086885 (program): Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.
  • A086892 (program): Greatest common divisor of 2^n-1 and 3^n-1.
  • A086893 (program): a(n) is the index of F(n+1) at the unique occurrence of the ordered pair of reversed consecutive terms (F(n+1),F(n)) in Stern’s diatomic sequence A002487, where F(k) denotes the k-th term of the Fibonacci sequence A000045.
  • A086894 (program): a(n) = (A000522(2*n) + 1)/2.
  • A086898 (program): a(n) = Sum_{d|n} tau(d-1).
  • A086901 (program): a(1) = a(2) = 1; for n>2, a(n) = 4*a(n-1) + 3*a(n-2).
  • A086902 (program): a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.
  • A086903 (program): a(n) = 8*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8, a(n) = (4+sqrt(15))^n + (4-sqrt(15))^n.
  • A086905 (program): a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,floor(k/2)).
  • A086910 (program): a[1]=1; a[n] =1+Abs[Prime[n]-a[n-1]]
  • A086914 (program): a(n) = ((n-1)^n/n)*Sum_{k>=1} (k^n/n^k).
  • A086915 (program): Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).
  • A086921 (program): Least number with at least n divisors that are at most its square root.
  • A086926 (program): Product of Fibonacci and (shifted) triangular numbers.
  • A086927 (program): a(n) = 10*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 10.
  • A086928 (program): a(n) = 12a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 12, a(n) = (6+sqrt(37))^n + (6-sqrt(37))^n.
  • A086931 (program): a(0) = 1, a(n) = spf(n)*a(n-spf(n)), where spf=A020639 (smallest prime factor).
  • A086932 (program): Number of non-congruent solutions of x^2 + y^2 == -1 (mod n).
  • A086936 (program): Number of primes between n and p(n) inclusive.
  • A086937 (program): Number of distinct zeros of x^2-x-1 mod prime(n).
  • A086940 (program): a(n) = k where R(k+4) = 2.
  • A086941 (program): a(n) = k where R(k+6) = 3.
  • A086942 (program): Integers k such that R(k+8) = 4.
  • A086943 (program): Integers k such that R(k+7) = 3.
  • A086944 (program): a(n) = k where R(k+8) = 5.
  • A086945 (program): a(n) = 7*10^n - 9.
  • A086946 (program): a(n) = k where R(k+6) = 2.
  • A086947 (program): Numbers k such that Reverse(k+9) = 3.
  • A086948 (program): a(n) = k where R(k+8) = 2.
  • A086949 (program): a(n) = k where R(k+9) = 5.
  • A086950 (program): Binomial transform of decagonal numbers A001107.
  • A086951 (program): a(n) = n*3^n*(4*n - 1)/9.
  • A086952 (program): a(n) = n^2*4^n/4.
  • A086953 (program): Binomial transform of (-1)^mod(n,3) (A257075).
  • A086954 (program): Binomial transform of A061800.
  • A086955 (program): a(n) = n^2 + 2*n + 2 - (-1)^n.
  • A086970 (program): Fix 1, then exchange the subsequent odd numbers in pairs.
  • A086971 (program): Number of semiprime divisors of n.
  • A086972 (program): a(n) = n*3^(n-1) + (3^n+1)/2.
  • A086975 (program): Numbers of the form p^2 * q * r with primes p < q < r.
  • A086983 (program): Primes of the form 2^r*p^s - 1, where p is an odd prime.
  • A086984 (program): Number of arrangements of n labeled balls in n labeled columns where only 1 column may have more than 1 ball.
  • A086988 (program): Sum of odd indexed divisors of n.
  • A086989 (program): Sum of even-indexed divisors of n.
  • A086992 (program): Product of nonzero digits in n-th row of Pascal’s triangle.
  • A086994 (program): Decimal expansion of Pi written in base 2.
  • A086995 (program): Number of 1’s in binary representation of n-th decimal digit in expansion of Pi.
  • A086996 (program): Decimal expansion of e (A001113) written in base 2.
  • A086997 (program): Number of 1’s in binary representation of n-th decimal digit in expansion of e.
  • A087000 (program): Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.
  • A087003 (program): a(2n) = 0 and a(2n+1) = mu(2n+1); also the sum of Mobius function values computed for terms of 3x+1 trajectory started at n, provided that Collatz conjecture is true.
  • A087005 (program): Divisors of 2310.
  • A087006 (program): Divisors of 30030.
  • A087009 (program): Least m such that omega(m) + Omega(m) = n, or 0 if no such m exists.
  • A087010 (program): Number of primes of form 4*k+1 between n and 2n (inclusive).
  • A087011 (program): Number of primes of form 4*k+3 between n and 2n (inclusive).
  • A087018 (program): Row sums of Fibonacci triangle shown below.
  • A087023 (program): Maximal exponent in prime factorization of n-th cyclic number.
  • A087030 (program): n “reflected” in the next prime: a(n)=2p-n, p is smallest prime > n.
  • A087031 (program): Numbers n such that 2p-n is prime, p is the smallest prime > n.
  • A087032 (program): a(n) = 1 if 2*A151800(n) - n is prime, otherwise 0, where A151800(n) is the smallest prime > n.
  • A087033 (program): Number of terms in each group of consecutive zeros in A087032.
  • A087035 (program): Maximum value taken on by f(P)=sum(i=1..n, p(i)*p(n+1-i) ) as {p(1),p(2),…,p(n)} ranges over all permutations P of {1,2,3,…n}.
  • A087039 (program): If n is prime then 1 else 2nd largest prime factor of n.
  • A087040 (program): 2nd largest prime factor of n-th composite number.
  • A087047 (program): a(n) = n*(n+1)*(n+2)*a(n-1)/6 for n >= 2; a(1) = 1.
  • A087049 (program): Characteristic sequence for numbers n>=0 that are either squares or have a square > 1 as factor.
  • A087050 (program): Square root of the largest square >1 dividing the n-th nonsquarefree number.
  • A087053 (program): Numbers of the form pq + qr + rp where p, q and r are distinct primes, with multiplicity.
  • A087055 (program): Largest square number less than 2*n^2.
  • A087056 (program): Difference between 2 * n^2 and the next smaller square number.
  • A087057 (program): Smallest number whose square is larger than 2*n^2.
  • A087058 (program): Smallest square number greater than 2*n^2.
  • A087059 (program): Difference between 2*n^2 and the next greater square number.
  • A087060 (program): Difference between 2n^2 and the nearest square number.
  • A087066 (program): a(n) = Sum_{k>=0} floor(n*(r^k)), where r = sqrt(5)-2.
  • A087068 (program): Sum{Floor(n*(r^k): r=2/3, k>=0).
  • A087069 (program): a(n) = Sum_{k >= 0} floor(n/(4^k)).
  • A087071 (program): Number of distinct prime 4-component links with crossing number n.
  • A087076 (program): Sums of the squares of the elements in the subsets of the integers 1 to n.
  • A087088 (program): Positive ruler-type fractal sequence with 1’s in every third position.
  • A087090 (program): Positive numbers n such that p=n^2+n+41 and p+2 are twin primes.
  • A087091 (program): Numbers k such that p = k^2 + k + 41 and p - 2 are twin primes.
  • A087098 (program): Partial sums of A087100.
  • A087099 (program): Partial sums of A063914.
  • A087100 (program): A000225 (2^n - 1) interlaced with A008593 (11n).
  • A087105 (program): (prime(n-1) + 1)*(prime(n+1) - 1).
  • A087112 (program): Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.
  • A087115 (program): Convolution of sum of cubes of divisors with itself.
  • A087116 (program): Number of maximal groups of consecutive zeros in binary representation of n.
  • A087117 (program): Number of zeros in the longest string of consecutive zeros in the binary representation of n.
  • A087118 (program): Numbers having exactly one maximal group of consecutive zeros in binary representation of n.
  • A087119 (program): Numbers having more than one maximal group of consecutive zeros in binary representation of n.
  • A087120 (program): Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.
  • A087123 (program): a(n) = Fibonacci(n+1) - (-1)^n*Fibonacci(n).
  • A087124 (program): a(n) = Fibonacci(n) + Fibonacci(2n+1).
  • A087125 (program): Indices n of hex numbers H(n) that are also triangular.
  • A087128 (program): a(1)=1 and, for n>1, a(n) is the smallest positive integer such that 1+Sum[k, k=a(n-1)+1,…,a(n)] is prime.
  • A087129 (program): First differences of A087128.
  • A087130 (program): a(n) = 5*a(n-1)+a(n-2) for n>1, a(0)=2, a(1)=5.
  • A087131 (program): a(n) = 2^n*Lucas(n), where Lucas = A000032.
  • A087133 (program): Number of divisors of n that are not greater than the greatest prime-factor of n; a(1)=1.
  • A087135 (program): Number of positive numbers m such that A073642(m) = n.
  • A087136 (program): Smallest positive number m such that A073642(m)=n.
  • A087137 (program): a(n) is the number of permutations in the symmetric group S_n that contain an odd cycle.
  • A087153 (program): Number of partitions of n into nonsquares.
  • A087156 (program): Nonnegative numbers excluding 1.
  • A087157 (program): Satisfies a(1)=1, a(A087158(n+1)) = a(n)+1, with a(m)=1 for all m not found in A087158, where A087158(n+2)=A087158(n+1)+a(n)+1.
  • A087158 (program): Satisfies a(n+2)=a(n+1)+A087157(n)+1.
  • A087159 (program): Satisfies a(1)=1, a(A087160(n+1)) = a(n)+1, with a(m)=2 for all m not found in A087160, where A087160(n+2)=A087160(n+1)+a(n)+1.
  • A087160 (program): Satisfies a(1)=1, a(2)=2, a(n+2)=a(n+1)+A087159(n)+1.
  • A087161 (program): Records in A087159, i.e., A087159(a(n)) = n, and satisfies the recurrence a(n+3) = 5*a(n+2) - 6* a(n+1) + 2*a(n) with a(1) = 1, a(2) = 2, and a(3) = 4.
  • A087164 (program): Records in A087162: A087162(a(n))=n.
  • A087165 (program): a(n)=1 when n == 1 (mod 4), otherwise a(n) = a(n - ceiling(n/4)) + 1. Removing all the 1’s results in the original sequence with every term incremented by 1.
  • A087168 (program): Expansion of (1 + 2*x)/(1 + 3*x + 4*x^2).
  • A087169 (program): Expansion of (1 + 3*x)/(1 + 5*x + 9*x^2).
  • A087170 (program): Expansion of (1 + 4*x)/(1 + 7*x + 16*x^2).
  • A087171 (program): Expansion of (1 + 5*x)/(1 + 9*x + 25*x^2).
  • A087172 (program): Greatest Fibonacci number that does not exceed n.
  • A087175 (program): Number of distinct primes dividing the n-th partition number.
  • A087177 (program): Number of even partition numbers <= P(n), where P=A000041.
  • A087180 (program): Number partition numbers <= P(n) of the form 3*k (P=A000041).
  • A087181 (program): Number partition numbers <= P(n) of the form 3*k+1 (P=A000041).
  • A087182 (program): Number partition numbers <= P(n) of the form 3*k+2 (P=A000041).
  • A087186 (program): Q(p)/p where p runs through the primes and Q(k) is the k-th central quadrinomial coefficient (A005190).
  • A087188 (program): Number of partitions of n into distinct squarefree parts.
  • A087192 (program): a(n) = ceiling(a(n-1)*4/3), with a(1) = 1.
  • A087204 (program): Period 6: repeat [2, 1, -1, -2, -1, 1].
  • A087205 (program): a(n) = -2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
  • A087206 (program): a(n) = 2*a(n-1) + 4*a(n-2); with a(0)=1, a(1)=4.
  • A087207 (program): A binary representation of the primes that divide a number, shown in decimal.
  • A087208 (program): Expansion of e.g.f.: exp(x)/(1-x^2).
  • A087211 (program): Floor((1+2^n+3^n)/3).
  • A087213 (program): Expansion of (1+x-4*x^2) / ((1+x)*(1-4*x^2)).
  • A087214 (program): Expansion of e.g.f.: exp(x)/(1-x^2/2).
  • A087215 (program): Lucas(6*n): a(n) = 18*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 18.
  • A087216 (program): a(n) = (6n)!/((3n)!(2n)!2^n).
  • A087217 (program): In decimal representation: smallest multiple of n containing it as substring.
  • A087218 (program): Satisfies A(x) = 1 + x*A(x)*f(x)^2, where f(x) = Sum_{k>=0} x^((3^n-1)/2) and f(x)^2 = 2 - f(x^2) + 2*Sum_{n>0} x^A023745(n).
  • A087219 (program): Satisfies A(x) = f(x) + x*A(x)*f(x)^2, where f(x) = Sum_{k>=0} x^((3^n-1)/2) and f(x)^2 = 2 - f(x^2) + 2*Sum_{n>0} x^A023745(n). Also, A(x) = f(x)*B(x), where B(x) = Sum_{k>=0} A087218(k)*x^k.
  • A087221 (program): Number of compositions (ordered partitions) of n into powers of 4.
  • A087222 (program): G.f. satisfies A(x) = 1 + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).
  • A087223 (program): G.f. satisfies A(x) = f(x) + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).
  • A087224 (program): G.f. satisfies A(x) = f(x)^2 + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).
  • A087227 (program): Number of distinct prime factors of A087226(n), the LCM of terms in trajectory of 3x+1 (function) initiated at n.
  • A087229 (program): Exponent of p=2 in 12n+4 = 3(4n+1)+1.
  • A087230 (program): a(n) is the 2-adic valuation of 6*n + 4.
  • A087231 (program): a(n) is the smallest number such that the exponent of p=2 factor in 6*a(n)+4 equals n.
  • A087232 (program): a(n) is the largest odd term in the 3x+1 trajectory initiated at n.
  • A087233 (program): a(n)=floor[sigma[A002110(n)]/A002110(n)]; integer quotient of divisor-sum of primorial numbers and primorials.
  • A087246 (program): Squarefree deficient numbers.
  • A087248 (program): Squarefree abundant numbers.
  • A087252 (program): Numbers that are divisible by 4, but cannot be the largest peak value in a 3x+1 trajectory, regardless of the initial value.
  • A087253 (program): Number of distinct initial values of various 3x+1 trajectories of which the peak-value equals 4n.
  • A087255 (program): Number of different initial values for 3x+1 trajectories of which the largest term appearing during the iteration equals n.
  • A087258 (program): a(n) = gcd(n, A025586(n)), greatest common divisor of n and largest value in 3x+1 iteration list started at n.
  • A087259 (program): a(n) = lcm(n, A025586(n)), least common multiple of n and largest value in 3x+1 iteration list started at n.
  • A087260 (program): a(n) = gcd(4n, A025586(4n)), greatest common divisor of 4n and largest value in 3x+1 iteration list started at 4n.
  • A087261 (program): a(n) = lcm(4n, A025586(4n)), least common multiple of 4n and the largest value in 3x+1 iteration list started at 4n.
  • A087262 (program): Integer quotient of largest and initial values in 3x+1 iteration, started at n.
  • A087265 (program): Lucas numbers L(8*n).
  • A087267 (program): a(n) = gcd(n, pi(n)) where pi is A000720.
  • A087268 (program): Solutions to gcd(x, pi(x)) = 1, where pi is A000720.
  • A087270 (program): Solutions to gcd(x,pi(x)) = gcd(x, A000720(x)) > 1. Numbers x such that x and pi(x) have common divisor larger than one.
  • A087273 (program): Largest prime factor of 3*prime(n) + 1.
  • A087274 (program): Prime index of largest prime factor of 3*prime(n)+1.
  • A087275 (program): Write n in binary: 1ab..yz, then a(n) = 1b..yz + … + 1yz + 1z + 1.
  • A087276 (program): Write n in binary: 1ab..yz, then a(n) = 1ab..yz + … + 1yz + 1z + 1.
  • A087278 (program): Nonnegative integers whose distance to the nearest square is not greater than 1.
  • A087279 (program): Nonnegative numbers whose distance to the nearest positive square equals exactly 1.
  • A087281 (program): a(n) = Lucas(7*n).
  • A087287 (program): a(n) = Lucas(9*n).
  • A087288 (program): a(n)=2a(n-1)+a(n-2)-2a(n-3).
  • A087289 (program): a(n) = 2^(2*n+1) + 1.
  • A087290 (program): Number of pairs of polynomials (f,g) in GF(3)[x] satisfying deg(f) <= n, deg(g) <= n and gcd(f,g) = 1.
  • A087291 (program): Number of pairs of polynomials (f,g) in GF(2)[x] satisfying 1 <= deg(f) <= n, 1 <= deg(g) <= n and gcd(f,g) = 1.
  • A087292 (program): Number of pairs of polynomials (f,g) in GF(3)[x] satisfying 1 <= deg(f) < =n, 1 <= deg(g) <= n and gcd(f,g) = 1.
  • A087297 (program): Add the next prime and multiply by the next prime.
  • A087298 (program): Exponent of 2 in factorization of (3n)!.
  • A087299 (program): Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).
  • A087301 (program): a(n) = n!*Sum_{i=1..n-1} (-1)^(i+1)/i.
  • A087320 (program): Smallest perfect power (at least a square) that is a multiple of n.
  • A087322 (program): Triangle T read by rows: T(n, 1) = 2*n + 1. For 1 < k <= n, T(n, k) = 2*T(n,k-1) + 1.
  • A087323 (program): a(n) = (n+1) * 2^n - 1.
  • A087327 (program): Independence numbers for KT_2 knight on triangular board.
  • A087330 (program): Sum of all digits of all integers less than or equal to 555…55 (with n 5’s) in base 10.
  • A087348 (program): a(n) = 10*n^2 - 6*n + 1.
  • A087349 (program): n + (smallest prime-factor of n+1).
  • A087350 (program): a(n) = Sum_{k=0..n} (3*n)!/(3*k)!.
  • A087355 (program): n^10 mod 10^n.
  • A087370 (program): Numbers n such that 3n - 1 is a prime.
  • A087374 (program): Smallest square >= n!.
  • A087375 (program): Smallest n-th power > n!.
  • A087376 (program): Leading diagonal of A087377.
  • A087380 (program): Let Pricom(n) be defined as the number obtained by replacing each prime digit (2,3,5,7) of n with a ‘0’ and a composite digit (0,4,6,8,9) with a ‘1’ . A 1 remains the same. a(n) = Pricom(n).
  • A087381 (program): Let Compri(n) be the number obtained by replacing each prime digit (2,3,5,7) of n with a ‘1’ and a composite digit( 0,4,6,8,9) with a ‘0’. A 1 remains the same. a(n) = Compri(n).
  • A087385 (program): a(n) = smallest prime == 1 (mod T(n)) where T(n) is the n-th triangular number (A000217).
  • A087386 (program): a(n) = smallest prime == 1 (mod P(n)) where P(n) is the n-th Palindrome.
  • A087397 (program): Smallest triangular number > 1 and == 1 (mod (prime(n)).
  • A087401 (program): Triangle of n*r-binomial(r+1,2).
  • A087404 (program): a(n) = 4a(n-1) + 5a(n-2).
  • A087405 (program): First differences of A087404: a(n)=A087404(n)-A087404(n-1), a(0)=A087404(0).
  • A087413 (program): a(n) = Sum_{k=1..n} C(3*k,k)/3.
  • A087417 (program): Sum of the cubes of A058182.
  • A087420 (program): a(n) is the sum of the squares of the sizes of the conjugacy classes in the dihedral group D_2n.
  • A087423 (program): a(n)=S(3*n,3)/S(n,3) where S(n,m)=sum(k=0,n,binomial(n,k)*fibonacci(m*k)).
  • A087426 (program): a(n) = S(n,4) where S(n,m) = sum(k=0,n,binomial(n,k)*Fibonacci(m*k)).
  • A087427 (program): Number of lattice points on or inside the rectangle formed by [1 <= x <= (q-1)/2] and [1 <= y <= (p-1)/2], where p = n-th prime, q = (n-1)-st prime.
  • A087428 (program): a(n) = [(p-1)/2 * (q-1)/2] mod p where p = (n+1)th odd prime, q = n-th odd prime.
  • A087429 (program): a(n) = 1 if gpf(n) < gpf(n+1), otherwise 0, where gpf = A006530 (greatest prime factor).
  • A087431 (program): a(n) = 0^n/2 + 2^n*(n^2+n+2)/4.
  • A087432 (program): Expansion of 1+x*(1-x-4*x^2)/((1+x)*(1-2*x)*(1-3*x)).
  • A087433 (program): Expansion of (1-2*x)*(1-4*x+x^2)/((1-x)*(1-3*x)*(1-4*x)).
  • A087436 (program): Number of odd prime factors of n, counted with repetitions.
  • A087438 (program): a(n) = 3*2^(2*(n-1)) + 2^(n-2)*(n+1).
  • A087439 (program): Expansion of (1-4x)/((1-x)(1-3x)(1-5x)).
  • A087440 (program): Expansion of (1-2x-3x^2)/((1-2x)(1-4x)).
  • A087444 (program): Numbers that are congruent to {1, 4} mod 9.
  • A087445 (program): Numbers that are congruent to 1 or 5 mod 12.
  • A087446 (program): Numbers that are congruent to {1, 6} mod 15.
  • A087447 (program): a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).
  • A087448 (program): 3^(n-1)(n+3)/2-(n-1)/2.
  • A087449 (program): a(n) = n * 4^(n-1) + (2*4^n + 1) / 3.
  • A087451 (program): G.f.: (2-x)/((1+2x)(1-3x)); e.g.f.: exp(3x)+exp(-2x); a(n)=3^n+(-2)^n.
  • A087452 (program): G.f.: (2-x)/((1+3x)(1-4x)); e.g.f.: exp(4x) + exp(-3x); a(n) = 4^n + (-3)^n.
  • A087453 (program): a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).
  • A087454 (program): Multiplicative inverse of the n-th prime prime(n) modulo prime(n-1).
  • A087455 (program): Expansion of (1 - x)/(1 - 2*x + 3*x^2) in powers of x.
  • A087457 (program): Number of odd length roads between any adjacent nodes in virtual optimal chordal ring of degree 3 (the length of chord < number of nodes/2).
  • A087458 (program): Greatest prime p such that prime(n)+p <= prime(n+1); a(1)=1.
  • A087461 (program): Arithmetic mean of n-th and 2n-th primes.
  • A087462 (program): Generalized mod 3 multiplicative Jacobsthal sequence.
  • A087463 (program): Generalized multiplicative Jacobsthal sequence.
  • A087464 (program): Generalized multiplicative Jacobsthal sequence.
  • A087466 (program): a(n) = number of the row (counting from initial row 0) of the array R in A087465 that contains n.
  • A087467 (program): a(n) = number of the row (counting from initial row 1) of the array R in A087465 that contains n.
  • A087468 (program): Dispersion, read by antidiagonals, of the complement of row 0 of the array R in A087465.
  • A087471 (program): Final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n.
  • A087472 (program): Number of iterations required for the function f(n) to reach a single digit, where f(n) is the product of the two numbers formed from the alternating digits of n.
  • A087475 (program): a(n) = n^2 + 4.
  • A087476 (program): Triangle read by rows where the n-th row has n terms: T(n,i)=(i+2)^2-4 mod n.
  • A087477 (program): Decimal expansion of sqrt(51)-4.
  • A087479 (program): Triangle read by rows where the n-th row has n terms: T(n,i)=i^2+4 mod n.
  • A087480 (program): Sum of all the primes raised to their corresponding powers.
  • A087483 (program): Row 0 of the order array of 3/2, i.e., row 0 of the transposable dispersion in A087465.
  • A087487 (program): 4*(2^(n^2-n+1)*(n^2-n)-n+3).
  • A087503 (program): a(n) = 3(a(n-2) + 1), with a(0) = 1, a(1) = 3.
  • A087504 (program): Composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type E_7 (binary octahedral group).
  • A087505 (program): Numbers k such that 5*k+3 is a prime.
  • A087507 (program): #{0<=k<=n: k*n is divisible by 3}.
  • A087508 (program): Number of k such that mod(k*n,3) = 1 for 0 <= k <= n.
  • A087509 (program): Number of k such that (k*n) == 2 (mod 3) for 0 <= k <= n.
  • A087525 (program): Primes p with the property that p-q does not divide p+q for all primes q < p.
  • A087539 (program): First differences of A011849.
  • A087540 (program): Let A(n) be the matrix in the group GL(n,2) such that for 1 <= i, j <= n: A[i,j] = 1 if i+j = n+1 A[i,j] = 0 if i+j != n+1. a(n) is the size of the centralizer of A(n) in GL(n,2).
  • A087547 (program): a(n) = n!*2^(n+1) * (Integral_{x = 0..1} 1/(1+x^2)^(n+1) dx - Pi*(2*n)!/(2^(n+1)*n!).
  • A087554 (program): a(n) = smallest number k >= n such that nk + 1 is a prime.
  • A087560 (program): Smallest m > n such that gcd(m, n^2) = n.
  • A087567 (program): a(n) = (1/5)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*5^k.
  • A087572 (program): Smallest prime of the form n + (n-1) + (n-2) + …(n-k), k < n, or 0 if no such prime exists.
  • A087579 (program): a(n)=(1/6)*sum(k=0,n,binomial(n,k)*Fibonacci(k)*6^k).
  • A087584 (program): a(n)=(1/7)*sum(k=0,n,binomial(n,k)*Fibonacci(k)*7^k).
  • A087603 (program): a(n) = (1/8)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*8^k.
  • A087611 (program): a(n) = (prime(n) - 1) mod n.
  • A087619 (program): a(n) = 137*a(n-1) + a(n-2), with a(0) = 2 and a(1) = 137.
  • A087620 (program): #{0<=k<=n: k*n is divisible by 4}.
  • A087621 (program): (1,1) entry of powers of the orthogonal design shown below.
  • A087624 (program): a(n)=0 if n is prime, A001221(n) otherwise.
  • A087625 (program): Number of primes in the ring Z_n.
  • A087626 (program): Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).
  • A087627 (program): Count …n,2n,2n…
  • A087628 (program): Generalized Jacobsthal sequence.
  • A087629 (program): Generalized Jacobsthal sequence.
  • A087635 (program): a(n) = S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)*fibonacci(m*k).
  • A087640 (program): To obtain a(n+1), take the square of the n-th partial sum, minus the sum of the first n squared terms, then divide this difference by a(n); for all n>1, starting with a(0)=1, a(1)=1.
  • A087645 (program): Third column of A071223.
  • A087648 (program): a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).
  • A087649 (program): a(n) = (1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)).
  • A087650 (program): a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).
  • A087652 (program): Product of the nonprime divisors of n.
  • A087654 (program): Decimal expansion of D(1) where D(x) is the Dawson function.
  • A087656 (program): Let f be defined on the rationals by f(p/q) =(p+1)/(q+1)=p_{1}/q_{1} where (p_{1},q_{1})=1. Let f^k(p/q)=p_{k}/q_{k} where (p_{k},q_{k})=1. Sequence gives least k such that p_{k}-q_{k} = 1 starting at n.
  • A087674 (program): Value of the n-th Eulerian polynomial (cf. A008292) evaluated at x=-2.
  • A087678 (program): Numbers n such that n + 9 and n - 9 are both prime.
  • A087679 (program): Numbers k such that both k+2 and k-2 are prime.
  • A087680 (program): Numbers n such that n + 4 and n - 4 are both prime.
  • A087681 (program): Numbers n such that n + 6 and n - 6 are both prime.
  • A087682 (program): Numbers n such that n + 8 and n - 8 are both prime.
  • A087683 (program): Numbers n such that n + 10 and n - 10 are both prime.
  • A087686 (program): Elements of A004001 that repeat consecutively.
  • A087688 (program): a(n) = number of solutions to x^3 - x == 0 (mod n).
  • A087689 (program): Numerators of successive partial sums of sum(1/(2^n-1)).
  • A087690 (program): Denominators of successive partial sums of sum(1/(2^n-1)).
  • A087691 (program): Squares of primes of the form 4*k+3.
  • A087692 (program): Number of cubes in multiplicative group modulo n.
  • A087695 (program): Numbers n such that n + 3 and n - 3 are both prime.
  • A087696 (program): Numbers n such that n + 5 and n - 5 are both prime.
  • A087697 (program): Numbers k such that k + 7 and k - 7 are both prime.
  • A087698 (program): Triangle read by rows, giving T(n,k) = maximum number of examples (Boolean inputs) at Hamming distance 2 for symmetric Boolean functions that can have different outputs.
  • A087701 (program): Maximal term in Collatz-iteration started at -1+2^n.
  • A087704 (program): Number of steps for iteration of map x -> (5/3)*floor(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.
  • A087705 (program): First integer > n reached under iteration of map x -> (5/3)*floor(x) when started at n, or -1 if no such integer is ever reached.
  • A087706 (program): A087705/5.
  • A087707 (program): Number of steps for iteration of map x -> (5/3)*ceiling(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.
  • A087708 (program): First integer > n reached under iteration of map x -> (5/3)*ceiling(x) when started at n, or -1 if no such integer is ever reached.
  • A087709 (program): A087708/5.
  • A087713 (program): Greatest prime factor of the product of the neighbors of the n-th prime.
  • A087718 (program): Semiprimes with greater factor less than twice the smaller factor.
  • A087719 (program): Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.
  • A087733 (program): Partial sums of A068639.
  • A087734 (program): a(n) = f(f(n)), where f() = A035327().
  • A087735 (program): Array read by antidiagonals: T(n,k) = o(n,k), where o(,) is a binary operation arising from counting the elements that are sums of m squares in a field of characteristic not equal to 2.
  • A087737 (program): Value of (n,n+1) concatenated in binary representation.
  • A087739 (program): a(1)=1; a(2)=2; for n>2 a(n) satisfies a(S(n))=n and a(k)=n-1 for S(n-1)< k <S(n) where S(n)=a(1)+a(2)+…+a(n).
  • A087743 (program): Numbers n >= 3 with property that the remainder when n is divided by k (for 3 <= k <= n-2) is not 1.
  • A087744 (program): Binary and decimal representation of n concatenated.
  • A087745 (program): Numbers A001317 repeated.
  • A087748 (program): Triangle formed by reading triangle of Stirling numbers of the first kind (A048994) mod 2.
  • A087751 (program): Weighted sum of the harmonic numbers.
  • A087752 (program): Powers of 49.
  • A087754 (program): a(n) = (C(2p,p)-2) / p^3, where p = prime(n).
  • A087755 (program): Triangle read by rows: Stirling numbers of the first kind (A008275) mod 2.
  • A087756 (program): a(n) = A087745(n+1).
  • A087774 (program): a[1] = 1, a[2] = a[3] = 2; a[n] = 3*a[a[n-1]] - 3*a[n-a[n-1]]+a[n-3].
  • A087775 (program): a[1] = 1, a[2] = 2, a[3] = 2; a[n] = 3*a[abs[a[n-2]]] - 3*a[n-abs[a[n-2]]] + a[n-3].
  • A087780 (program): Number of non-congruent solutions to x^2 == 2 mod n.
  • A087781 (program): Number of non-congruent solutions to x^2 - x - 1 == 0 mod n.
  • A087782 (program): a(n) = number of solutions to x^3 + x == 0 (mod n).
  • A087787 (program): a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).
  • A087790 (program): Partial sums of A085068.
  • A087791 (program): Indices n such that S^n(1) is an integer where S(x) = 3/2*ceiling(x).
  • A087793 (program): Least k such that S^k(n)=n^2 where S(x)=n*ceiling(sqrt(x)).
  • A087794 (program): Products of prime-indices of factors of semiprimes.
  • A087797 (program): Primes, squares of primes and cubes of primes.
  • A087798 (program): a(n) = 9*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9.
  • A087799 (program): a(n) = 10*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10.
  • A087800 (program): a(n) = 12*a(n-1) - a(n-2), with a(0) = 2 and a(1) = 12.
  • A087801 (program): Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.
  • A087802 (program): a(n) = Sum_{d|n, d nonprime} mu(d), where mu = A008683.
  • A087804 (program): Binomial transform of squares of Catalan numbers.
  • A087805 (program): Partial sums of b(k) where {b(k)}_{k>=0} = limit n ->infty {A080578(k)-2k : k=2^n,2^n+1,2^n+2,……}.
  • A087806 (program): Inverse binomial transform of squares of Catalan numbers.
  • A087808 (program): a(0) = 0; a(2n) = 2a(n), a(2n+1) = a(n) + 1.
  • A087810 (program): First differences of A029931.
  • A087811 (program): Numbers k such that ceiling(sqrt(k)) divides k.
  • A087812 (program): Sum of MoebiusMu for numbers between n and 2n inclusive.
  • A087815 (program): Terms in A087816 that occur in a run of length more than 1.
  • A087816 (program): a(n) = a(a(n-1)) + a(n - 1 - a(n-1)) with a(1) = a(2) = 1.
  • A087839 (program): a[n] =a[a[a[a[a[n-2]]]]]+ a[n - a[n-2]].
  • A087847 (program): a(n) = a(|n - a(n-1)|) + a(a(a(|n - a(n-4)|))).
  • A087851 (program): a(n)=Abs(a(n-1)-floor(n*phi)), where phi=(1+sqrt(5))/2.
  • A087857 (program): Primes of the form 16*m^2 + 25 for m=1,2,3,…
  • A087860 (program): Expansion of e.g.f.: (1-exp(x/(x-1)))/(1-x).
  • A087861 (program): Primes of the form 16*m^2 + 81, m=1,2,3,…
  • A087862 (program): Primes of the form 16*m^2 + 169, m=1,2,3,…
  • A087863 (program): (n^3+24*n^2+65*n+36)/6.
  • A087866 (program): Composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type E_8 (binary icosahedral group).
  • A087871 (program): Primes of the form (4*k + 1)^2 + (4*k + 2)^2 where k=0,1,2,3,…
  • A087872 (program): Primes of the form (4*k + 3)^2 + (4*k + 2)^2 where k=0,1,2,3,…
  • A087887 (program): a(n) = 18n^3 + 6n^2.
  • A087890 (program): Given a sequence u consisting just of 1’s and 2’s, let f(u)(n) be the length of n-th run. Then we may define a sequence u = {a(n)} by a(n)=f^(n-1)(u)(1) (starting with n=1).
  • A087893 (program): Number of numbers m satisfying 1 < m < n such that m^2 == m (mod n).
  • A087895 (program): Primes p such that 10^p - 9^p is composite.
  • A087897 (program): Number of partitions of n into odd parts greater than 1.
  • A087906 (program): a(n) = Sum_{d|n} (n-1)!/(d-1)!.
  • A087908 (program): Largest integer not expressible as a nonnegative linear combination of n and n^2 + 1.
  • A087909 (program): a(n) = Sum_{d|n} (n/d)^(d-1).
  • A087910 (program): Exponent of the greatest power of 2 dividing the numerator of 2^1/1 + 2^2/2 + 2^3/3 + … + 2^n/n.
  • A087912 (program): Exponential generating function is exp(2*x/(1-x))/(1-x).
  • A087915 (program): Even numbers n such that 2*n+3 is a prime.
  • A087923 (program): Number of ways of arranging the numbers 1 … 2n into a 2 X n array so there is exactly one local maximum.
  • A087935 (program): Perrin sequence of order 5.
  • A087936 (program): Perrin sequence of order 6.
  • A087940 (program): a(n) = Sum_{k=0..n} binomial(n+(-1)^k, k).
  • A087943 (program): Numbers n such that 3 divides sigma(n).
  • A087944 (program): Expansion of (1-4*x+3*x^2)/((1-2*x)*(1-4*x+x^2)).
  • A087945 (program): Expansion of (1-2x-x^2)/((1-2x)(1-4x+x^2)).
  • A087946 (program): Expansion of (1-3x+x^2)/((1-2x)(1-4x+x^2)).
  • A087952 (program): Smallest prime == 1 (mod n) and > n^2.
  • A087953 (program): a(n) = floor((Fibonacci(2*n+1)+1)/2).
  • A087954 (program): (2 + phi)/a(n) is the sum of successive remainders when computing the Euclidean algorithm for (1, A088166(n)/phi) with phi being the golden ratio, where n >= 2.
  • A087955 (program): a(n) is the square of the n-th partial sum minus the n-th partial sum of the squares, divided by a(n-1), for all n>=1, starting with a(0)=1, a(1)=2.
  • A087956 (program): a(n) is the square of the n-th partial sum minus the n-th partial sum of the squares, divided by a(n-1), for all n>=1, starting with a(0)=1, a(1)=3.
  • A087957 (program): a(n) is the square of the n-th partial sum minus the n-th partial sum of the squares, divided by a(n-1), for all n>=1, starting with a(0)=1, a(1)=4.
  • A087958 (program): a(n) is the square of the n-th partial sum minus the n-th partial sum of the squares, divided by a(n-1), for all n>=1, starting with a(0)=1, a(1)=5.
  • A087960 (program): a(n) = (-1)^binomial(n+1,2).
  • A087963 (program): Exponent of highest power of 2 dividing 3*prime(n)+1.
  • A087966 (program): a(n) = gcd(-1 + 2^n, n^2).
  • A087968 (program): a(n) = gcd(1 + 2^n, n^2).
  • A087970 (program): Maximal term in Collatz-iteration started at 1+2^n.
  • A087971 (program): Maximal term in Collatz-iteration started at -1+3^n.
  • A087972 (program): Maximal term in Collatz-iteration started at 1+3^n.
  • A087973 (program): Maximal term in Collatz-iteration started at 3^n.
  • A087976 (program): a(n) = A001221(A025586(n)), the number of distinct prime-factors of maximal term in 3x+1 iteration list started at n.
  • A087981 (program): E.g.f.: exp(-2*x) / (1-x)^2.
  • A087990 (program): Number of palindromic divisors of n.
  • A087991 (program): Number of non-palindromic divisors of n.
  • A087996 (program): Residues when (n+rev[n]) is divided by (abs(n-rev[n]); if n=rev[n] (when n is palindromic), or when the quotient is integer (see A087993).
  • A088000 (program): a(n) is the sum of the palindromic divisors of n.
  • A088001 (program): a(n) is the sum of non-palindromic divisors of n.
  • A088002 (program): Expansion of (1+x^2)/(1+x^2+x^5).
  • A088003 (program): Take the list t(n,0) = {1,…,n}; denote by t(n,j) this list after rotating to left (or right) by j positions. Calculate inner product of t(n,0) and t(n,j) and denote the value by s(n,j). Compute this inner product for all j = 1..n and choose the smallest. This is a(n).
  • A088009 (program): Number of “sets of odd lists”, cf. A000262.
  • A088013 (program): Binomial transform of A001541 (with interpolated zeros).
  • A088014 (program): Expansion of e.g.f.: cosh(sqrt(2)*x)*(1+exp(x)).
  • A088015 (program): Expansion of e.g.f. cosh(sqrt(2)*x) + exp(x)*(cosh(sqrt(2*x) - 1).
  • A088016 (program): To obtain a(n+1), add the square of the n-th partial sum to the n-th partial sum of the squares, then divide this result by a(n), for all n >= 0, with a(0)=1.
  • A088018 (program): Number of twin-prime pairs between n and 2n (inclusive).
  • A088019 (program): Number of twin primes between n and 2n (inclusive).
  • A088023 (program): Set a(1) = 1. Then take the list of defined initial terms, reverse their order, add 1, 2, 3, … to the reversed list in succession and append this new list to the right of the previously defined terms. Repeat this process indefinitely.
  • A088026 (program): Number of “sets of even lists” for even n, cf. A000262.
  • A088031 (program): Smallest n-th power k such that k-1 is divisible by an n-th power. Smallest n-th power == 1 (mod some n-th power).
  • A088032 (program): Smallest number k such that k^n -1 is divisible by an n-th power. a(n) = A088031(n)^(1/n).
  • A088033 (program): Even squares k such that k-1 is divisible by a square.
  • A088034 (program): Even numbers k such that k^2-1 is divisible by a square.
  • A088037 (program): Smallest square k == 1 (mod some n-th power), k > 1.
  • A088038 (program): Smallest cube k == 1 (mod some n-th power), k > 1.
  • A088039 (program): Smallest k such that k^3 == 1 (mod some n-th power), k > 1.
  • A088040 (program): Smallest fourth power k such that k-1 is divisible by an n-th power, k > 1.
  • A088041 (program): Smallest k such that k^4 - 1 is divisible by an n-th power, k > 1.
  • A088055 (program): a(n) = (n^(n+1)-1)/(n-1) - 1 - n!*n^n, or A031972(n) - A061711(n): sums of geometric progressions minus products of arithmetic progressions.
  • A088070 (program): Numbers sandwiched between two numbers having the same number of prime divisors.
  • A088071 (program): Number sandwiched between two numbers having only one prime divisor.
  • A088113 (program): a(n) = digit reversal of (11^n) divided by 11.
  • A088121 (program): Smallest prime obtained as a sum of n terms of a geometric progression + the common ratio, or 0 if no such terms exists. Smallest prime of the form (a +ar +ar^2 + ar^3 +… ) + r.
  • A088127 (program): E.g.f. exp(-x)*cosh(x)/(1-x)^2.
  • A088128 (program): Expansion of e.g.f.: cosh(x)/(1-x)^2.
  • A088129 (program): Expansion of e.g.f. sinh(x)/(1-x)^2.
  • A088131 (program): a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>=1, with a(0)=1, a(1)=1.
  • A088132 (program): a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>1, with a(0)=1, a(1)=3.
  • A088133 (program): Sum of first and last digits of n. Different from A115299.
  • A088134 (program): Numbers n such that sum of first and last digits is prime.
  • A088135 (program): Sum of first and last digits of n-th prime.
  • A088137 (program): Generalized Gaussian Fibonacci integers.
  • A088138 (program): Generalized Gaussian Fibonacci integers.
  • A088139 (program): a(n) = 2*a(n-1) - 6*a(n-2), a(0)=0, a(1)=1.
  • A088140 (program): Duplicate of A005451 (for n >= 3).
  • A088146 (program): n-th prime rotated one binary place to the right.
  • A088147 (program): n-th prime rotated one binary place to the left.
  • A088150 (program): Value of n-th digit (counting from the right) in binary representation of n^n.
  • A088151 (program): Value of n-th digit in ternary representation of n^n.
  • A088153 (program): Value of n-th digit in decimal representation of n^n.
  • A088161 (program): n rotated one binary place to the right less n rotated one binary place to the left.
  • A088162 (program): n-th prime rotated one binary place to the right less the n-th prime rotated one binary place to the left.
  • A088163 (program): Numbers for which rotating one binary place to the right less rotating one binary place to the left is equal to zero.
  • A088166 (program): Smallest integer divisible by Fibonacci(2n) such that the second partial quotient in the continued fraction expansion of a(n)/phi is 2 (phi is the golden ratio), n >= 2.
  • A088172 (program): First differences of A019300.
  • A088175 (program): Primes such that the next two primes are a twin prime pair.
  • A088176 (program): Primes such that the previous two primes are a twin prime pair.
  • A088181 (program): E.g.f.: 1/(1-sinh(x)-x).
  • A088186 (program): Sums of twin primes and their indices.
  • A088190 (program): Largest quadratic residue modulo prime(n).
  • A088191 (program): First differences of A088190.
  • A088192 (program): Distance between prime(n) and the largest quadratic residue modulo prime(n).
  • A088202 (program): Chromatic number of the n X n queen graph.
  • A088207 (program): a(n) = Sum_{k=0..n} floor(k*phi^2)) where phi=(1+sqrt(5))/2.
  • A088208 (program): Table read by rows where T(0,0)=1; n-th row has 2^n terms T(n,j),j=0 to 2^n-1. For j==0 mod 2, T(n+1,2j)=T(n,j) and T(n+1,2j+1)=T(n,j)+2^n. For j==1 mod 2, T(n+1,2j+1)=T(n,j) and T(n+1,2j)=T(n,j)+2^n.
  • A088209 (program): Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,…(n 1’s)…,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], …
  • A088210 (program): Numerators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,…(n 2’s)…,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], …
  • A088211 (program): Denominators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,…(n 2’s)…,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], …
  • A088212 (program): Smallest k>1 such that n+k^2 is prime.
  • A088213 (program): Primes appeared in A088212.
  • A088218 (program): Total number of leaves in all rooted ordered trees with n edges.
  • A088224 (program): Numbers that fill in the gaps between primes in a list of twin primes.
  • A088225 (program): Solutions to x^n == 7 (mod 11).
  • A088226 (program): a(1)=0, a(2)=0, a(3)=1; for n>3, a(n)=abs(a(n-1)-a(n-2)-a(n-3)).
  • A088227 (program): Solutions x to x^n == 7 mod 13.
  • A088229 (program): Number of n X n (0,1) matrices with distinct rows.
  • A088232 (program): Numbers k such that 3 does not divide phi(k).
  • A088233 (program): First differences of roots of consecutive prime powers; a(1)=1.
  • A088234 (program): First differences of exponents of consecutive prime powers.
  • A088235 (program): Total number of digits (in base 10) in all preceding terms in the sequence.
  • A088236 (program): Total number of digits (in base 2) in all preceding terms in the sequence.
  • A088239 (program): Triangle read by rows: T(n,k) is the number of primes not less than n-k and not greater than n+k, 0<=k<n.
  • A088244 (program): Number of 3-smooth divisors of n!.
  • A088245 (program): Decimal expansion of 9/(2*Pi^2).
  • A088246 (program): Decimal expansion of 21/(2*Pi^2).
  • A088265 (program): Numbers of the form 10^n + 1, 3, 7, or 9 for n>=1.
  • A088305 (program): a(0) = 1, a(n) = Fibonacci(2*n). It has the property: a(n) = 1*a(n-1) + 2*a(n-2) + 3*a(n-3) + 4*a(n-4) + …
  • A088307 (program): Triangle read by rows, 1 <= k <= n: T(n,k) = n^2 + k^2 if gcd(n,k)=1, otherwise 0.
  • A088308 (program): 2 followed by list of composite numbers mod 10.
  • A088311 (program): Number of sets of lists with distinct list sizes, cf. A000262.
  • A088312 (program): Number of sets of lists (cf. A000262) with even number of lists.
  • A088313 (program): Number of “sets of lists” (cf. A000262) with an odd number of lists.
  • A088316 (program): a(n) = 13*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 13.
  • A088317 (program): a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.
  • A088320 (program): a(n) = 10a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 5.
  • A088323 (program): Number of numbers b>1 such that n is a repunit in base b representation.
  • A088333 (program): A version of Josephus problem: a(n) is the surviving integer under the following elimination process. Arrange 1,2,3,…,n in a circle, increasing clockwise. Starting with i=1, delete the integer 3 places clockwise from i. Repeat, counting 3 places from the next undeleted integer, until only one integer remains.
  • A088336 (program): Number of permutations in the symmetric group S_n that have even number of transpositions in their cycle decomposition.
  • A088338 (program): Numbers n such that frac(x^n)=frac(x*frac(x^(n-1))) where x=3/2 and frac(x) denotes the fractional part of x.
  • A088339 (program): Numbers n such that n multiplied by (the 9’s complement of n) + 1 is a prime. If n has d digits then 9’s complement of n is 10^d -(n+1).
  • A088340 (program): Numbers k such that frac(x^k) = frac(x*frac(x^(k-1))) where x=5/2 and frac(x) denotes the fractional part of x.
  • A088342 (program): Let T = Sum_{k >= 1} k^(k-1)*x^k be the g.f. for rooted labeled trees (A000169); sequence has g.f. T/(1-T).
  • A088347 (program): This sequence needs a definition.
  • A088359 (program): Numbers which occur only once in A004001.
  • A088371 (program): Position where n is inserted into the n-th row of triangle A088370, where the n-th row differs from the prior row only by the presence of n.
  • A088372 (program): Table read by rows where T(0,0)=0; n-th row has 2^n terms T(n,j),j=0 to 2^n-1. T(n,T088208(n,j))=2^n-j, where T088208 is the table described in A088208.
  • A088377 (program): (Smallest prime-factor of n)^2.
  • A088378 (program): (Smallest prime-factor of n)^3.
  • A088379 (program): (Smallest prime-factor of n)^4.
  • A088380 (program): Numbers not exceeding the cube of their smallest prime-factor.
  • A088381 (program): Numbers greater than the cube of their smallest prime factor.
  • A088386 (program): a(n) = 2^n*(n!)^3.
  • A088387 (program): Prime corresponding to largest prime power factor of n, a(1)=1.
  • A088388 (program): Exponent of the largest prime power factor of n, a(1)=0.
  • A088404 (program): a(n) = A069537(n)/2.
  • A088405 (program): a(n) = A052217(n)/3.
  • A088407 (program): a(n) = A069540(n)/5.
  • A088408 (program): a(n) = A062768(n)/6.
  • A088409 (program): a(n) = A063416(n)/7.
  • A088410 (program): a(n) = A069543(n)/8.
  • A088417 (program): a(n) = A088339(n)/3.
  • A088420 (program): Number of primes in arithmetic progression starting with 3 and with d = 2n.
  • A088421 (program): Number of primes in arithmetic progression starting with 5 and with d=2n.
  • A088422 (program): Number of primes in arithmetic progression starting with 7 and with d=2n.
  • A088423 (program): a(n) is the number of primes in arithmetic progression starting with 11 and with d = 2n.
  • A088424 (program): Number of primes in arithmetic progression starting with 13 and with d=2n.
  • A088425 (program): Number of primes in arithmetic progression starting with 17 and with d=2n.
  • A088426 (program): Number of primes in arithmetic progression starting with 19 and with d=2n.
  • A088427 (program): Number of primes in arithmetic progression starting with 23 and with d=2n.
  • A088428 (program): Number of primes in arithmetic progression starting with 29 and with d=2n.
  • A088429 (program): Number of primes in arithmetic progression starting with 31 and with d=2n.
  • A088431 (program): Half of the (n+1)-st component of the continued fraction expansion of Sum_{k>=1} 1/2^(2^k).
  • A088434 (program): Number of ways to write n as n = u*v*w with 1 <= u < v < w.
  • A088435 (program): 1/2 + half of the (n+1)-st component of the continued fraction expansion of sum(k>=1,1/3^(2^k)).
  • A088436 (program): Number of permutations in the symmetric group S_n that have exactly one transposition in their cycle decomposition.
  • A088437 (program): Number of n X n orthogonal matrices over GF(2) modulo permutations of rows.
  • A088439 (program): a(3n) = 3n, otherwise a(n) = 1.
  • A088440 (program): a(4n) = 4n, otherwise a(n) = 1.
  • A088441 (program): A one third Cantor set as a factorial product function.
  • A088442 (program): A linear version of the Josephus problem.
  • A088459 (program): Triangle read by rows: T(n,k) represents the number of lozenge tilings of an (n,1,n)-hexagon which include the non-vertical tile above the main diagonal starting in position k+1.
  • A088462 (program): a(1)=1, a(n) = ceiling((n - a(a(n-1)))/2).
  • A088472 (program): Numbers n such that the lunar sum of the distinct lunar prime divisors of n is < n.
  • A088475 (program): Numbers n such that the lunar sum of the distinct lunar prime divisors of n is >= n.
  • A088476 (program): Numbers n such that the lunar sum of the distinct lunar prime divisors of n is > n.
  • A088480 (program): Numbers n such that the lunar product of the distinct lunar prime divisors of n is >= n.
  • A088481 (program): Numbers n such that the lunar product of the distinct lunar prime divisors of n is > n.
  • A088482 (program): A four-level self-similar Sierpinski chaotic integer sequence.
  • A088485 (program): Numbers n such that n^2 + n - 1 and n^2 + n + 1 are twin primes.
  • A088486 (program): Primes p of the form k*(k + 1) - 1 such that p and p + 2 are twin primes.
  • A088487 (program): a(n) = Sum_{k=1..8} floor(A254864(n,k)/A254864(n-1,k)), where A254864(n,k) = n! / (n-floor(n/3^k))!.
  • A088491 (program): a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).
  • A088492 (program): a(2n+1)=2n+1, a(2n) = floor(2*n/A005185(n)), a weighted inverse of Hofstadter’s Q-sequence.
  • A088499 (program): Doubly (3)-perfect numbers.
  • A088500 (program): Expansion of e.g.f. 1/(1+2*log(1-x)).
  • A088501 (program): Expansion of e.g.f. 1/(1-2*log(1+x)).
  • A088502 (program): Numbers n such that (n^2 - 5)/4 is prime.
  • A088503 (program): Numbers n such that (n^2 + 3)/4 is prime.
  • A088504 (program): Sum of even entries in row n of Pascal’s triangle.
  • A088505 (program): a(n) = (2^(3*n-1))/(integral_{x=0..1} (1-x^4)^n dx).
  • A088506 (program): Number of permutations in the symmetric group S_n that have odd number of transpositions in their cycle decomposition.
  • A088512 (program): Number of partitions of n into two parts whose xor-sum is n.
  • A088517 (program): First differences of Golomb’s sequence.
  • A088520 (program): Permutation of natural numbers generated by 3-rowed array shown below.
  • A088521 (program): a(1) = 1; for n > 1, a(n) = (a(n-1) + n) mod prime(n).
  • A088522 (program): a(1) = 2; for n > 1, a(n) = (a(n-1) + n) mod prime(n).
  • A088527 (program): Define a Fibonacci-type sequence to be one of the form s(1) = s_1 >= 1, s(2) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.
  • A088529 (program): Numerator of Bigomega(n)/Omega(n).
  • A088534 (program): Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.
  • A088536 (program): Number of unimodal functions [1..n]->[1..n].
  • A088538 (program): Decimal expansion of 4/Pi.
  • A088543 (program): Decimal expansion of sqrt(15)/2.
  • A088545 (program): Quotient Fib(5n)/(5*Fib(n)), where Fib(n) = A000045(n).
  • A088547 (program): Primes of the form x^3+x^2+x+2.
  • A088551 (program): Fibonacci winding number: the number of ‘mod n’ operations in one cycle of the Fibonacci sequence modulo n.
  • A088555 (program): Primes of the form 5*p + 6 where p is a prime.
  • A088556 (program): Numbers of the form (4^n + 4^(n-1) + … + 1) + (n mod 2).
  • A088557 (program): Least even leg of primitive Pythagorean triangles with odd leg 2n+1.
  • A088558 (program): Least odd leg of primitive Pythagorean triangles with even leg 4n.
  • A088559 (program): Decimal expansion of R^2 where R^2 is the real root of x^3 + 2*x^2 + x - 1 = 0.
  • A088560 (program): Sum of odd entries in row n of Pascal’s triangle.
  • A088561 (program): A088555 indexed by A000040.
  • A088564 (program): a(n)=sum(i=0,n,binomial(2*i,i) (mod 3)).
  • A088567 (program): Number of “non-squashing” partitions of n into distinct parts.
  • A088568 (program): 3*n - 2*(partial sums of Kolakoski sequence A000002).
  • A088569 (program): Anti-Kolakoski sequence (sequence of run lengths never coincides with the sequence itself).
  • A088570 (program): Sum of terms in n-th block of Kolakoski sequence.
  • A088572 (program): Numbers n such that (2n+1)^2 - 2 is prime.
  • A088575 (program): Bisection of A088567.
  • A088578 (program): a(n) = n*x^n + (n-1)*x^(n-1) + . . . + x + 1 for x=2.
  • A088580 (program): a(n) = 1 + sigma(n).
  • A088581 (program): a(n) = n*x^n + (n-1)*x^(n-1) + . . . + x + 1 for x=3.
  • A088582 (program): a(n) = n*x^n + (n-1)*x^(n-1) + . . . + x + 1 for x=4.
  • A088585 (program): Bisection of A088567.
  • A088592 (program): Let p be the n-th 4k+3 prime (A002145), g be any primitive root of p. The mapping x->g^x mod p gives a permutation of {1,2,…,p-1}. a(n) is 0 if the permutation is even for each g, 1 if odd for each g.
  • A088601 (program): Number of steps to reach 0 when iterating A261424(x) = x - (the largest palindrome less than x), starting at n.
  • A088613 (program): Smallest nonsquarefree multiple of n.
  • A088617 (program): Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.
  • A088625 (program): 14*C(n,8).
  • A088626 (program): 42*C(n,10).
  • A088631 (program): Largest number m < n such that m+n is a prime.
  • A088633 (program): P-n where P = smallest prime > 2n.
  • A088644 (program): Smallest n-digit number divisible by n!.
  • A088645 (program): a(n) = A088644(n)/n!.
  • A088648 (program): a(1) = 1, then the smallest odd number not occurring earlier such that the concatenation a(r), a(s) is composite for all s > r.
  • A088650 (program): a(n) = smallest value of x pertaining to A020498, or the smallest x such that A020498(k) + x is prime for all k = 1 to n.
  • A088659 (program): a(n) = n*(p-1) where p is the largest prime factor of n.
  • A088660 (program): A logarithmic scale Sierpinski self-similar sequence.
  • A088662 (program): Number of peaks at even level in all symmetric Dyck paths of semilength n+2.
  • A088666 (program): a(n) = (n^4 + 1) mod 10.
  • A088667 (program): n^4 + 6 mod 10.
  • A088668 (program): Number of n X n matrices over GF(2) with characteristic polynomial x^(n-1) * (x-1).
  • A088669 (program): Number of partitions of n into decimal repdigit numbers.
  • A088671 (program): Number of n X n matrices over GF(3) with characteristic polynomial x^(n-1) * (x-1).
  • A088673 (program): n mod A002024(n), where A002024 is “n appears n times”: 1, 2, 2, 3, 3, 3, …
  • A088677 (program): Numbers that can be represented as j^6 + k^6, with 0 < j < k, in exactly one way.
  • A088679 (program): a(n) = a(n-1)^2 * n / (n-1), n>1, a(0) = 0, a(1) = 1.
  • A088680 (program): a(n) = prime(2n+1) - prime(2n).
  • A088682 (program): a(n) = prime(3*n+1) - prime(3*n-1).
  • A088683 (program): a(n) = prime(3*n+2) - prime(3*n).
  • A088684 (program): Prime(3n+3)-prime(3n+1).
  • A088688 (program): Binomial transform of A088689.
  • A088689 (program): Jacobsthal numbers modulo 3.
  • A088690 (program): E.g.f.: A(x) = f(x*A(x)), where f(x) = (1+x)*exp(x).
  • A088692 (program): E.g.f: A(x) = f(x*A(x)), where f(x) = (1+2*x)*exp(x).
  • A088695 (program): E.g.f: A(x) = f(x*A(x)), where f(x) = exp(x+x^2).
  • A088696 (program): Triangle read by rows, giving number of partial quotients in continued fraction representation of terms in the left branch of the infinite Stern-Brocot tree.
  • A088697 (program): Replace 0 with 10 in binary representation of n.
  • A088698 (program): Replace 1 with 11 in binary representation of n.
  • A088699 (program): Array read by antidiagonals of coefficients of generating function exp(x)/(1-y-xy).
  • A088700 (program): Number of primes between successive semiprimes.
  • A088705 (program): First differences of A000120. One minus exponent of 2 in n.
  • A088707 (program): Semiprimes + 1.
  • A088719 (program): Numbers that can be represented as a^7 + b^7, with 0 < a < b, in exactly one way.
  • A088720 (program): Unique monotone sequence satisfying a(a(a(n))) = 2n.
  • A088721 (program): Unique monotone sequence satisfying a(a(a(a(n)))) = 2n.
  • A088722 (program): Number of divisors d>1 of n such that d+1 also divides n.
  • A088723 (program): Numbers k with at least one divisor d>1 such that d+1 also divides k.
  • A088724 (program): Numbers having exactly one divisor d>1 such that also d+1 is a divisor.
  • A088725 (program): Numbers having no divisors d>1 such that also d+1 is a divisor.
  • A088730 (program): Numbers of the form p^p - 1, where p is a prime.
  • A088731 (program): Numbers of the form p^p - 3 where p is prime.
  • A088732 (program): First prime in the arithmetic progression n+k*(n+1) with k>0.
  • A088733 (program): n-th prime in the arithmetic progression n+k*(n+1) with k>0.
  • A088736 (program): 10^p - p for prime p.
  • A088737 (program): Number of semiprime divisors of n-th composite number.
  • A088738 (program): Sum of semiprime divisors of n-th composite number.
  • A088739 (program): Smallest semiprime divisor of n-th composite number.
  • A088740 (program): Greatest semiprime divisor of n-th composite number.
  • A088741 (program): Number of connected strongly regular simple graphs on n nodes.
  • A088742 (program): Run lengths of A088023.
  • A088743 (program): a(n) = 2*A088023(n) - 1.
  • A088744 (program): a(n) = 3*A088023(n) - 2.
  • A088748 (program): a(n) = 1 + Sum_{k=0..n-1} 2 * A014577(k) - 1.
  • A088758 (program): Numbers k such that (4*k + 1)^2 + (4*k + 2)^2 is prime.
  • A088759 (program): Numbers n such that (4n+3)^2 + (4n+2)^2 is prime.
  • A088762 (program): Numbers n such that (2n-1, 2n+3) is a cousin prime pair.
  • A088763 (program): a(n) = A087695(n)/2.
  • A088764 (program): a(n) = (A087680(n)-1)/2.
  • A088765 (program): a(n) = A087696(n)/2.
  • A088766 (program): a(n) = (A087681(n)-1)/2.
  • A088767 (program): a(n) = A087697(n)/2.
  • A088768 (program): a(n) = (A087682(n)-1)/2.
  • A088769 (program): a(n) = A087678(n)/2.
  • A088770 (program): a(n) = (A087683(n)-1)/2.
  • A088783 (program): Numbers n such that 10*n^k + 1 is composite for all k > 0.
  • A088789 (program): E.g.f.: REVERT(2*x/(1+exp(x))) = Sum_{n>=0} a(n)*x^n/n!.
  • A088795 (program): Fibonacci(n) as n runs through the quarter-squares.
  • A088802 (program): Denominators of the coefficients of powers of n^(-1) in the Romanovsky series expansion of the mean of the standard deviation from a normal population.
  • A088805 (program): 10^p + p for prime p.
  • A088808 (program): Number of subsets of {1, …, n} that are not double-free.
  • A088816 (program): Numbers of the form p^p - 2 where p is prime.
  • A088821 (program): a(n) is the sum of smallest prime factors of numbers from 1 to n.
  • A088822 (program): a(n) is the sum of largest prime factors of numbers from 1 to n.
  • A088827 (program): Even numbers with odd abundance: even squares or two times squares.
  • A088828 (program): Nonsquare positive odd numbers.
  • A088829 (program): Even numbers with even abundance.
  • A088835 (program): a(n) = lcm(A020639(n), A006530(n)).
  • A088836 (program): a(n) = (A020639(2n+1) + A006530(2n+1))/2, arithmetic mean of max and min prime factor applied to odd numbers (i.e., when this mean is integer).
  • A088837 (program): Numerator of sigma(2*n)/sigma(n). Denominator see in A038712.
  • A088838 (program): Numerator of the quotient sigma(3n)/sigma(n).
  • A088839 (program): Numerator of sigma(4n)/sigma(n).
  • A088840 (program): Denominator of sigma(4n)/sigma(n).
  • A088841 (program): Numerator of quotient=sigma[7n]/sigma[n].
  • A088842 (program): Denominator of the quotient sigma(7n)/sigma(n).
  • A088852 (program): Number of n X n matrices over GF(4) with characteristic polynomial x^(n-1) * (x-1).
  • A088853 (program): Number of n X n matrices over GF(5) with characteristic polynomial x^(n-1) * (x-1).
  • A088854 (program): a(n) = (2^(n-1))*(Integral_{x=0..1} (1+x^2)^n dx)/(Integral_{x=0..1} (1-x^2)^n dx).
  • A088858 (program): Define a Fibonacci-type sequence to be one of the form s(0) = s_1 >= 1, s(1) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.
  • A088859 (program): a(n) = L(n) + 2^n where L(n) = A000032(n) (the Lucas numbers).
  • A088860 (program): Twice the primorials (first definition), 2*A002110(n).
  • A088865 (program): (Sum of distinct prime factors)^(sum of prime exponents).
  • A088878 (program): Prime numbers p such that 3p - 2 is a prime.
  • A088879 (program): Numbers n such that 3n + 5 is a prime.
  • A088889 (program): Polynexus numbers of order 8.
  • A088890 (program): Polynexus numbers of order 8.
  • A088891 (program): Polynexus numbers of order 9.
  • A088892 (program): Polynexus numbers of order 14.
  • A088896 (program): Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.
  • A088905 (program): Numbers m having exactly one representation m = x^i + x^j with 1<x<=m and 0<=i<=j.
  • A088911 (program): Period 6: repeat [1, 1, 1, 0, 0, 0].
  • A088914 (program): a(n) = (Fibonacci(2n+1) + Fibonacci(2n+2)*phi)/kappa(phi/Fibonacci(4n+2)) where kappa(x) is the sum of successive remainders by computing the Euclidean algorithm for (1,x).
  • A088917 (program): Central Delannoy numbers (mod 3); Characteristic function for Cantor set.
  • A088920 (program): Solutions k to the Diophantine equation k = 2n^2 = m^2+1.
  • A088921 (program): The number of 321- and 2143-avoiding permutations of length n.
  • A088922 (program): Consider the n X n matrix with entries (i*j mod n), where i,j=0..n-1; a(n) = rank of this matrix over the real numbers.
  • A088924 (program): Number of “9ish numbers” with n digits.
  • A088927 (program): Antidiagonal sums of table A088925, which lists coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.
  • A088931 (program): G.f.: Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ).
  • A088932 (program): G.f.: 1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8)).
  • A088938 (program): Occurrences of 2’s in A088936.
  • A088941 (program): a(n)=12*sum(1<=i<=j<=k<=n,i*j/k).
  • A088948 (program): Numbers k such that (A006530(k) + A020639(k))/2 is an integer; that is, arithmetic mean of least and largest prime factor is an integer.
  • A088954 (program): G.f.: 1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8)*(1-x^16)).
  • A088955 (program): Primes of the form 60*n + 1.
  • A088956 (program): Triangle, read by rows, of coefficients of the hyperbinomial transform.
  • A088957 (program): Hyperbinomial transform of the sequence of 1’s.
  • A088958 (program): Numbers n such that 60*n+1 is prime.
  • A088967 (program): Numbers n such that n+9 is a prime.
  • A088978 (program): Number of Pythagorean triangles having the n-th prime prime(n) as one of their sides.
  • A088979 (program): a(n) = n! - ((n-1)!!)^2.
  • A088980 (program): G.f.: 1 + x + Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ).
  • A088981 (program): a(n+2) = a(n+1) + a(n) - [(2*n)+1] where a(0)=7, a(1)=11.
  • A088982 (program): Primes that are between consecutive prime-indexed primes.
  • A088984 (program): a(n) = (n!!)^2 - n!.
  • A088985 (program): Numbers of the form prime(prime(n)+1), with n satisfying prime(n)+2 = prime(n+1).
  • A088987 (program): Triples of primes between two consecutive prime-indexed primes.
  • A088991 (program): Derangement numbers d(n,4) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.
  • A088992 (program): Derangement numbers d(n,5) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.
  • A088993 (program): Primes such that exactly five of them occur between consecutive prime-indexed primes.
  • A089000 (program): Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by :T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.
  • A089001 (program): Numbers n such that 2*n^2 + 1 is prime.
  • A089008 (program): Numbers k such that 18*k^2 + 1 is prime.
  • A089010 (program): a(n) = 1 if n is an exponent of the Weyl group W(E_8), 0 otherwise.
  • A089011 (program): a(n) = 1 if n is an exponent of the Weyl group W(E_7), 0 otherwise.
  • A089012 (program): a(n) = 1 if n is an exponent of the Weyl group W(E_6), 0 otherwise.
  • A089013 (program): a(n) = (A088567(8n) mod 2).
  • A089022 (program): Number of walks of length 2n on the 3-regular tree beginning and ending at some fixed vertex.
  • A089024 (program): 1’s separated by d(n) 0’s, where d(n) = n-th digit of decimal expansion of Pi.
  • A089026 (program): a(n) = n if n is a prime, otherwise a(n) = 1.
  • A089027 (program): a(n) =1 if the prime gap A001223(n) is <=2, otherwise a(n)=n+1.
  • A089028 (program): a(n) = n+1 where the Hofstadter-Conway Delta A093878(n) >0, otherwise a(n) = 1.
  • A089033 (program): Numbers n such that 7*n+3 is prime.
  • A089034 (program): a(n) = (prime(n)^4 - 1) / 240.
  • A089038 (program): Nonnegative numbers k such that 2k+5 is prime.
  • A089039 (program): Number of circular permutations of 2n letters that are free of jealousy.
  • A089041 (program): Inverse binomial transform of squares of factorial numbers.
  • A089043 (program): a(n) = n!^2 + (-1)^n.
  • A089045 (program): a(n) = a(n-1) + (-1)^floor(n/2)*a(floor(n/2)) with a(1) = 1.
  • A089054 (program): Solution to the non-squashing boxes problem (version 1).
  • A089055 (program): Solution to the non-squashing boxes problem (version 2).
  • A089057 (program): n! divided by prime whose index is the integer part of log(n).
  • A089061 (program): a(0) = 5, a(1) = 7; for n>1, a(n) = a(n-1)+a(n-2)-(2n-2).
  • A089064 (program): Expansion of e.g.f. log(1-log(1-x)).
  • A089067 (program): a(n) = 2*a(n-1) + (-1)^n*a(floor(n/2)); a(1)=1.
  • A089068 (program): a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.
  • A089071 (program): Number of liberties a big eye of size n gives in the game of Go.
  • A089072 (program): Triangle read by rows: T(n,k) = k^n, n >= 1, 1 <= k <= n.
  • A089073 (program): Number of symmetric non-crossing connected graphs on n equidistant nodes on a circle.
  • A089074 (program): Expansion of x*(1 + x + x^2)/(1 - 2*x + x^5).
  • A089079 (program): Numbers n such that 7*n - 23 is prime.
  • A089080 (program): Sequence is S(infinity) where S(1)={1,2} and S(n)=S(n-1)S’(n-1), where S’(k) is obtained from S(k) by replacing the single 1 with the least integer not occurring in S(k).
  • A089081 (program): 26th powers: a(n) = n^26.
  • A089083 (program): T(n,k) = (floor(k*n/2) * ceiling(k*n/2))^2, triangular array read by rows, 1 <= k <= n.
  • A089086 (program): Greatest common divisor of n^2-5 and n^2+5.
  • A089087 (program): Triangular array of coefficients multiplied by n! of polynomials in e. These give the expected number of trials needed for the sum of uniform random variables from the interval [0,1] to exceed n+1.
  • A089089 (program): Convoluted convolved Fibonacci numbers G_j^(2).
  • A089090 (program): a(n) is the smallest composite number coprime to n.
  • A089091 (program): a(n) is the smallest composite number coprime to n and n+1.
  • A089092 (program): Convoluted convolved Fibonacci numbers G_j^(5).
  • A089094 (program): Convoluted convolved Fibonacci numbers G_j^(7).
  • A089098 (program): Sign twisted convoluted convolved Fibonacci numbers H_j^(2).
  • A089101 (program): a(n) = (n - 4 + prime(n) mod 9) mod 10.
  • A089103 (program): a(n) = Mod[n+Prime[n],10]
  • A089105 (program): Values taken by least witness function W(n).
  • A089108 (program): Convoluted convolved Fibonacci numbers G_4^(r).
  • A089109 (program): Convoluted convolved Fibonacci numbers G_5^(r).
  • A089110 (program): Sign twisted convoluted convolved Fibonacci numbers H_5^(r).
  • A089111 (program): Convoluted convolved Fibonacci numbers G_6^(r).
  • A089115 (program): Convoluted convolved Fibonacci numbers G_8^(r).
  • A089118 (program): Nonnegative numbers in (3*A005836 - 1) [A005836 are the numbers with base representation containing no 2].
  • A089119 (program): Complement of ((3*A005836) union (3*A005836 - 1) union (3*A005836 - 2)).
  • A089120 (program): Smallest prime factor of n^2 + 1.
  • A089123 (program): Smallest prime factor of numbers of the form A^2 + 3.
  • A089124 (program): Largest prime factor of numbers of the form A^2 + 3.
  • A089125 (program): a(n+2) = a(n+1) + F(n+1)*a(n), where F = Fibonacci number (A000045) and a(0) = a(1) = 1.
  • A089126 (program): a(n+2) = F(n+1)*a(n+1) + F(n)*a(n) where F(n) = Fibonacci number (A000045), a(0) = a(1) = 1.
  • A089127 (program): a(n+2) = F(n)*a(n+1) + F(n+1)*a(n) where F(n) = Fibonacci number (A000045) and a(0) = a(1) = 1.
  • A089128 (program): a(n) = gcd(6,n).
  • A089129 (program): Greatest common divisor of n^2 - 7 and n^2 + 7.
  • A089134 (program): The matrix sequence made by the lowest fifth power Pisot that is similar to the 5 Bonacci ( Pentafibonacci ).
  • A089138 (program): a(n) = (3^(2*n))*(integral_{x=0 to 1} (1+x^3)^n dx)/(integral_{x=0 to 1} (1-x^3)^n dx).
  • A089143 (program): a(n) = 9*2^n - 6.
  • A089145 (program): Greatest common divisor of n^2-3 and n^2+3.
  • A089146 (program): Greatest common divisor of n^2 - 4 and n^2 + 4.
  • A089148 (program): Expansion of e.g.f.: 1/(exp(x) - x).
  • A089151 (program): Primes p such that 6*p - 7 and 6*p - 5 are twin primes.
  • A089154 (program): A palindromic matrix version of the alternating matrix sequence at the X^3 level 3 X 3: BAAB.
  • A089155 (program): a(n) = (2*n)!*(Integral_{x=0..sqrt(2/3)} 1/(1-x^2)^(n+1/2) dx)/((n!*2^n)*sqrt(2)).
  • A089156 (program): a(n) = A069722(n+1)^2.
  • A089160 (program): Numbers k such that 30*k + 11 and 30*k + 13 are twin primes.
  • A089161 (program): Numbers k such that 30*k + 17 and 30*k + 19 are twin primes.
  • A089164 (program): Number of steps in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0).
  • A089165 (program): Partial sums of the central Delannoy numbers (A001850).
  • A089166 (program): Number of primes between squares of successive odd numbers.
  • A089171 (program): Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).
  • A089175 (program): a(n) = Floor[Sqrt[Prime[n]*PrimePi[n]]]
  • A089181 (program): (1,3) entry of powers of the orthogonal design shown in A090592.
  • A089182 (program): Prime digit palindromes 2,…,23577532 continued by adding 10^(n-k) and 10^(k-1) times prime(k).
  • A089186 (program): Decreases from 9 * 10^k down to 1, restarting at 9 * 10^(k+1).
  • A089189 (program): Primes p such that p-1 is cubefree.
  • A089190 (program): a(n) = Floor[(Prime[n]+PrimePi[n])/2]
  • A089191 (program): Primes p such that p+1 is cubefree.
  • A089192 (program): Numbers n such that 2n - 7 is a prime.
  • A089193 (program): Odd numbers n such that 2*n-7 is a prime of the form 4*k+3.
  • A089196 (program): Floor(n / (smallest prime factor of n+1)).
  • A089205 (program): a(n) = n^n * (n-1).
  • A089207 (program): a(n) = 4n^3 + 2n^2.
  • A089211 (program): 10*n^n+(n-1).
  • A089214 (program): Let u(1)=0, u(2)=1; for k>2, u(k)= A010060(k)*u(k-1) + u(k-2) (mod 2); then a(n)=4n-b(n) where sequence (b(k)) gives values such that u(b(k))=0.
  • A089215 (program): Thue-Morse sequence on the integers.
  • A089217 (program): n-2 is a prime of the form 4*k+3.
  • A089224 (program): In binary representation: number of zeros of number of zeros of n.
  • A089225 (program): Triangle T(n,k) read by rows, defined by T(n,k) = (n-k)*T(n-1,k)+Sum(k=1..n, T(n-1,k)); T(1,1) = 1, T(1,k)= 0 if k >1.
  • A089227 (program): Numbers k such that 1 + k*ds(k) is prime, where ds(k) is the sum of digits of k.
  • A089229 (program): Neither primes nor square numbers.
  • A089231 (program): Triangular array A066667 or A008297 unsigned and transposed.
  • A089233 (program): Number of coprime pairs of divisors > 1 of n.
  • A089237 (program): List of primes and squares.
  • A089238 (program): Numbers n such that 3*n^2/2 - 1 is a prime.
  • A089241 (program): Even numbers k such that k/2 - 1 is prime.
  • A089242 (program): Sequence is S(infinity), where S(1) = 1, S(m+1) = concatenation S(m), a(m)+1, S(m) and a(m) is the m-th term of S(m). a(m) is also the m-th term of the sequence.
  • A089249 (program): Triangular array read by rows illustrating the connection between A000522 and A008292.
  • A089250 (program): Add 2 (mod 10) to each decimal digit of Pi.
  • A089252 (program): a(n) = ((2*n-1)!!/sqrt(3))*(integral_{x=0..sqrt(3/4)} 1/(1-x^2)^(n+1/2) dx).
  • A089253 (program): Numbers n such that 2n - 5 is a prime.
  • A089255 (program): Odd numbers n such that 2*n-5 is a prime.
  • A089257 (program): Even numbers n such that 2n-5 is a prime of the form 4k+3.
  • A089258 (program): Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.
  • A089262 (program): a(n) = 2^floor(log_2(n)) - 2^floor(log_2(n*2/3)).
  • A089263 (program): First differences of A080791.
  • A089265 (program): a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.
  • A089266 (program): Rational knots of determinant 2n+1, counting chiral pairs twice.
  • A089268 (program): Odd semiprimes m such that m-2 is composite.
  • A089269 (program): Squarefree numbers congruent to 1 or 2 mod 4.
  • A089270 (program): Positive numbers represented by the integer binary quadratic form x^2 + x*y - y^2 with x and y relatively prime.
  • A089271 (program): Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).
  • A089279 (program): a(n) = 2 + sum(k=1 to n) [(-1)^k A001511(k)].
  • A089280 (program): Tower of Hanoi game: a(n) is the number of pegs occupied by already-moved disks after move #n.
  • A089290 (program): Digital root of floor(Pi*10^n), Pi=3.14….
  • A089293 (program): Sum of digits in the mixed-base enumeration system n=…d(4)d(3)d(2)d(1), where the digits satisfy 0<=d(i)<=1 if i is odd, 0<=d(i)<=2 if i is even.
  • A089303 (program): a(n) = floor( (10^n - 1) / (9*n) ).
  • A089304 (program): Sum of all digits in all even numbers from 0 to 222…2 (with n 2’s).
  • A089306 (program): Smallest prime of the form n + (n+1)+ (n+2)+…+(n+k), or 0 if no such prime exists.
  • A089309 (program): Write n in binary; a(n) = length of the rightmost run of 1’s.
  • A089310 (program): Write n in binary; a(n) = number of 1’s in second block of 1’s from right.
  • A089311 (program): Write n in binary; a(n) = number of 0’s in rightmost block of zeros, after dropping any trailing 0’s.
  • A089312 (program): Write n in binary; a(n) = number represented by rightmost block of 1’s.
  • A089314 (program): Sum of all digits in all even numbers from 0 to 444…4 (with n 4’s).
  • A089341 (program): Numbers n with spf(n) < gpf(n) < 2*spf(n), where spf=A020639, gpf=A006530.
  • A089342 (program): Numerator of sqrt(2) * Integral_{x=0..sqrt(1/3)} 1/(1-x^2)^(n+3/2) dx.
  • A089343 (program): Sum of all digits in all even numbers from 0 to 6(10^(k+1)-1)/9 (with (k+1) 6’s).
  • A089348 (program): Primes of the form smallest multiple of n followed by a 1.
  • A089350 (program): Sum of all digits in all even numbers from 0 to 8(10^(k+1)-1)/9 (with (k+1) 8’s).
  • A089352 (program): Numbers that are divisible by the sum of their distinct prime factors (A008472).
  • A089354 (program): Number of generalized {(1,2),(1,-1)}-Dyck paths of length 3n with no peaks at level 2.
  • A089357 (program): a(n) = 2^(6*n).
  • A089358 (program): Numbers n such that n^2 - 3n + 3 is prime.
  • A089359 (program): Primes which can be partitioned into distinct factorials. 0! and 1! are not considered distinct.
  • A089361 (program): Numbers of pairs (i, j), i, j > 1, such that i^j <= n.
  • A089362 (program): Numbers n such that n^2 - 5n + 5 is prime.
  • A089368 (program): Least k such that 2*pi(n) = pi(n+k), where pi(n) = number of primes up to n. (The number of primes between 1 to n is the same as the number of primes between n+1 and n+k.
  • A089371 (program): Number of divisors of n that do not exceed the abundance of n.
  • A089372 (program): Number of Motzkin paths of length n with no peaks at level 1.
  • A089376 (program): Primes of the form k^2 - 7*k + 7.
  • A089382 (program): Total number of triangles in all the dissections of a convex (n+3)-gon by nonintersecting diagonals.
  • A089383 (program): Number of peaks at even level in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the axis) from (0,0) to (2n+4,0).
  • A089384 (program): Greatest squarefree proper divisor of n, a(1) = 1.
  • A089387 (program): Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no UD, UHD, UHHD, UHHHD, … starting at level zero.
  • A089389 (program): Sum of the smallest and the largest nontrivial divisor of n or 0 if n is 1 or a prime.
  • A089398 (program): a(n) = n-th column sum of binary digits of k*2^(k-1), where summation is over k>=1, without carrying between columns.
  • A089400 (program): a(n) = m - A089398(2^m - n) for m>=n.
  • A089401 (program): a(n) = m - A089398(2^m + n) for m>=n.
  • A089402 (program): Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089864.
  • A089408 (program): Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089864.
  • A089410 (program): Least common multiple of all cycle sizes (also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A074679/A074680.
  • A089418 (program): Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A082333/A082334.
  • A089422 (program): Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A082335/A082336 (and also of A082349/A082350, to be proved).
  • A089423 (program): Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A082335/A082336 (and also of A082349/A082350, to be proved).
  • A089425 (program): Least common multiple of all cycle sizes (and also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A082351/A082352.
  • A089431 (program): Even-indexed terms of A089423.
  • A089432 (program): Numbers n such that n concatenated with floor(n/2) is prime.
  • A089433 (program): Number of noncrossing connected graphs on n nodes having exactly two interior faces.
  • A089436 (program): Number of non-crossing connected graphs on n nodes on a circle in which a fixed (distinguished) node has degree one.
  • A089438 (program): Primes p such that 6p+11 is also a prime.
  • A089439 (program): 6p+13 and p are primes.
  • A089440 (program): 14*p+13 and p are primes.
  • A089441 (program): Primes p such that 16*p+17 is a prime.
  • A089442 (program): Primes p such that (p-11)/10 is also a prime.
  • A089443 (program): Primes p such that 12*p + 13 is prime.
  • A089451 (program): a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).
  • A089460 (program): Triangle, read by rows, of coefficients for the second iteration of the hyperbinomial transform.
  • A089461 (program): Hyperbinomial transform of A088957. Also the row sums of triangle A089460, which lists the coefficients for the second hyperbinomial transform.
  • A089462 (program): 2nd hyperbinomial transform of A001858.
  • A089463 (program): Triangle, read by rows, of coefficients for the third iteration of the hyperbinomial transform.
  • A089464 (program): Hyperbinomial transform of A089461. Also the row sums of triangle A089463, which lists the coefficients for the third hyperbinomial transform.
  • A089465 (program): 3rd hyperbinomial transform of A001858; also the hyperbinomial transform of A089462.
  • A089466 (program): Inverse hyperbinomial transform of A089467.
  • A089467 (program): Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.
  • A089468 (program): Hyperbinomial transform of A089467 and also the 2nd hyperbinomial transform of A089466.
  • A089491 (program): Decimal expansion of Buffon’s constant 3/Pi.
  • A089495 (program): a(n) = mu(prime(n)+1), where mu is the Moebius function.
  • A089499 (program): a(0)=0; a(1)=1; a(2n) = 4*Sum_{k=0..n} a(2k-1); a(2n+1) = a(2n) + a(2n-1).
  • A089503 (program): Triangle of numbers used for basis change between certain falling factorials.
  • A089507 (program): Second column of triangle A089504 and second column of array A078741 divided by 18.
  • A089508 (program): Solution to a binomial problem together with companion sequence A081016(n-1).
  • A089510 (program): A periodic sequence with period length 30.
  • A089512 (program): Denominators used in the computation of the column sequences of array A078739 ((2,2)-Stirling2).
  • A089513 (program): Third column of triangle A089504.
  • A089514 (program): Fourth column of triangle A089504.
  • A089530 (program): A023204 indexed by A000040.
  • A089531 (program): Primes p such that (p-3)/2 is also prime.
  • A089532 (program): A089531 indexed by A000040.
  • A089533 (program): a(n)=(A089348(n)-1)/10n.
  • A089534 (program): A089348 indexed by A000040.
  • A089559 (program): Nonnegative numbers n such that 2*n + 15 is prime.
  • A089574 (program): Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).
  • A089581 (program): a(n) = prime(2*n-1)*prime(2*n).
  • A089582 (program): From Gilbreath’s conjecture.
  • A089591 (program): “Lazy binary” representation of n. Also called redundant binary representation of n.
  • A089593 (program): Numbers k such that k^2 + 2k + 2 is prime.
  • A089594 (program): Alternating sum of squares to n.
  • A089598 (program): G.f.: (1+x^2+x^3)/(1-x^3)^2.
  • A089600 (program): Another lazy binary representation of n: similar to A089591 except that the single carry is performed before the increment instead of after.
  • A089601 (program): Interleaving of A089591 and A089600.
  • A089607 (program): a(n)=((-1)^(n+1)*A002425(n)) modulo 4.
  • A089608 (program): a(n) = ((-1)^(n+1)*A002425(n)) modulo 6.
  • A089610 (program): Number of primes between n^2 and (n+1/2)^2.
  • A089611 (program): Number of primes between n^2 and (n+1/3)^2.
  • A089612 (program): a(n) = ((-1)^(n+1)*A002425(n)) modulo 5.
  • A089614 (program): Number of primes between n^2 and (n+1/4)^2.
  • A089615 (program): Number of primes between n^2 and (n+1/5)^2.
  • A089616 (program): Number of primes between n^2 and (n+1/6)^2.
  • A089617 (program): Number of primes between n^2 and (n+1/7)^2.
  • A089619 (program): Greatest prime factor of n^2 + (n+1)^2.
  • A089620 (program): n^3 + n-th prime.
  • A089621 (program): n^4 + n-th prime.
  • A089622 (program): a(n) = n^n + n-th prime.
  • A089623 (program): Numbers n such that n^2 + 2n - 1 is prime.
  • A089625 (program): Replace 2^k in binary expansion of n with (k+1)-st prime.
  • A089627 (program): T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.
  • A089631 (program): a(n) = (Product_{p is a prime factor of n} p)) mod (Product_{p is a prime factor of n} p-1).
  • A089633 (program): Numbers having no more than one 0 in their binary representation.
  • A089638 (program): Numerator of (5/2)*Sum_{i=1..n} (-1)^(i-1)/(i^3*C(2*i,i)).
  • A089639 (program): Denominator of (5/2)*Sum_{i=1..n} (-1)^(i-1)/(i^3*C(2*i,i)).
  • A089640 (program): Enumeration of partial sums of 1+[1,2]+[2,3]+[1,2]+[2,3]+…
  • A089643 (program): 3^a(n) divides C(3n,n); 3-adic valuation of A005809.
  • A089644 (program): Numbers k such that 7 divides the numerator of B(2*k) where B(k) = the k-th Bernoulli number.
  • A089646 (program): a(n) = Sum(a(floor(n/p)): p prime and p<=n); a(1) = 1.
  • A089648 (program): Numbers whose numbers of zeros and ones in binary representation differ at most by 1.
  • A089649 (program): a(1)=1, a(2)=2, a(n) = a(n-1) + a(floor((n+1)/3)).
  • A089650 (program): a(n) = A089649(n) mod 3.
  • A089651 (program): Partial sums of the sequence : a(1)=1, a(1), a(1), a(1), a(1), a(2), a(2), a(2), a(2), a(3), a(3), a(3), a(3), a(4), …each terms (not a(1)) repeated 4 times.
  • A089652 (program): Partial sums, modulo 4, of the sequence: a(1)=1, a(1), a(1), a(1), a(1), a(2), a(2), a(2), a(2), a(3), a(3), a(3), a(3), a(4), … each term (not a(1)) repeated 4 times.
  • A089655 (program): a(1)=1 and for n>=2 a(n) is the denominator of A(n) (see comment for A(n) definition).
  • A089658 (program): a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
  • A089659 (program): a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
  • A089660 (program): a(n) = S1(n,3), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
  • A089661 (program): a(n) = S1(n,4), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).
  • A089662 (program): a(n) = S1(n,5), where S1(n,t) = Sum_{k=0..n} k^t * Sum_{j=0..k} binomial(n,j).
  • A089663 (program): a(n) = S1(n, 6), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).
  • A089664 (program): a(n) = S2(n,1), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
  • A089665 (program): a(n) = S2(n,2), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
  • A089666 (program): a(n) = S2(n,3), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
  • A089667 (program): a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
  • A089668 (program): a(n) = S2(n,5), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
  • A089669 (program): a(n) = S3(n,1), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
  • A089670 (program): a(n) = S3(n,2), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
  • A089671 (program): a(n) = S3(n,3), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
  • A089672 (program): a(n) = S3(n,4), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.
  • A089676 (program): a(n) is the maximal size of a set S of points in {0,1}^n in real n-dimensional Euclidean space such that every angle determined by three points in S is acute.
  • A089677 (program): Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.
  • A089681 (program): Numbers n such that 3n^2 - 1 is prime.
  • A089682 (program): Primes of the form 3*m^2 - 1.
  • A089683 (program): a(n) = 3^(4n).
  • A089692 (program): a(n) = phi(2n)/2^omega(n).
  • A089693 (program): Numbers n such that phi(n) = 2^bigomega(n).
  • A089708 (program): a(1) = 1, a(2) = 2, a(n) = a(n-1) + d where d is the sum of the absolute differences between all pairs of previous terms.
  • A089709 (program): a(1) = 1, a(2) = 2; for n>2, a(n) = sum_{r=1..n} {sum of all previous terms taken r at a time}.
  • A089716 (program): Primes that are both congruent to 1 mod 10 and congruent to 1 or 2 mod 9.
  • A089717 (program): Triangular numbers with palindromic indices.
  • A089719 (program): Pseudofactor sets of primes ending in 1: 9 greater than 3.
  • A089720 (program): Primes == 1 or 11 (mod 70).
  • A089721 (program): Primes ending in 1 such that floor(p/7) ends in a digit > 3.
  • A089723 (program): a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.
  • A089724 (program): Numbers p satisfying the following conditions: p is a prime of form 10k+9 and the function f[p,7,10] applied to p is not greater than 5, where f[p,7,10]=10*fractionalpart[n/70]=1*.((n/70)-Floor[n/70]).
  • A089726 (program): Smallest prime of the form nk+1, k > n.
  • A089727 (program): Largest prime of the form n*k+1, k <= n.
  • A089732 (program): Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n having k (1,1) steps (can be easily translated into RNA secondary structure terminology). Except for row 0, row n has ceiling(n/2) entries.
  • A089735 (program): Self-convolution of A004148 (the RNA secondary structure numbers) with itself.
  • A089739 (program): Let b(m)= base ten expansion of prime(n); then a(n)=Sum[Mod(10-b(l),10)*10^l,{l,1,10}]
  • A089742 (program): Number of subwords UHH…HD in all peakless Motzkin paths of length n+3, where U=(1,1), D=(1,-1) and H=(1,0).
  • A089745 (program): a(n) = direuler(p=2,n,1/(1-X)/(1-p*n*X))[n].
  • A089747 (program): Numbers n such that n^2 - 2n + 5 is prime.
  • A089756 (program): a(1)=1 and a(i+1)=a(i)+9 if a(i)<=35, and a(i+1)=a(i)-35 if a(i)>35.
  • A089759 (program): Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.
  • A089767 (program): Squares which when concatenated with a 1 gives prime.
  • A089772 (program): a(n) = Lucas(11*n).
  • A089775 (program): Lucas numbers L(12n).
  • A089781 (program): Successive coprime numbers with distinct successive differences: gcd(a(k+1),a(k)) = gcd(a(m+1),a(m)) = 1 and a(k+1)-a(k) = a(m+1)-a(m) <==> m=k.
  • A089792 (program): a(n) = n-(exponent of highest power of 3 dividing n!).
  • A089798 (program): Expansion of Jacobi theta function theta_4(q^2).
  • A089799 (program): Expansion of Jacobi theta function theta_2(q^(1/2))/q^(1/8).
  • A089800 (program): Expansion of Jacobi theta function theta_2(q)/q^(1/4).
  • A089801 (program): a(n) = 0 unless n = 3j^2+2j or 3j^2+4j+1 for some j>=0, in which case a(n) = 1.
  • A089802 (program): Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.
  • A089803 (program): Expansion of Jacobi theta function theta_4(q^5).
  • A089805 (program): Expansion of Jacobi theta function (theta_4(q^6) - theta_4(q^(2/3)))/2/q^(2/3).
  • A089806 (program): Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).
  • A089807 (program): Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.
  • A089808 (program): a(n) = floor(1/((n*r) mod 1)), where r = phi^(-2) = (3 - sqrt(5))/2.
  • A089809 (program): Complement of A078588.
  • A089810 (program): Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.
  • A089811 (program): Expansion of Jacobi theta function (3*theta_4(q^18) - theta_4(q^2))/2.
  • A089812 (program): Expansion of Jacobi theta function q^(-1/8) * (theta_2(q^(1/2)) - 3 * theta_2(q^(9/2))) / 2 in powers of q.
  • A089813 (program): Expansion of Jacobi theta function (theta_2(q) - 3*theta_2(q^9))/(2 q^(1/4)) in powers of q.
  • A089815 (program): a(n) = floor((n+2)^(n+2)/((n+2)^2-1)).
  • A089816 (program): a(n) = floor((n+3)^(n+2)/((n+3)^2-1)).
  • A089817 (program): a(n) = 5*a(n-1) - a(n-2) + 1 with a(0)=1, a(1)=6.
  • A089819 (program): Number of subsets of {1,2,…,n} containing no primes.
  • A089820 (program): Number of subsets of {1,..,n} containing at least one prime.
  • A089821 (program): Number of subsets of {1,.., n} containing exactly one prime.
  • A089822 (program): Number of subsets of {1,.., n} containing exactly two primes.
  • A089826 (program): Decimal expansion of real root of 2*x^3+x^2-1.
  • A089830 (program): Expansion of (1-3*x+6*x^2-5*x^3+3*x^4-x^5)/(1-x)^6.
  • A089833 (program): a(n) = A000108(n)*(A000142(n+1)-1).
  • A089835 (program): a(n) = (A000108(n)^2)*(n+1)!.
  • A089836 (program): INVERT transform of A089835.
  • A089845 (program): Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A089865/A089866.
  • A089849 (program): Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A069772.
  • A089880 (program): Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A069772.
  • A089883 (program): Expansion of (1-4*x+6*x^2-3*x^3)/((1-3*x)*(1-2*x)*(1-3*x+x^2)).
  • A089885 (program): Triangle A046899 read mod 2.
  • A089887 (program): Number of subsets of {1,.., n} containing no squares.
  • A089888 (program): Number of subsets of {1,.., n} containing at least one square.
  • A089889 (program): Number of subsets of {1,.., n} containing exactly one square.
  • A089890 (program): Number of subsets of {1,.., n} containing exactly two squares.
  • A089893 (program): a(n) = (A001317(2n)-1)/4.
  • A089898 (program): Product of (digits of n each incremented by 1).
  • A089899 (program): Square array, read by antidiagonals, where the n-th row is the binomial transform of (1+x+x^2)^n, starting with n=0.
  • A089900 (program): Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,…}.
  • A089901 (program): Main diagonal of A089900, also the inverse hyperbinomial of A000312 (offset 1).
  • A089902 (program): Antidiagonal sums of array A089900.
  • A089903 (program): Sum of digits of numbers between 0 and (1/9)*(10^n-1).
  • A089904 (program): Sum of digits of numbers between 0 and (2/9)*(10^n-1).
  • A089905 (program): Sum of digits of numbers between 0 and (3/9)*(10^n-1).
  • A089906 (program): Sum of digits of numbers between 0 and (4/9)*(10^n-1).
  • A089907 (program): Sum of digits of numbers between 0 and (6/9)*(10^n-1).
  • A089908 (program): Sum of digits of numbers between 0 and (7/9)*(10^n-1).
  • A089909 (program): Sum of digits of numbers between 0 and (8/9)*(10^n-1).
  • A089910 (program): Indices n at which blocks (1;1) occur in infinite Fibonacci word, i.e., such that A005614(n) = A005614(n+1) = 1.
  • A089911 (program): a(n) = Fibonacci(n) mod 12.
  • A089913 (program): Table T(n,k) = lcm(n,k)/gcd(n,k) = n*k/gcd(n,k)^2 read by antidiagonals (n >= 1, k >= 1).
  • A089914 (program): a(n) = 3^n *n! *L_{n}^{-1/3}(-1), where L_n^{alpha}(x) are generalized Laguerre polynomials.
  • A089915 (program): Special values of generalized Laguerre polynomials L_n^(alpha)(x): a(n) = 4^n *n! *L_n^(-1/4)(-1).
  • A089917 (program): a(n) = 6^n *n! *L_n^{-1/6}(-1), where L_n^(alpha)(x) are generalized Laguerre polynomials.
  • A089918 (program): (n+1)*a(n) equals the (n+1)-th term of the n-th binomial transform of this sequence.
  • A089926 (program): a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.
  • A089927 (program): Expansion of 1/((1-x^2)(1-5x+x^2)).
  • A089928 (program): a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=4, a(4)=10.
  • A089929 (program): Algebraic degree of cot(Pi/n).
  • A089931 (program): a(n) = 3*a(n-1) + 3*a(n-3) + a(n-4).
  • A089935 (program): Order of recurrence generating row (or column) n of A089934.
  • A089940 (program): Triangle read by rows: T(n,k)=binomial(n+k,floor((n-k)/2))
  • A089941 (program): Row sums of triangle A089940.
  • A089942 (program): Inverse binomial matrix applied to A039599.
  • A089944 (program): Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.
  • A089945 (program): Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.
  • A089946 (program): Secondary diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.
  • A089950 (program): Partial sums of A001652.
  • A089953 (program): Numbers n such that 3*n+7 is prime.
  • A089960 (program): Positions in A089959 where 4 occurs.
  • A089962 (program): Triangle, read by rows, that equals the matrix inverse of A071207 when treated as a lower triangular matrix.
  • A089965 (program): Both n + 1 and n/2 + 1 are primes.
  • A089968 (program): Decimal primes whose decimal representation in base 16 is also prime.
  • A089974 (program): (n+1)-st term of the n-th binomial transform of this sequence equals 1 for all n>=0.
  • A089977 (program): Expansion of 1/((1-2*x)*(1+x+2*x^2)).
  • A089978 (program): Expansion of 1/(1-3x-3x^3).
  • A089979 (program): Expansion of 1/(1-4x-4x^3).
  • A089982 (program): Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.
  • A089983 (program): 1, 1, 1, 1, … a, b, c, d, ab-cd, …
  • A089984 (program): 1, 1, 1, 1, … a, b, c, d, ac-bd, …
  • A089985 (program): a(n) = A089709(n+1)/A089709(n).
  • A089986 (program): Numbers n such that 4n + 7 is prime.
  • A089992 (program): Second prime divisor of numbers that are not powers of primes (A024619).
  • A089993 (program): Penultimate prime divisor of numbers that are not powers of primes (A024619).
  • A089994 (program): Number of primes between factors of n-th semiprime.
  • A089995 (program): Products of pairs of distinct, non-consecutive primes.
  • A090000 (program): Length of longest contiguous block of 1’s in binary expansion of n-th prime.
  • A090001 (program): Length of longest contiguous block of 1’s in binary expansion of n^2.
  • A090002 (program): Length of longest contiguous block of 1’s in binary expansion of n-th triangular number.
  • A090003 (program): Length of longest contiguous block of 1’s in binary expansion of n^3.
  • A090010 (program): Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
  • A090012 (program): Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
  • A090013 (program): Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
  • A090014 (program): Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
  • A090015 (program): Permanent of (0,1)-matrix of size n X (n+d) with d=5 and n-1 zeros not on a line.
  • A090016 (program): Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
  • A090017 (program): a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1.
  • A090018 (program): a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6.
  • A090019 (program): a(n) = (3*10^n + 2*0^n)/5.
  • A090020 (program): Number of distinct lines through the origin in the n-dimensional lattice of side length 4.
  • A090021 (program): Number of distinct lines through the origin in the n-dimensional lattice of side length 5.
  • A090022 (program): Number of distinct lines through the origin in the n-dimensional lattice of side length 6.
  • A090023 (program): Number of distinct lines through the origin in the n-dimensional lattice of side length 7.
  • A090024 (program): Number of distinct lines through the origin in the n-dimensional lattice of side length 8.
  • A090040 (program): (3*6^n + 2^n)/4.
  • A090041 (program): a(n) = 10*a(n-1) - 20*a(n-2), a(0)=1, a(1)=6.
  • A090042 (program): a(n) = 2*a(n-1) + 11*a(n-2) for n > 1, a(0) = a(1) = 1.
  • A090044 (program): Triangle read by rows: T(n,k) = A083093 with 1’s and 2’s interchanged.
  • A090046 (program): Length of longest contiguous block of 0’s in binary expansion of n-th prime.
  • A090047 (program): Length of longest contiguous block of 0’s in binary expansion of n^2.
  • A090048 (program): Length of longest contiguous block of 0’s in binary expansion of n-th triangular number.
  • A090049 (program): Length of longest contiguous block of 0’s in binary expansion of n^3.
  • A090075 (program): (Presumed) number of palindromes in the Reverse and Add! trajectory of 10^n.
  • A090076 (program): a(n) = prime(n)*prime(n+2).
  • A090077 (program): In binary expansion of n: reduce contiguous blocks of 1’s to 1.
  • A090078 (program): In binary expansion of n, reduce contiguous blocks of 0’s to 0.
  • A090079 (program): In binary expansion of n: reduce contiguous blocks of 0’s to 0 and contiguous blocks of 1’s to 1.
  • A090080 (program): In binary expansion of n-th prime: reduce contiguous blocks of 0’s to 0 and contiguous blocks of 1’s to 1.
  • A090090 (program): a(n) = prime(n)*prime(n+3).
  • A090092 (program): a(n) is the smallest composite number coprime to n, n+1 and n+2.
  • A090093 (program): a(n) is the smallest composite number coprime to n, n+1, n+2, n+3.
  • A090094 (program): a(n) is the smallest composite number coprime to n, n+1, n+2, n+3 and n+4.
  • A090095 (program): a(n) is the smallest composite number coprime to n, n+1, n+2, n+3, n+4 and n+5.
  • A090097 (program): Bases n such that the smallest prime-power-pseudoprime to base n is 9.
  • A090112 (program): a(n)=pi(n)*[pi(n)-pi(n-2)] where pi(n)=A000720(n).
  • A090113 (program): a(n)=pi(n)*[pi(n)-pi(n-3)] where pi(n)=A000720(n).
  • A090114 (program): a(n) = Product_{j=1..n} (prime(n)-j).
  • A090115 (program): a(n)=Product[p(n)-j, j=1..n]/n!=A090114(n)/n!.
  • A090120 (program): Numbers x such that nextprime(x^2) - prevprime(x^2) = 4.
  • A090129 (program): Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.
  • A090131 (program): Expansion of (1+x)/(1 - 2*x + 2*x^2).
  • A090132 (program): Expansion of (1+2*x)/(1+2*x+2*x^2).
  • A090133 (program): Expansion of (1+4x)/(1+4x+5x^2).
  • A090134 (program): a(n) = (6*n!/(n+5)) *binomial(n+5,n-1)* 6F6(-n+1, 1/5*n+1, 1/5*n+9/5, 1/5*n+8/5, 1/5*n+7/5, 1/5*n+6/5; 7/6, 4/3, 3/2, 5/3, 11/6, 2; -3125/46656), where 6F6(;;) is the generalized hypergeometric series.
  • A090137 (program): Numerator of the probability that the sum of n uniform picks on [0,1] is first greater than 2 (i.e., the sum of n-1 is not).
  • A090138 (program): Denominator of the probability that the sum of n uniform picks on [0,1] is first greater than 2 (i.e., the sum of n-1 is not).
  • A090139 (program): a(n) = 10*a(n-1) - 20*a(n-2), a(0)=1,a(1)=5.
  • A090142 (program): Decimal expansion of e^2 - e.
  • A090145 (program): Even-indexed terms of the binomial transform equal 1 and the odd-indexed terms of the second binomial transform equal 1.
  • A090158 (program): Odd-indexed terms of the binomial transform equals 1 and the even-indexed terms of the second binomial transform equals 1.
  • A090161 (program): A023219 indexed by A000040.
  • A090168 (program): Floor( 3n/2 ) - floor( 2n/3 ).
  • A090169 (program): a(n) = floor( 3*n/2 ) + floor( 2*n/3 ).
  • A090176 (program): G.f.: (1+x^9)/((1-x^4)(1-x^6)(1-x^12)).
  • A090178 (program): a(1) = 2; for n > 1, a(n) = n + prime(n-1).
  • A090181 (program): Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.
  • A090183 (program): a(n) = Mod[10-Mod[Prime[n+3],10],4]-Mod[n,4]+3.
  • A090184 (program): Number of partitions of the n-th 3-smooth number into parts 2 and 3.
  • A090187 (program): Primes of the form 11*n+2.
  • A090192 (program): Carlitz-Riordan q-Catalan numbers (recurrence version) for q = -1.
  • A090193 (program): a(n) = A053838(n) + 1 modulo 3.
  • A090196 (program): Odd integers with two divisors a, b such that a < b <= 2a.
  • A090197 (program): a(n) = n^3 + 6*n^2 + 6*n + 1.
  • A090198 (program): a(n) = N(5,n), where N(5,x) is the 5th Narayana polynomial.
  • A090199 (program): a(n) = N(6,n), where N(6,x) is the 6th Narayana polynomial.
  • A090200 (program): a(n) = N(7,n), where N(7,x) is the 7th Narayana polynomial.
  • A090205 (program): n^n * (n+1)^(n+1).
  • A090211 (program): Alternating row sums of array A078739 ((2,2)-Stirling2).
  • A090223 (program): Nonnegative integers with doubled multiples of 4.
  • A090237 (program): Numerators of the partial sums of the reciprocals of the lower members of twin prime pairs.
  • A090239 (program): a(n) = A053838(n) + 2 modulo 3.
  • A090241 (program): a(n) = F(n)^F(n+1), where F is the Fibonacci sequence A000045.
  • A090242 (program): a(n) = F(n)^F(n-1), where F is the Fibonacci sequence A000045.
  • A090244 (program): a(0) = 1; a(1) = 2; a(n) = { a(n-1) + a(n-2) for n even, a(n-1) - a(n-2) for n odd }.
  • A090247 (program): a(n) = 26*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26.
  • A090248 (program): a(n) = 27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27.
  • A090249 (program): a(n) = 28a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 28.
  • A090251 (program): a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.
  • A090273 (program): Numbers with a palindromic digital product.
  • A090274 (program): Numbers with a nonzero palindromic digital product (contains no zeros).
  • A090277 (program): “Plain Bob Minimus” in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(2,1,4,3), .. which runs through all permutations of {1,2,3,4} with period 24; sequence gives number in position 1 of n-th permutation.
  • A090281 (program): “Plain Bob Minimus” in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(2,1,4,3), … which runs through all permutations of {1,2,3,4} with period 24; sequence gives position of bell 1 (the treble bell) in n-th permutation.
  • A090285 (program): Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(n,k)=0 if k>n, T(n,0) = A000108(n); T(n+1,k)= Sum_{j=0..n} T(n-j,k-1)*binomial(2j+1,j+1).
  • A090288 (program): a(n) = 2*n^2 + 6*n + 2.
  • A090294 (program): a(n) = K_3(n) = Sum_{k>=0} A090285(3,k)*2^k*binomial(n,k). a(n) = (4*n^3+30*n^2+56*n+15)/3.
  • A090295 (program): Let f(0) = 0, f(1) = 1 and for n > 1 let f(n) = (-1)*sum((-1)^(n+r)*f(r),r=0..n-2)/(n*(n-1)); sequence gives numerator of f(n).
  • A090296 (program): a(n) = K_4(n) = Sum_{k>=0} A090285(4,k)*2^k*binomial(n,k). a(n) = 2*(n^4+14*n^3+62*n^2+91*n+21)/3.
  • A090297 (program): a(n) = K_5(n) = Sum_{k>=0} A090285(5,k)*2^k*binomial(n,k). a(n) = 2*(2*n^5+45*n^4+360*n^3+1215*n^2+1528*n+315)/15.
  • A090299 (program): Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.
  • A090300 (program): a(n) = 14*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 14.
  • A090301 (program): a(n) = 15*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 15.
  • A090302 (program): Begin with n and consider numbers obtained by successively subtracting 0, 1, 2, 3, …; a(n) = largest prime that arises in the process, i.e., largest prime of the form n - T(r), where T(r) is the r-th triangular number; or 0 if no such number exists.
  • A090305 (program): a(n) = 16*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 16.
  • A090306 (program): a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.
  • A090307 (program): a(n) = 18*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 18.
  • A090308 (program): a(n) = 19*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 19.
  • A090309 (program): a(n) = 20*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 20.
  • A090310 (program): a(n) = 21*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 21.
  • A090313 (program): a(n) = 22*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 22.
  • A090314 (program): a(n) = 23*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 23.
  • A090316 (program): a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.
  • A090317 (program): Row sums of triangle in A090285.
  • A090321 (program): T(n,k) = prime(n+1) - prime(n-k+1), 1<=k<=n, triangular array read by rows.
  • A090324 (program): Second in a series of triangular arrays generating the natural numbers (cf. A079946).
  • A090326 (program): Number of rules of a context-free grammar in Chomsky normal form that generates all permutations of n symbols.
  • A090327 (program): Number of rules of a context-free grammar in Chomsky normal form that generates all permutations of n symbols.
  • A090328 (program): Number of rules of a context-free grammar in Chomsky normal form that generates all permutations of n symbols.
  • A090336 (program): Odd-indexed terms of the first binomial transform equals 1 and the even-indexed terms of the third binomial transform equals 1, with a(0)=1.
  • A090337 (program): Let b(0) = 1, b(n) = b(n-1) + (-1)^(n-1)*b(n-1)/10; sequence gives numerator of b(n).
  • A090340 (program): Difference between the sums of the prime factors, including multiplicity, of n and those of n + 1.
  • A090341 (program): Difference between the sums of the prime factors, including multiplicity, of n and those of n + 2.
  • A090342 (program): Difference between the sums of the prime factors, including multiplicity, of n and those of n + 3.
  • A090343 (program): Difference between the sums of the prime factors, including multiplicity, of n and those of n + 4.
  • A090344 (program): Number of Motzkin paths of length n with no level steps at odd level.
  • A090345 (program): Number of Motzkin paths of length n with no level steps at even level.
  • A090346 (program): Number of divisors of prime(n) + prime(n+1).
  • A090347 (program): Number of labeled trees with n nodes and even number of leaves minus number of labeled trees with n nodes and odd number of leaves.
  • A090351 (program): Satisfies A^3 = BINOMIAL(A^2).
  • A090352 (program): Satisfies A^3 = BINOMIAL(A)^2, where A = A090351^2.
  • A090353 (program): Satisfies A^4 = BINOMIAL(A^3).
  • A090355 (program): Satisfies A^4 = BINOMIAL(A)^3.
  • A090356 (program): Satisfies A^5 = BINOMIAL(A^4).
  • A090357 (program): Satisfies A^5 = BINOMIAL(A)^4; also equals A090356^4.
  • A090358 (program): Satisfies A^6 = BINOMIAL(A^5).
  • A090362 (program): Satisfies A^6 = BINOMIAL(A)^5 and also equals A090358^5.
  • A090364 (program): Convolution of this sequence with its binomial transform equals the second iteration of the binomial transform upon this sequence.
  • A090368 (program): a(1) = 1; for n > 1, smallest divisor > 1 of 2n-1.
  • A090369 (program): Smallest divisor of 2n that is > 2, or 0 if no such divisor exists.
  • A090370 (program): Least m > 3 such that gcd(n-1, m*n - 1) = m-1.
  • A090374 (program): Number of rooted planar 4-constellations with n quadrangles: rooted planar maps with bicolored faces having n black quadrangular faces and an arbitrary number of white faces of degrees multiple to 4.
  • A090381 (program): Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).
  • A090383 (program): Minimal number of vertices of polytope P_T associated with any binary tree having 2n+1 nodes.
  • A090384 (program): Maximal number of vertices of polytope P_T associated with any binary tree having 2n+1 nodes.
  • A090386 (program): Fifth diagonal (m=4) of triangle A084938; a(n) = A084938(n+4,n) = (n^4 + 18*n^3 + 131*n^2 + 426*n)/24.
  • A090387 (program): Numerator of d(n)/n, where d(n) (A000005) is the number of divisors of n.
  • A090388 (program): Decimal expansion of 1 + sqrt(3).
  • A090390 (program): Repeatedly multiply (1,0,0) by ([1,2,2],[2,1,2],[2,2,3]); sequence gives leading entry.
  • A090391 (program): a(n) = n*(n^4 + 30*n^3 + 395*n^2 + 2910*n + 11064)/120.
  • A090395 (program): Denominator of d(n)/n, where d(n) is the number of divisors of n (A000005).
  • A090396 (program): Remainder when the sum of the first n primes is divided by n.
  • A090398 (program): a(n) = Floor[(2*Pi/E)*n^2]
  • A090399 (program): Expansion of 1/(1-2x+2x^4).
  • A090400 (program): Expansion of 1/(1-3x+3x^3) in powers of x.
  • A090401 (program): Expansion of 1/(1-3x+3x^4).
  • A090405 (program): a(n) = PrimePi(n+2) - PrimePi(n).
  • A090406 (program): a(n) = PrimePi(n+3) - PrimePi(n).
  • A090407 (program): a(n) = Sum_{k = 0..n} C(4*n + 1, 4*k).
  • A090408 (program): a(n) = Sum_{k=0..n} binomial(4n+3,4k).
  • A090409 (program): 7*8^n/9+2(-1)^n/9.
  • A090411 (program): G.f.: (1-x)/(1-16x).
  • A090412 (program): A Chebyshev transform of 2^n.
  • A090413 (program): A Chebyshev transform of 3^n.
  • A090415 (program): a(n) = Floor[4*Pi*n/(E)]
  • A090417 (program): Primes of the form floor(2*Pi*n/(e*log(n))).
  • A090431 (program): Difference between sums of digits of n and n-th prime.
  • A090438 (program): Generalized Stirling2 array (4,2).
  • A090439 (program): Alternating row sums of array A090438 ((4,2)-Stirling2).
  • A090442 (program): Row sums of array A090452 (s2_{3,2}, scaled (3,2)-Stirling2).
  • A090443 (program): a(n) = (n+2)! * (n+1)! * n! / 2.
  • A090444 (program): Fifth column (m=4) of triangle A090441.
  • A090447 (program): Triangle of partial products of binomials.
  • A090448 (program): Fourth column (m=3) of triangle A090447.
  • A090449 (program): Fifth column (m=4) of triangle A090447.
  • A090450 (program): Row sums of triangle A090447.
  • A090451 (program): Alternating row sums of triangle A090447.
  • A090453 (program): Third column (m=4) of array A090452.
  • A090458 (program): Decimal expansion of (3 + sqrt(21))/2.
  • A090461 (program): Numbers k for which there exists a permutation of the numbers 1 to k such that the sum of adjacent numbers is a square.
  • A090466 (program): Regular figurative or polygonal numbers of order greater than 2.
  • A090467 (program): Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + k*n(n-1)/2 - (n-1)^2 where n >= 2 and k >= 2.
  • A090470 (program): E.g.f.: 1/((1-4*x)*sqrt(1-2*x)).
  • A090488 (program): Decimal expansion of 2 + 2*sqrt(2).
  • A090492 (program): G.f.: (1+x^10)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
  • A090498 (program): Number of divisors of all the numbers from (1/2)n(n-1)+1 to n(n+1)/2, i.e., tau(1), tau(2)+tau(3), tau(4)+tau(5)+tau(6), tau(7)+tau(8)+tau(9)+tau(10), …, where tau(j) is the number of divisors of j.
  • A090529 (program): a(n) is the smallest positive m such that n <= m!.
  • A090530 (program): Least multiple k of prime(n) such that (k-1,k+1) forms a twin prime pair, or 0 if no such number exists.
  • A090532 (program): Let f(n) = n - pi(n). Then a(n) = least number of steps such that f(f(…(n)))=1.
  • A090541 (program): a(n) = floor((Sum_{r=1..n} r)*(Sum_{r=1..n} 1/r)).
  • A090550 (program): Decimal expansion of solution to n/x = x - n for n = 5.
  • A090551 (program): Decimal expansion of sqrt(3) - Pi/2.
  • A090561 (program): Least n-th power that begins with k^(n-1) for some k > 1.
  • A090562 (program): Primes of the form 5k^2 + 5k + 1.
  • A090563 (program): Numbers k such that 5*k^2 + 5*k + 1 is prime.
  • A090568 (program): Least m such that m^n begins with k^(n-1) for some k > 1.
  • A090569 (program): The survivor w(n,2) in a modified Josephus problem, with a step of 2.
  • A090570 (program): Numbers that are congruent to {0, 1} mod 9.
  • A090582 (program): T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.
  • A090585 (program): Numerator of (Sum_{k=1..n} k) / (Product_{k=1..n} k).
  • A090586 (program): Denominator of Sum/Product of first n numbers.
  • A090588 (program): Number of labeled idempotent groupoids.
  • A090590 (program): (1,1) entry of powers of the orthogonal design shown below.
  • A090591 (program): Expansion of g.f.: 1/(1 - 2*x + 8*x^2).
  • A090592 (program): (1,1) entry of powers of the orthogonal design shown below.
  • A090597 (program): a(n) = - a(n-1) + 5(a(n-2) + a(n-3)) - 2(a(n-4) + a(n-5)) - 8(a(n-6) + a(n-7)).
  • A090598 (program): Numerator of ((integral_{x = 0..1/2} 1/(1+x^2)^(n + 1/2) dx) * sqrt(1/5)).
  • A090599 (program): Number of n-element labeled commutative groupoids with an identity.
  • A090602 (program): Number of n-element labeled groupoids with an identity.
  • A090605 (program): Numbers m such that m-th prime is of the form 60*k+1.
  • A090613 (program): Numbers k such that the k-th prime is congruent to 3 mod 7.
  • A090614 (program): Numbers n such that 14n+3 is prime.
  • A090616 (program): Exponent of highest power of 4 dividing n!.
  • A090617 (program): Exponent of highest power of 8 dividing n!.
  • A090618 (program): Highest power of 9 dividing n!.
  • A090620 (program): Highest power of 13 dividing n!.
  • A090621 (program): Exponent of highest power of 16 dividing n!.
  • A090624 (program): If n = Product(pj^ej), i.e., written in its prime factorization, then a(n) = max_j{(pj-1)*ej}.
  • A090628 (program): Square array T(n,k) (row n, column k) read by antidiagonals defined by: T(n,k) is the permanent of the n X n matrix with 1 on the diagonal and k elsewhere; T(0,k)=1.
  • A090629 (program): a(n) = abs(numerator of 2n-th Bernoulli number) modulo 3.
  • A090631 (program): Given n boxes labeled 1..n, such that box i weighs 2i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.
  • A090632 (program): Given n boxes labeled 1..n, such that box i weighs 3i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.
  • A090633 (program): Start with the sequence [1, 1/2, 1/3, …, 1/n]; form new sequence of n-1 terms by taking averages of successive terms; repeat until reach a single number F(n); a(n) = numerator of F(n).
  • A090634 (program): Start with the sequence [1, 1/2, 1/3, …, 1/n]; form new sequence of n-1 terms by taking averages of successive terms; repeat until reach a single number F(n); a(n) = denominator of F(n).
  • A090639 (program): a(0) = 0; a(2n) = 3*a(n), a(2n+1) = a(n) + 1.
  • A090640 (program): a(0) = 0; a(2n) = 4*a(n), a(2n+1) = a(n) + 1.
  • A090642 (program): Triangle read by rows: T(n,k) = binomial(n^2, k), 0 <= k <= n.
  • A090648 (program): a(n)=2*(4^n-1)/denominator(B(2n)) where B(k) denotes the k-th Bernoulli number.
  • A090650 (program): n^(n+6).
  • A090654 (program): Decimal expansion of 4 + 2*sqrt(6).
  • A090655 (program): Decimal expansion of solution to n/x = x-n for n = 9.
  • A090656 (program): Decimal expansion of 5 + sqrt(35).
  • A090657 (program): Triangle read by rows: T(n,k) = number of functions from [1,2,…,n] to [1,2,…,n] such that the image contains exactly k elements (0<=k<=n).
  • A090658 (program): Numbers n such that n-1 is a prime of the form 4k+3.
  • A090663 (program): Second term in continued fraction for the n-th root of n.
  • A090670 (program): Odd numbers k such that 2*k-3 is a prime of the form 4*j+3.
  • A090671 (program): Decreases from 10^k - 1 down to 1, restarting at 10^(k+1) - 1, for k >= 1.
  • A090672 (program): a(n) = (n^2-1)*n!/3.
  • A090677 (program): Number of ways to partition n into sums of squares of primes.
  • A090678 (program): a(n) = A088567(n) mod 2.
  • A090679 (program): Integer part of the hypotenuse of a right triangle with twin prime legs.
  • A090681 (program): Expansion of (sec(x/2)^2 + sech(x/2)^2)/2 in powers of x^4.
  • A090682 (program): Integer part of one leg of a right triangle where the other leg and hypotenuse are twin primes.
  • A090684 (program): Primes of the form 8*n^2 - 1.
  • A090685 (program): Primes of the form 8*k^2 + 1.
  • A090686 (program): Primes of the form 6n^2 - 1.
  • A090687 (program): Primes of the form 6*k^2 + 1.
  • A090691 (program): 5x - 1 sequence starting at 50.
  • A090692 (program): Expansion of 2*(x^2-9*x+15) / ((1+x)*(1-3*x+x^2)).
  • A090693 (program): Positive numbers n such that n^2 - 2n + 2 is a prime.
  • A090696 (program): Numbers k such that k^2 - 11 is a prime.
  • A090697 (program): Numbers n such that n^2/2 - 1 is a prime.
  • A090698 (program): Primes of the form 2*n^2+1.
  • A090701 (program): a(n) is the minimal number k such that every binary word of length n can be divided into k palindromes.
  • A090702 (program): a(n) is the minimal number k such that every binary word of length n can be transformed into a palindrome or an antipalindrome by deleting at most k letters.
  • A090706 (program): Number of numbers having in binary representation the same number of zeros and ones as n has.
  • A090707 (program): Primes whose decimal representation is a valid number in base 4 and interpreted as such is again a prime.
  • A090711 (program): Primes whose base-11 expansion is a (valid) decimal expansion of a prime.
  • A090712 (program): Primes whose base-13 expansion is a (valid) decimal expansion of a prime.
  • A090722 (program): a(n) = if 10 - Mod(Prime(n),10) == {1,3,7,9} respectively then {1,2,3,0}.
  • A090725 (program): Primes whose representation in base 16 has no alphabetic characters.
  • A090727 (program): a(n) = 16a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 16.
  • A090728 (program): a(n) = 20*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 20.
  • A090729 (program): a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.
  • A090730 (program): a(n) = 22*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 22.
  • A090731 (program): a(n) = 23a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 23.
  • A090732 (program): a(n) = 24a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 24.
  • A090733 (program): a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.
  • A090735 (program): Number of positive squarefree numbers <= n that can be expressed as a sum of 2 squares > 0.
  • A090736 (program): Number of positive integers <= n that can be expressed as a sum of 2 coprime squares > 0.
  • A090737 (program): Triangle of Eulerian numbers modulo 3.
  • A090739 (program): Exponent of 2 in 9^n - 1.
  • A090740 (program): Exponent of 2 in 3^n - 1.
  • A090749 (program): a(n) = 12 * C(2n+1,n-5) / (n+7).
  • A090754 (program): Numerator of the expansion of e^(1 + x + x^2 + x^3 + x^4).
  • A090755 (program): Denominator of the expansion of e^(1 + x + x^2 + x^3 + x^4).
  • A090763 (program): a(n) = (3*n+3)!/(3*n!*(2*n+2)!).
  • A090764 (program): Number of partitions of n with two sorts of part 1.
  • A090770 (program): a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).
  • A090771 (program): Numbers that are congruent to {1, 9} mod 10.
  • A090772 (program): Numbers that are congruent to {2, 8} mod 10.
  • A090773 (program): Numbers that are congruent to {4, 6} mod 10.
  • A090778 (program): Numbers k such that phi(k) divides k*(k - phi(k)).
  • A090780 (program): a(n) = n*Product_{p prime, p|n} (p - 1)/2.
  • A090792 (program): a(0)=1; for n>0, a(n)=a([n/2])+a([n/4])+a([n/8]).
  • A090802 (program): Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.
  • A090805 (program): A simple recurrence with one error.
  • A090809 (program): Coefficient of the irreducible character of S_m indexed by (m-2n+2,2n-2) in the n-th Kronecker power of the representation indexed by (m-2,2).
  • A090815 (program): a(n)=denominator(B(2*prime(n))) where prime(n)=n-th prime and B(k) denotes the k-th Bernoulli number.
  • A090816 (program): a(n) = (3*n+1)!/((2*n)! * n!).
  • A090817 (program): a(n)=numerator(B(2*prime(n))) where prime(n)=n-th prime and B(k) denotes the k-th Bernoulli number.
  • A090821 (program): Numbers that are products of two consecutive nonprimes.
  • A090826 (program): Convolution of Catalan and Fibonacci numbers.
  • A090830 (program): Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A085163/A085164.
  • A090831 (program): a(n) = prime(n)^1 + prime(n-1)^2 + prime(n-2)^3 + … + prime(1)^n.
  • A090842 (program): Square array of numbers read by antidiagonals where T(n,k)=((k+3)(k+2)^n-2)/(k+1)
  • A090843 (program): Number of nodes on a tree with degree 11 interior nodes and degree 1 boundary nodes.
  • A090844 (program): Square array of numbers read by antidiagonals with T(n,k)=((k+3)(k+1)^n-2*0^n)/(k+1)
  • A090848 (program): Positions of the terms of A090847^4 in A090847, where A090847 is equal to the union of the self-convolutions A090847^2 and A090847^4 when ordered by size.
  • A090850 (program): Clark’s triangle with f=6 read by row.
  • A090860 (program): Number of ways of 4-coloring a map in which there is a central circle surrounded by an annulus divided into n-1 regions. There are n regions in all.
  • A090864 (program): Complement of generalized pentagonal numbers (A001318).
  • A090866 (program): Primes p == 1 (mod 4) such that (p-1)/4 is prime.
  • A090878 (program): Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.
  • A090879 (program): a(n) = Sum_{d|n} d*2^(n-d).
  • A090880 (program): Suppose n=(p1^e1)(p2^e2)… where p1,p2,… are the prime numbers and e1,e2,… are nonnegative integers. Then a(n) = e1 + (e2)*3 + (e3)*9 + (e4)*27 + … + (ek)*(3^(k-1)) + …
  • A090881 (program): Suppose n=(p1^e1)(p2^e2)… where p1,p2,… are the prime numbers and e1,e2,… are nonnegative integers. Then a(n) = e1 + (e2)*4 + (e3)*16 + (e4)*64 + … + (ek)*(4^(k-1)) + …
  • A090885 (program): Sum of the squares of the exponents in the prime factorization of n.
  • A090889 (program): Double partial sums of (n * its dyadic valuation).
  • A090892 (program): Solutions x to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r = sqrt(2).
  • A090893 (program): Solutions x to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r=sqrt(3).
  • A090894 (program): Numbers in n-th downward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; …
  • A090895 (program): a(1)=1 then a(n)=a(n-1)/2 if a(n-1) is even, a(n)=a(n-1)+n otherwise.
  • A090902 (program): a(n) = floor((product of first n triangular numbers)/(sum of first n factorials)).
  • A090908 (program): Terms a(k) of A073869 for which a(k)=a(k+1).
  • A090909 (program): Terms a(k) of A073869 for which a(k-1), a(k) and a(k+1) are distinct.
  • A090914 (program): Reciprocal of (n+1)! times determinant of n X n matrix whose (i,j)-th element is 1/(i+j).
  • A090926 (program): Least odd prime divisor of prime(n)-1, or a(n) = 2 if no odd prime divisors are encountered.
  • A090932 (program): a(n) = n! / 2^floor(n/2).
  • A090937 (program): a(1) = 1, a(2) = 2; for n > 2, a(n) = a(n-1) + (smallest integer >= n which is coprime to a(n-1)).
  • A090939 (program): Least multiple of n == -1 (mod prime(n)).
  • A090942 (program): n-th arithmetic mean = prime(n).
  • A090946 (program): Non-Lucas numbers: complement of A000204.
  • A090949 (program): a(n) = (1/24)*(n+1)*(3*n^3+59*n^2+358*n+648).
  • A090950 (program): a(n) = (1/24)*(n+1)*(n+3)*(n^2+22*n+88).
  • A090951 (program): LCM of the first n numbers of the form p^q, where p and q are 1 or prime.
  • A090957 (program): a(n) = 1/(Integral_{x=0..1} (x^4 - x^5)^n dx).
  • A090964 (program): Permutation of natural numbers generated by 2-rowed array shown below.
  • A090965 (program): a(n) = 8*a(n-1) - 4*a(n-2), where a(0) = 1, a(1) = 4.
  • A090967 (program): Given the sequence of the sums of the divisors of the semiprimes, this is the subsequence where each sum is an even number.
  • A090969 (program): a(n) = 1/Integral_{x=0..1} (x^5 - x^6)^n.
  • A090970 (program): Number of primes strictly between T(n) and T(n+1), where T(n) = n-th triangular number.
  • A090971 (program): Sierpiński’s triangle, read by rows, starting from 1: T(n,k) = (T(n-1,k) + T(n-1,k-1)) mod 2.
  • A090972 (program): Least k such that there are at least n primes between n and n*k.
  • A090973 (program): a(n) = ceiling((prime(n)/n).
  • A090976 (program): a(n) = 100 reduced mod n.
  • A090982 (program): a(n) = partitions(n)*partitions(n+1).
  • A090988 (program): a(n) = 2^A004736(n).
  • A090989 (program): Number of meaningful differential operations of the n-th order on the space R^4.
  • A090990 (program): Number of meaningful differential operations of the n-th order on the space R^5.
  • A090991 (program): Number of meaningful differential operations of the n-th order on the space R^6.
  • A090992 (program): Number of meaningful differential operations of the n-th order on the space R^7.
  • A090993 (program): Number of meaningful differential operations of the n-th order on the space R^8.
  • A090994 (program): Number of meaningful differential operations of the n-th order on the space R^9.
  • A090995 (program): Number of meaningful differential operations of the n-th order on the space R^10.
  • A090996 (program): Number of leading 1’s in binary expansion of n.
  • A091000 (program): Number of closed walks of length n on the Petersen graph.
  • A091001 (program): Number of walks of length n between adjacent nodes on the Petersen graph.
  • A091002 (program): Number of walks of length n between non-adjacent nodes on the Petersen graph.
  • A091003 (program): Expansion of (1-3*x^2)/((1-2*x)*(1+3*x)).
  • A091004 (program): Expansion of x*(1-x)/((1-2*x)*(1+3*x)).
  • A091005 (program): Expansion of x^2/((1-2*x)*(1+3*x)).
  • A091018 (program): Numbers in n-th upward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; …
  • A091019 (program): Denominators of the Taylor series of arccosh(z)/sqrt(2(x-1)) about 1.
  • A091022 (program): Semiprimes with semiprime indices.
  • A091024 (program): Let v(0) be the column vector (1,0,0,0)’; for n>0, let v(n) = [1 1 1 1 / 1 1 1 0 / 1 1 0 0/ 1 0 0 0] v(n-1). Sequence gives third entry of v(n).
  • A091025 (program): Smallest positive k such that phi(1+k*2^m) <= phi(k*2^m) for all m = n (mod 12), where phi is Euler’s totient function.
  • A091030 (program): Partial sums of powers of 13 (A001022).
  • A091031 (program): Third to last entries in rows of array A090452 (scaled (3,2)-Stirling2).
  • A091032 (program): Second column (k=3) of array A090438 ((4,2)-Stirling2) divided by 8.
  • A091033 (program): Third column (k=4) of array A090438 ((4,2)-Stirling2).
  • A091034 (program): Fourth column (k=5) of array A090438 ((4,2)-Stirling2) divided by 24.
  • A091035 (program): Fifth column (k=6) of array A090438 ((4,2)-Stirling2).
  • A091036 (program): Sixth column (k=7) of array A090438 ((4,2)-Stirling2) divided by 48=4!*2.
  • A091037 (program): Second column (k=5) of array A090214 ((4,4)-Stirling2) divided by 4*4!=96.
  • A091038 (program): Second column (k=6) of array A090216 ((5,5)-Stirling2) divided by 5*5! = 600.
  • A091042 (program): Triangle of even numbered entries of odd numbered rows of Pascal’s triangle A007318.
  • A091044 (program): One half of odd-numbered entries of even-numbered rows of Pascal’s triangle A007318.
  • A091045 (program): Partial sums of powers of 17 (A001026).
  • A091050 (program): Number of divisors of n that are perfect powers.
  • A091051 (program): Sum of divisors of n that are perfect powers.
  • A091052 (program): Record values in A091023.
  • A091053 (program): Where records occur in A091023.
  • A091054 (program): Expansion of (1 - 5*x - 2*x^2) / ((1 - x)*(1 + 2*x)*(1 - 6*x)).
  • A091055 (program): Expansion of x*(1-2*x)/((1-x)*(1+2*x)*(1-6*x)).
  • A091056 (program): Expansion of x^2/((1-x)*(1+2*x)*(1-6*x)).
  • A091067 (program): Numbers whose odd part is of the form 4k+3.
  • A091068 (program): a(0) = 0; for n>0, a(n) = a(n-1) - n if that is >= 0, else a(n) = a(n-1) + n - 1.
  • A091069 (program): Moebius mu sequence for real quadratic extension sqrt(2).
  • A091072 (program): Numbers whose odd part is of the form 4k+1. The bit to the left of the least significant bit of each term is unset.
  • A091074 (program): Fibonacci sequence beginning 12, 67.
  • A091084 (program): a(n) = A001045(n) mod 10.
  • A091085 (program): a(n) = mod(A078008(n),10).
  • A091086 (program): a(n) = A000975(n) mod 10.
  • A091087 (program): a(n) = floor(r*n) + floor(n/r), where r=sqrt(2).
  • A091090 (program): In binary representation: number of editing steps (delete, insert, or substitute) to transform n into n + 1.
  • A091095 (program): Expansion of (1+4x-24x^2)/((1-4x)(1+4x)).
  • A091096 (program): Expansion of (1+5x-40x^2)/((1-5x)(1+5x)).
  • A091097 (program): Expansion of (1+6x-60x^2)/((1-6x)(1+6x)).
  • A091103 (program): Expansion of (1-3x+12x^2)/((1-3x)(1+3x)).
  • A091104 (program): Expansion of (1-4x+24x^2)/((1-4x)(1+4x).
  • A091105 (program): Expansion of (1-5x+40x^2)/((1-5x)(1+5x)).
  • A091106 (program): a(0)=a(1)=-1. For n>1: a(n)=Sum(i!i^2 Stirling2[n-1,i],i=2,..,n-1).
  • A091113 (program): Nonprimes of the form 4*k+1.
  • A091114 (program): Number of partitions of n-th composite number containing the smallest prime factor: a(n) = A027293(A002808(n), A056608(n)).
  • A091131 (program): Decimal expansion of e - 1.
  • A091132 (program): Decimal expansion of e^2 - 2e.
  • A091135 (program): Number of Dyck paths of semilength n+4, having exactly two long ascents (i.e., ascents of length at least two).
  • A091137 (program): Largest number m such that number of times m divides k! is almost k/n for large k, i.e., largest m with A090624(m)=n.
  • A091140 (program): a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 3, 9.
  • A091141 (program): a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 4, 13.
  • A091142 (program): a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 2, 6.
  • A091143 (program): Number of Pythagorean triples mod 2^n; i.e., number of solutions to x^2 + y^2 = z^2 mod 2^n.
  • A091144 (program): a(n) = binomial(n^2, n)/(1+(n-1)*n).
  • A091147 (program): Expansion of (1-x-sqrt(1-2x-15x^2))/(8x^2).
  • A091148 (program): Expansion of (1-x-sqrt(1-2x-19x^2))/(10x^2).
  • A091149 (program): Expansion of (1 - x - sqrt(1 - 2*x - 23*x^2))/(12*x^2).
  • A091154 (program): Numerator of Maclaurin expansion of (t*sqrt(t^2+1) + arcsinh(t))/2, the arc length of Archimedes’ spiral.
  • A091177 (program): Numbers m such that the m-th prime is of the form 3*k-1.
  • A091178 (program): Numbers k such that k-th prime is of the form 6*m+1.
  • A091179 (program): A088878 indexed by A000040.
  • A091180 (program): Primes of the form 3*p - 2 such that p is a prime.
  • A091181 (program): A091180 indexed by A000040.
  • A091185 (program): a(n) = A090938(n)/n.
  • A091186 (program): Triangle read by rows, in which n-th row gives expansion of x^n/((1-x)(1-x-x^2)^n).
  • A091187 (program): Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.
  • A091194 (program): Number of abundant numbers <= n.
  • A091199 (program): Numbers n such that (6n-3)^2 + 2 is prime.
  • A091236 (program): Nonprimes of form 4k+3.
  • A091258 (program): Denominator of sigma(3,n)/sigma(1,n).
  • A091259 (program): Numerator of sigma_3(n)/sigma(n).
  • A091260 (program): Denominator of sigma_3(n)/sigma_2(n).
  • A091262 (program): Sum of totient function values of powers of n, as exponent runs from 1 to n.
  • A091264 (program): Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.
  • A091265 (program): Take sequence of prime numbers (A000040) and reverse successive subsequences of lengths 1,2,3,4,…
  • A091270 (program): Smallest number having in binary representation a prefix of length n that is also a suffix of its successor.
  • A091271 (program): Numbers k such that 4*k^2-11 is a prime.
  • A091272 (program): Primes of the form n^2 - 11.
  • A091276 (program): A090939(n)/n.
  • A091282 (program): Exponent of 2 in prime(n)^2 - 1.
  • A091283 (program): Exponent of 2 in -1+prime[n]^s, if s is an exponent of the form s=8k-4.
  • A091284 (program): Exponent of 2 in -1+prime[n]^s, if s is an exponent of form 16k-8. Except a(1)=0, a(n)=1+A091283(n).
  • A091296 (program): Semiprimes with odd digits.
  • A091297 (program): A fixed point of the morphism 0 -> 02, 1 -> 02, 2 -> 11, starting from 0.
  • A091300 (program): Nonprimes of the form 6k + 1.
  • A091304 (program): a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).
  • A091306 (program): Sum of squares of unitary, squarefree divisors of n, including 1.
  • A091307 (program): a(n)=6*2^n+4 (Bode Number A003461(n+2)) except for a(1)=6.
  • A091311 (program): Partial sums of 3^A007814(n).
  • A091330 (program): a(n) = ((p-1)!/p) - ((p-1)*(p-1)!/p!), where p is the n-th prime.
  • A091335 (program): Number of prime divisors of n-th term of Sylvester’s sequence A000058.
  • A091336 (program): Number of prime divisors of A000058(n)-1 = A000058(0)*…*A000058(n-1).
  • A091337 (program): a(n) = (2/n), where (k/n) is the Kronecker symbol.
  • A091338 (program): a(n) = (3/n), where (k/n) is the Kronecker symbol.
  • A091339 (program): a(n) = a(n-1)*a(n-2) + (n-2); a(1) = 1, a(2) = 2.
  • A091342 (program): Given (1) f(h,j,a) = ( [ ((a/gcd(a,h)) / gcd(j+1,(a/gcd(a,h)))) * (h(j+1)) ] - [ ((a/gcd(a,h)) / gcd(j+1,(a/gcd(a,h)))) * (ja) ] ) / a then let (2) a(h) = d(h,j) = lcm( f(h,j,1) … f(h,j,h) ).
  • A091344 (program): a(n) = 2*3^n - 3*2^n + 1.
  • A091346 (program): Binomial convolution of A069321(n), where A069321(0)=0, with the sequence of all 1’s alternating in sign.
  • A091347 (program): a(n) = 6*4^n - 12*3^n + 7*2^n - 1.
  • A091356 (program): Number of planar partitions of n with exactly 2 rows.
  • A091360 (program): Partial sums of A000219.
  • A091361 (program): Numbers n such that A001840(n) == 0 (mod n).
  • A091363 (program): a(n) = n!*n^3.
  • A091364 (program): a(n) = n! * n^4.
  • A091369 (program): a(n) = Sum_{i=1..n} phi(i)*ceiling(n/i).
  • A091370 (program): Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having a k-gon over a fixed edge (base).
  • A091371 (program): Smallest prime factor of n - number of prime factors of n with multiplicity.
  • A091373 (program): Number of numbers <= n having exactly as many prime factors as the value of their smallest prime factor.
  • A091374 (program): Number of numbers <= n having fewer prime factors than the value of their smallest prime factor.
  • A091376 (program): Numbers k with property that the number of prime factors of k (counted with repetition) equals the smallest prime factor of k.
  • A091377 (program): Numbers having fewer prime factors than the value of their smallest prime factor.
  • A091379 (program): a(n) = Product_{ p | n } (1 + Legendre(-1,p) ).
  • A091392 (program): Product_{ p | n } (1 + Legendre(-2,p) ).
  • A091393 (program): a(n) = Product_{ p | n } (1 + Legendre(-3,p) ).
  • A091394 (program): a(n) = Product_{ p | n } (1 + Legendre(-5,p) ).
  • A091395 (program): a(n) = Product_{ p | n } (1 + Legendre(-7,p) ).
  • A091396 (program): a(n) = Product_{ p | n } (1 + Legendre(2,p) ).
  • A091397 (program): a(n) = Product_{ p | n } (1 + Legendre(3,p) ).
  • A091398 (program): a(n) = Product_{ p | n } (1 + Legendre(5,p) ).
  • A091400 (program): a(n) = Product_{ odd primes p | n } (1 + Legendre(-1,p) ).
  • A091428 (program): Numbers n such that A092673(n)=+/-1.
  • A091429 (program): Numerator of a(n) = (integral_{x=0..1/3} (1-x^2)^n dx).
  • A091433 (program): a(n)=2*3^n - 18*4^n + 24*5^n.
  • A091435 (program): Array T(n,k) = n*(n+k), read by antidiagonals.
  • A091441 (program): Table (by antidiagonals) of permutations of two types of objects such that each cycle contains at least one object of each type. Each type of object is labeled from its own label set.
  • A091453 (program): Triangular array T(n,k) read by rows in which row n consists of the numbers floor(2n/k), k=1,2,…,2n+1.
  • A091454 (program): Integers n such that 3*phi(n) < n.
  • A091468 (program): Number of unlabeled alternating octupi with n black nodes and n white nodes.
  • A091469 (program): Number of unlabeled alternating octupi with n black nodes.
  • A091476 (program): Decimal expansion of Pi^2/4.
  • A091478 (program): Table of graphs with n (>=1) nodes and k (>=0) edges. Each type of object labeled from its own label set.
  • A091479 (program): Number of graphs with n nodes. Nodes and edges labeled each from their own label set.
  • A091480 (program): Table of multigraphs (by antidiagonals) with n (>=1) nodes and k (>=0) edges. Each type of object labeled from its own label set.
  • A091481 (program): Number of labeled rooted 2,3 cacti (triangular cacti with bridges).
  • A091482 (program): a(n) = (3*n)^n.
  • A091483 (program): a(n) = (4*n)^n.
  • A091485 (program): Number of labeled 2,3 cacti (triangular cacti with bridges).
  • A091491 (program): Triangle, read by rows, where the n-th diagonal is generated from the n-th row by the sum of the products of the n-th row terms with binomial coefficients.
  • A091496 (program): a(n) = ((5n)!/(n!(2n)!))(gamma(1+n/2)/gamma(1+5n/2)).
  • A091507 (program): Product of the anti-divisors of n.
  • A091512 (program): a(n) is the largest integer m such that 2^m divides (2*n)^n, i.e., the exponent of 2 in (2*n)^n.
  • A091518 (program): Decimal expansion of the hyperbolic volume of the figure eight knot complement.
  • A091519 (program): G.f.: Sum_{k>=0} (2^k*t*(1+t)/(1-t)^3, t=x^2^k).
  • A091520 (program): Expansion of 1 / ((1 - 4*x) * sqrt(1 + 4*x)) in powers of x.
  • A091522 (program): Graham-Pollak sequence with initial term 5.
  • A091523 (program): Graham-Pollak sequence with initial term 8.
  • A091524 (program): a(m) is the multiplier of sqrt(2) in the constant alpha(m) = a(m)*sqrt(2) - b(m), where alpha(m) is the value of the constant determined by the binary bits in the recurrence associated with the Graham-Pollak sequence.
  • A091526 (program): Coefficient of x^n in 1/((1+x)*(1-x)^(n-1)).
  • A091527 (program): a(n) = ((3*n)!/n!^2)*(Gamma(1+n/2)/Gamma(1+3n/2)).
  • A091528 (program): Sum {k=1 to n} H(k) k! (n-k)! (mod {n+1}), where H(k) is the k-th harmonic number.
  • A091529 (program): When sum {k=1 to n} H(k) k! (n-k)! (mod {n+1}) (A091528) is not zero.
  • A091530 (program): Sum {k=1 to n} H(k) k! (n-k)!, where H(k) is the k-th harmonic number.
  • A091535 (program): First column (k=2) of array A091534 ((5,2)-Stirling2).
  • A091540 (program): Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).
  • A091541 (program): Four times triple factorials (3*n-2)!!! with leading 1 added.
  • A091543 (program): Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m >= 0.
  • A091544 (program): First column sequence of array A091746 ((6,2)-Stirling2).
  • A091545 (program): First column sequence of the array (7,2)-Stirling2 A091747.
  • A091546 (program): First column of the array A092077 ((8,2)-Stirling2).
  • A091547 (program): Row sums of triangle A091543.
  • A091549 (program): Second column (k=3) sequence of array A078740 ((3,2)-Stirling2) divided by 6.
  • A091555 (program): Partial sums of Mertens’s function (A002321).
  • A091561 (program): Expansion of (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).
  • A091565 (program): Expansion of (1 - x - sqrt(1 - 2*x + x^2 - 8*x^3)) / (4*x^2) in powers of x.
  • A091567 (program): Primes p such that p^2-p-1 is prime.
  • A091568 (program): Primes of the form p^2 - p - 1, where p is prime.
  • A091570 (program): Sum of odd proper divisors of n. Sum of the odd divisors of n that are less than n.
  • A091571 (program): Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_8.
  • A091572 (program): Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_7.
  • A091573 (program): Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_6.
  • A091574 (program): Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type D_4.
  • A091577 (program): Poincaré series [or Poincare series] of the preprojective algebra of a Dynkin diagram of type E_6.
  • A091580 (program): Number of partitions of n into decimal palindromes.
  • A091591 (program): Number of pairs of twin primes between n^2 and (n+1)^2.
  • A091593 (program): Reversion of Jacobsthal numbers A001045.
  • A091594 (program): Triangle read by rows: T(n,m) := sum{k=0..floor((n-m)/2), binomial(n-2k,m)binomial(n-m-k,k)}.
  • A091595 (program): Triangle read by rows: T(n,m) := sum{k=0..floor((n-m)/2), binomial(n-2k,m)binomial(n-m-k,k)2^k}.
  • A091596 (program): Expansion of x(1-2x^2)/(1-x-2x^2)^2.
  • A091597 (program): Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).
  • A091598 (program): Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).
  • A091601 (program): Number of compositions (ordered partitions) of n with designated summands.
  • A091626 (program): Number of ordered integer pairs (b,c) with 0 <= b, c <= n such that both roots of x^2+bx+c=0 are integers.
  • A091627 (program): Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.
  • A091628 (program): Concatenation of n 2’s followed by 3.
  • A091629 (program): Product of digits associated with A091628(n). Essentially the same as A007283.
  • A091630 (program): Numbers n + product of digits associated with A091628.
  • A091650 (program): Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 -1 3 2]. Set seed vector = [1 1 1 1] = first row, then take M*[1 1 1 1] = [1 1 1 3] then M * [1 1 1 3], etc. Sequence gives terms in rightmost column.
  • A091651 (program): Decimal expansion of (5 - Pi)/4.
  • A091661 (program): Coefficients in a 10-adic square root of 1.
  • A091663 (program): 10-adic integer x=…..93380022607743740081787109375 satisfying x^3 = x.
  • A091664 (program): 10-adic integer x=…..06619977392256259918212890624 satisfying x^3 = x.
  • A091666 (program): Difference between prime(n)^2 and the next prime.
  • A091669 (program): a(n) = (2^(n-1)/n!) * Product_{k=1..n-1} (2^k-1).
  • A091681 (program): Decimal expansion of BesselJ(0,2).
  • A091682 (program): Decimal expansion of 2*(18 + sqrt(3)*Pi)/27.
  • A091684 (program): a(n) = 0 if n is divisible by 3, otherwise a(n) = n.
  • A091685 (program): Sieve out 6n+1 and 6n-1.
  • A091686 (program): 0^n+((n-9)/9)(1-10^n).
  • A091691 (program): (10^(n-1)-1) * (n-10) / 9.
  • A091692 (program): (10^n-1) * (n+9) / 9.
  • A091693 (program): (n*10^n - n + 9)/9.
  • A091695 (program): E.g.f.: exp(x/(1-x)^3).
  • A091696 (program): Number of classes of compositions of n equivalent under reflection or cycling.
  • A091698 (program): Matrix inverse of triangle A063967.
  • A091699 (program): Row sums of triangle A091698.
  • A091701 (program): Row sums of triangle A091700.
  • A091702 (program): Column 0 of triangle A091700.
  • A091703 (program): Count, setting 5n to zero.
  • A091704 (program): Number of Barker codes (or Barker sequences) of length n up to reversals and negations.
  • A091711 (program): Exponent of 2 in (n^2)!.
  • A091712 (program): a(n)=6(2n-4)!/((n-2)!n!), if n>2. a(0)=1,a(1)=a(2)=2.
  • A091719 (program): Greatest common divisors of consecutive partition numbers.
  • A091720 (program): Babylonian sexagesimal (base 60) expansion of 1/7.
  • A091721 (program): Babylonian sexagesimal (base 60) expansion of 1/11.
  • A091722 (program): Babylonian sexagesimal (base 60) expansion of 1/13.
  • A091732 (program): Iphi(n): infinitary analog of Euler’s phi function.
  • A091733 (program): a(n) is the least m > 1 such that m^3 = 1 (mod n).
  • A091735 (program): Primes arising in the first row of array in A091734.
  • A091738 (program): Primes arising in the second row of array in A091734.
  • A091753 (program): Fourth column (m=5) of array A091752 ((-1,2)Stirling2) divided by -6.
  • A091755 (program): Sixth column (m=7) of array A091752 ((-1,2)Stirling2) divided by -12.
  • A091759 (program): a(n) = 0^n + 2((n+1)^n - (-1)^n) / (n+2).
  • A091760 (program): 0^n+3((n+2)^n/(n+3)-(-1)^n/(n+3)).
  • A091761 (program): a(n) = Pell(4n) / Pell(4).
  • A091772 (program): a(n) = gcd(A000108(n), A000110(n)).
  • A091773 (program): G.f.: Product((1+x^i)/(1-x^i),i=1..n-1)/(1-x^n), with n = 5.
  • A091774 (program): G.f.: Product((1+x^i)/(1-x^i),i=1..n-1)/(1-x^n), with n = 6.
  • A091775 (program): 1 + 4^n + 9^n + 16^n.
  • A091777 (program): Product of the numbers from (n-1)^2+1 to n^2.
  • A091785 (program): Evil numbers (see A001969) in A003159.
  • A091793 (program): Triangle read by rows in which row n contains the smallest nontrivial set of n consecutive numbers divisible by the next n numbers respectively. The next n numbers are numbers from n(n-1)/2 +1 up to n(n+1)/2.
  • A091794 (program): First column of A091793.
  • A091795 (program): Diagonal of A091793.
  • A091796 (program): Smallest k such that for 0 <= i < n, n*(n+1)/2-i divides k+i.
  • A091797 (program): Smallest k such that for 0 <= i < n, 1+i+n*(n-1)/2 divides k-i.
  • A091799 (program): a(1) = 3. To get a(n+1), write the string a(1)a(2)…a(n) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence so far. Then a(n+1) = max(k,3).
  • A091801 (program): Largest n-digit multiple of the n-th prime.
  • A091802 (program): 10^n - (largest n-digit multiple of the n-th prime).
  • A091803 (program): (Smallest n-digit multiple of the n-th prime) - 10^(n-1).
  • A091811 (program): Array read by rows: T(n,k) = binomial(n+k-2,k-1)*binomial(2*n-1,n-k).
  • A091814 (program): Numerator of I(n) = (integral_{x=0..1/2}(1-x^2)^n,dx), where the denominator is b(n) = 2^n*(2*n+2)!/(n+1)!.
  • A091818 (program): Sum of the even proper divisors of 2n. Sum of the even divisors of 2n that are less than 2n.
  • A091822 (program): a(n)=2*sum(k=0,n,sum(i=0,k,sum(j=0,i, A010060(j))))-(1/6)*(n^3+6*n^2+11*n-6).
  • A091823 (program): a(n) = 2*n^2 + 3*n - 1.
  • A091825 (program): Integers of the form ((p-1)!*2^(p-1) + 1)/p.
  • A091826 (program): a(n)=(1/n)*(1+A000254(n)-n) as n runs through the primes.
  • A091829 (program): a(1)=1; a(2n)=a(n)+1, a(2n+1)=a(n) mod 2.
  • A091830 (program): a(1)=1; a(2n)=(a(n)+1) mod 2, a(2n+1)=a(2n)+1.
  • A091835 (program): Double factorial of primes.
  • A091841 (program): Records in A091840.
  • A091844 (program): a(1) = 4. To get a(n+1), write the string a(1)a(2)…a(n) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence so far. Then a(n+1) = max(k,4).
  • A091848 (program): Johnson bound J(n,4,2).
  • A091849 (program): Maximal size of (n,4,2) optical orthogonal code.
  • A091855 (program): Odious numbers (see A000069) in A003159.
  • A091856 (program): Beginning with 1, minimum value such that gcd(a(2n-1),a(2n)) = 1, gcd(a(2n),a(2n+1))>1 and a(n) > a(n-1).
  • A091858 (program): a(n) = n! mod prime(n).
  • A091860 (program): a(1)=1, a(n)=sum(i=1,n-1,b(i)) where b(i)=0 if a(i) and a(n-i) are both even, b(i)=1 otherwise.
  • A091862 (program): a(n) = 1 if the sum of all exponents of the prime-factorization of n has no carries when summed in base 2, or a(n) = 0 if there are any carries.
  • A091863 (program): Partial sums of A091862.
  • A091867 (program): Triangle read by rows: T(n,k) = number of Dyck paths of semilength n having k peaks at odd height.
  • A091868 (program): a(n) = (n!)^(n+1).
  • A091869 (program): Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks at even height.
  • A091870 (program): A trinomial transform of Fibonacci(3n).
  • A091880 (program): A049232 indexed by A000040.
  • A091881 (program): Expansion of (1-11x)/((1-x)(1-16x)).
  • A091882 (program): Expansion of (1-10x)/(1-15x).
  • A091883 (program): Expansion of (1-5x)/((1+5x)(1-10x)).
  • A091884 (program): Triangle of numbers defined by Knuth.
  • A091903 (program): Expansion of x/((1+5x)(1-10x)).
  • A091904 (program): Expansion of x/((1+4x)(1-8x)).
  • A091905 (program): Expansion of (1-4x)/((1+4x)(1-8x)).
  • A091912 (program): Numerators of Taylor series for log(tan(x)+1/cos(x)).
  • A091913 (program): Triangle read by rows: a(n,k) = C(n,k)*(2^(n-k) - 1) for k<n, a(n,k) = 0 for k >= n, where k=0..max(n-1,0).
  • A091914 (program): a(n) = 2*a(n-1) + 12*a(n-2).
  • A091915 (program): Maximum of even products of partitions of n.
  • A091916 (program): Maximum of odd products of partitions of n.
  • A091917 (program): Coefficient array of polynomials (z-1)^n-1.
  • A091918 (program): Inverse of number triangle A091917.
  • A091919 (program): Expansion of 1/((1-2*x)*(1-x^2)^2).
  • A091921 (program): Sum of odd proper distinct prime divisors of n. That is, the sum of odd distinct prime divisors of n that are less than n.
  • A091925 (program): Decimal expansion of Pi^3.
  • A091927 (program): Expansion of (1-6x)/(1-6x-5x^2).
  • A091928 (program): a(0)=1, a(1)=5; a(n) = 6*a(n-1) + 5*a(n-2) for n > 1.
  • A091929 (program): Expansion of (1-6x)/(1-6x-11x^2).
  • A091931 (program): Change the first bit to 0 in binary notation for the n-th prime.
  • A091933 (program): Decimal expansion of e^3.
  • A091934 (program): Number of dual isomorphisms on [ n,n* ].
  • A091940 (program): Given n colors, sequence gives number of ways to color the vertices of a square such that no edge has the same color on both of its vertices.
  • A091946 (program): a(n) = floor(11^n/10^n).
  • A091947 (program): (Fractional part of 1.1^n) * 10^n.
  • A091949 (program): a(n) = A087659(n) mod 2.
  • A091952 (program): a(1)=1, a(2n)=(a(n)+1) mod 2; a(2n+1)=2*a(2n).
  • A091953 (program): a(1)=1, a(2n)=1+a(n)(mod 2); a(2n+1)=2*a(2n)+1.
  • A091954 (program): Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.
  • A091957 (program): a(1)=0, a(2)=1, a(n)=A000217(a(n-1)) + A000217(a(n-2)).
  • A091959 (program): a(1)=1, a(2n)=(a(n)+1) mod 2, a(2n+1)=2*a(2n)+1.
  • A091960 (program): a(1)=1, a(2n)=a(2n-1)+(a(n)mod 2), a(2n+1)=a(2n)+1.
  • A091962 (program): From enumerating paths in the plane.
  • A091964 (program): Number of left factors of peakless Motzkin paths of length n.
  • A091965 (program): Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).
  • A091968 (program): Primes congruent to 3 (mod 16).
  • A091971 (program): G.f.: (1+x^3)*(1+x^5)*(1+x^6)/((1-x^4)*(1-x^5)*(1-x^6)).
  • A091972 (program): G.f.: (1 + x^5 ) / ( (1-x^3)*(1-x^4)).
  • A091979 (program): Number of odd proper distinct prime divisors of n. That is, the number of odd distinct prime divisors of n that are less than n.
  • A091983 (program): a(0) = 1, a(n) = 20*sigma3.
  • A091985 (program): Number of steps required for the initial value p = 10^n to reach 0 in the recurrence p = pi(p).
  • A091986 (program): a(0)=1, a(n) = sigma_3(2n).
  • A091987 (program): Number of steps required for initial p = 2^n to reach 0 in the recurrence p = pi(p).
  • A091992 (program): Numbers n such that 5*n-3 and 5*n+3 are both primes.
  • A091993 (program): Numerator of I(n) = Integral_{x=0 to 1/3} (1+x^2)^n dx.
  • A091994 (program): Numerator of I(n) = sqrt(10)*(Integral_{x=0 to 1/3} 1/(1+x^2)^(n+1/2) dx).
  • A091995 (program): Permutation of the natural numbers.
  • A091998 (program): Numbers that are congruent to {1, 11} mod 12.
  • A091999 (program): Numbers that are congruent to {2, 10} mod 12.
  • A092022 (program): Numbers n such that 16n + 3 is prime.
  • A092028 (program): a(n) is the smallest m > 1 such that m divides n^m-1.
  • A092032 (program): Arises in partition theory.
  • A092037 (program): A092255 mod 3.
  • A092038 (program): a(n+1) = a(n) + (a(n) mod 2)^(n mod a(n)), a(1) = 1.
  • A092041 (program): Decimal expansion of cube root of e.
  • A092042 (program): Decimal expansion of e^(1/4).
  • A092043 (program): a(n) = numerator(n!/n^2).
  • A092053 (program): Denominators of the convergents of the continued fraction expansion [1;1/2,1/3,1/4,…,1/n,…].
  • A092054 (program): Base-2 logarithm of the sum of numerator and denominator of the convergents of the continued fraction expansion [1; 1/2, 1/3, 1/4, …, 1/n, …].
  • A092055 (program): C(2+2^n,3).
  • A092056 (program): Square table by antidiagonals where T(n,k)=C(n+2^k-1,n).
  • A092057 (program): Primes of the form 2*p^2 - 1, where p is prime.
  • A092058 (program): Numbers n such that 2*prime(n)^2 - 1 is prime.
  • A092067 (program): a(n) is the smallest number m such that m > 1 and m divides n^m + 1.
  • A092073 (program): Boustrophedon transform (first version) of Fibonacci numbers 1, 1, 2, 3, 5, 8, …
  • A092074 (program): Primes congruent to 3 mod 17.
  • A092076 (program): Expansion of (1+4*x^3+x^6)/((1-x)*(1-x^3)^2).
  • A092080 (program): Triangle read by rows in which row n lists the partition numbers of the first n positive integers.
  • A092081 (program): Triangle of certain double factorials.
  • A092086 (program): Row sums of triangle A092083 (s2(7)).
  • A092087 (program): Alternating row sums of triangle A092083 (s2(7)).
  • A092089 (program): Number of odd-length palindromes among the k-tuples of partial quotients of the continued fraction expansions of n/r, r = 1, …, n.
  • A092090 (program): Boustrophedon transform of Fibonacci numbers 1, 2, 3, 5, 8, …
  • A092092 (program): Back and Forth Summant S(n, _3): a(n) = Sum_{i=0..floor(2n/3)} (n-3i).
  • A092093 (program): Back and Forth Summant S(n, _5): a(n) = sum_{i = 0..floor(2n/5)} n-5i.
  • A092094 (program): a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=3.
  • A092095 (program): a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=4.
  • A092104 (program): Primes of form p*q + 4, with prime p and q.
  • A092106 (program): Fractal mountains in base 3.
  • A092109 (program): Primes p such that p+3 is a semiprime.
  • A092136 (program): Number of spanning trees in S_5 x P_n.
  • A092137 (program): Lower bound for A005842(n).
  • A092143 (program): Cumulative product of all divisors of 1..n.
  • A092144 (program): A092143(n!)/n!.
  • A092145 (program): Numerator of I(n) = 2*(Integral_{x=0..1/2} (1+x^2)^n dx).
  • A092146 (program): Primes of the form p + 10 where p is a prime.
  • A092147 (program): Number of even-length palindromes among the k-tuples of partial quotients of the continued fraction expansions of n/r, r=1,…,n.
  • A092148 (program): Expansion of e.g.f. 1/(exp(x)-x*exp(2*x)).
  • A092149 (program): Partial sums of A092673.
  • A092150 (program): Partial sums of A092674.
  • A092151 (program): A092673(n)*A092674(n).
  • A092152 (program): Sign of A092673(n).
  • A092154 (program): First differences of A092674.
  • A092155 (program): First differences of A092673.
  • A092163 (program): a(n) = Prime(2n) + prime(2n+1).
  • A092164 (program): Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (1,1) entry of M^n.
  • A092165 (program): Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (1,2) entry of M^n.
  • A092166 (program): Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (2,1) entry of M^n.
  • A092167 (program): Let M = 2 X 2 matrix [ 1 2 / 5 4 ]; a(n) = (2,2) entry of M^n.
  • A092168 (program): Primes congruent to 3 (modulo 19).
  • A092169 (program): A000217(n)-A092674(n).
  • A092170 (program): Sum of squares of alternating factorials : n!^2 - (n-1)!^2 + (n-2)!^2 - … 1!^2.
  • A092176 (program): A067076 + A000079/2.
  • A092178 (program): Primes congruent to 8 mod 13.
  • A092180 (program): Permutation of primes generated by triangle shown below.
  • A092181 (program): Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).
  • A092184 (program): Sequence S_6 of the S_r family.
  • A092185 (program): a(n) = (5/6)*n^3+(5/2)*n^2+(8/3)*n.
  • A092186 (program): a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1.
  • A092187 (program): A092186(n)/2.
  • A092191 (program): Numbers n such that sum of n-th and (n+1)-th semiprimes is a semiprime.
  • A092192 (program): Semiprimes that are the sum of two successive semiprimes.
  • A092196 (program): Number of letters in “old style” Roman numeral representation of n (e.g., IIII rather than IV).
  • A092200 (program): Expansion of (1+2x)/((1-x)(1-x^3)).
  • A092202 (program): Expansion of (x - x^3) / (1 - x^5) in powers of x.
  • A092205 (program): Number of units in the imaginary quadratic field Q(sqrt(-n)).
  • A092206 (program): Positive integers that are not of the form n^2 or 3n^2.
  • A092207 (program): Semiprimes k such that k+2 is also a semiprime.
  • A092216 (program): Primes of the form p + 12 where p is a prime.
  • A092220 (program): Expansion of x*(1-x)/ ((1+x)*(1-x+x^2)) in powers of x.
  • A092236 (program): a(n) = (3^n + 2*3^(n/2)*cos(n*Pi/6))/3.
  • A092242 (program): Numbers that are congruent to {5, 7} (mod 12).
  • A092243 (program): Score at stage n in “tug of war” between prime gap increases vs. prime gap decreases: start with score = 0 at n = 1 and at stage n = k > 1, increase (resp. decrease) the score by 1 if the k-th prime gap is greater (resp. less) than the previous prime gap.
  • A092246 (program): Odd “odious” numbers (A000069).
  • A092248 (program): Parity of number of distinct primes dividing n (function omega(n)) parity of A001221.
  • A092249 (program): Positions of the integers in the standard diagonal enumeration of the rationals (with the integers in the first column and diagonals moving up to the right).
  • A092256 (program): Nonprimes of form 6k+5.
  • A092259 (program): Numbers that are congruent to {4, 8} mod 12.
  • A092261 (program): Sum of unitary, squarefree divisors of n, including 1.
  • A092263 (program): a(1)=1, a(n+1)=ceiling(phi*a(n))+1 if a(n) is odd, a(n+1)=ceiling(phi*a(n)) if a(n) is even, where phi=(1+sqrt(5))/2.
  • A092264 (program): a(n)*a(n-5) = a(n-1)*a(n-4)+a(n-2)+a(n-3), with initial terms a(1) = … = a(5) = 1.
  • A092266 (program): Expansion of (1+4x)/AGM(1+4x,1-4x) where AGM denotes the arithmetic-geometric mean.
  • A092270 (program): If n mod 2 == 0 then 3^n else 2^n.
  • A092271 (program): Triangle read by rows. First in a series of triangular arrays counting permutations of partitions.
  • A092276 (program): Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.
  • A092277 (program): a(n) = 7*n^2 + n.
  • A092278 (program): Floor( (3*n+4)/16 ).
  • A092279 (program): a(n) = floor(7*n/16) + 5.
  • A092283 (program): Triangular array read by rows: T(n,k)=n+k^2, 1<=k<=n.
  • A092286 (program): Fourth diagonal (m=3) of triangle A084938; a(n) = A084938(n+3,n) = (n^3 + 9*n^2 + 26*n)/6.
  • A092287 (program): a(n) = Product_{j=1..n} Product_{k=1..n} gcd(j,k).
  • A092290 (program): Decimal expansion of solution to n/x = x-n for n = 7.
  • A092292 (program): a(n) = 3*n + A053838(n).
  • A092293 (program): a(n) = 3*n + A090239(n).
  • A092294 (program): Decimal expansion of 3 + sqrt(15).
  • A092296 (program): a(n) = 3*n + A090193(n).
  • A092297 (program): Number of ways of 3-coloring an annulus consisting of n zones joined like a pearl necklace.
  • A092305 (program): Length of period of sequence of Genocchi number of first kind read modulo (2n+1).
  • A092323 (program): 2^m - 1 appears 2^m times.
  • A092327 (program): a(n) = (1/12)*(n+1)*(n^3+19*n^2+118*n+228).
  • A092338 (program): a(n) = number of numbers d with n mod d <= 1.
  • A092339 (program): Number of adjacent identical digits in the binary representation of n.
  • A092341 (program): a(0)=1, a(n) = sigma_3(3n).
  • A092342 (program): a(n) = sigma_3(3n+1).
  • A092343 (program): a(n) = sigma_3(3n+2).
  • A092344 (program): a(0)=1; a(n) = sigma_2(n) + sigma_3(n).
  • A092345 (program): a(0)=1; a(n) = sigma_1(n) + sigma_3(n).
  • A092346 (program): a(0)=1; a(n) = sigma_1(n) + sigma_2(n).
  • A092347 (program): a(0)=1; a(n) = sigma_1(n) + sigma_2(n) + sigma_3(n).
  • A092348 (program): a(n) = sigma_3(n) - sigma_1(n).
  • A092349 (program): a(n) = sigma_3(n) - sigma_2(n).
  • A092350 (program): a(n) = sigma_3(n) - sigma_2(n) - sigma_1(n).
  • A092352 (program): G.f.: (1+3*x^3)/((1-x)^2*(1-x^3)^2).
  • A092353 (program): Expansion of (1+x^3)/((1-x)^2*(1-x^3)^2).
  • A092364 (program): a(n) = n^2*binomial(n,2).
  • A092365 (program): Coefficient of X^2 in expansion of (1 + n*X + n*X^2)^n.
  • A092366 (program): Coefficient of x^n in expansion of (1+n*x+n*x^2)^n.
  • A092370 (program): Triangle read by rows: T(n,k)=(1/2)*C(n+k,k)*C(n,n-k).
  • A092371 (program): Triangle read by rows: T(n,k)=C(n,k)*C(n+k,n-k).
  • A092383 (program): Sum of digits of n if n odd, else sum of digits of 2n.
  • A092384 (program): Sum of digits of n if n even, else sum of digits of 2n.
  • A092387 (program): a(n) = Fibonacci(2*n+1) + Fibonacci(2*n-1) + 2.
  • A092390 (program): a(n) = prime(n) + prime(2n).
  • A092391 (program): a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight of n.
  • A092392 (program): Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.
  • A092393 (program): Triangle read by rows: T(n,k) = (n+k)*binomial(n,k) (for k=0,…,n-1).
  • A092396 (program): Row 2 of array in A288580.
  • A092400 (program): Fixed point of the morphism 1 -> 1121211, 2 -> 1121212121211, starting from a(1) = 1.
  • A092401 (program): List of pairs n, 3n, where n is the least unused number so far.
  • A092402 (program): Primes of the form p+8 where p is a prime.
  • A092403 (program): a(n) = sigma(n) + sigma(n+1).
  • A092404 (program): phi(n)+phi(n+1).
  • A092405 (program): a(n) = tau(n) + tau(n+1), where tau(n) = A000005(n), the number of divisors of n.
  • A092406 (program): a(1)=1, a(n) = sigma(n) if sigma(n) >= a(n-1), otherwise a(n) = a(n-1) + sigma(n).
  • A092410 (program): a(n) = moebius(n)+moebius(n+1).
  • A092411 (program): a(n) = sigma(n,2) + sigma(n+1,2).
  • A092412 (program): Fixed point of the morphism 0->11, 1->12, 2->13, 3->10, starting from a(1) = 1.
  • A092420 (program): a(n+2) = 9*a(n+1) - a(n) + 1, with a(1)=1, a(2)=10.
  • A092425 (program): Decimal expansion of Pi^4.
  • A092426 (program): Decimal expansion of e^4.
  • A092428 (program): Numbers n such that A092255(n) == 2 (mod 3).
  • A092431 (program): Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.
  • A092433 (program): Positive numbers from the children’s game “Buzz” or “Sevens”: positive integers which are divisible by seven, or which contain a seven as a digit.
  • A092434 (program): Number of words X=x(1)x(2)x(3)…x(n) of length n in three digits {0,1,2} that are invariant under the mapping X -> Y, where y(i)=((AD)^(i-1))x(1) and where (AD) denotes the absolute difference (AD)x(i)=abs(x(i+1)-x(i)) (in other words, y(i) is the i-th element in the diagonal of leading entries in the table of absolute differences of {x(1), x(2),…,x(n)).
  • A092435 (program): Prime factorials divided by their corresponding primorials.
  • A092436 (program): a(n) = 1/2 + (-1)^n*(1/2 - A010060(floor(n/2))).
  • A092438 (program): Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
  • A092439 (program): Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
  • A092440 (program): a(n) = 2^(2n+1) - 2^(n+1) + 1.
  • A092441 (program): Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
  • A092442 (program): Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
  • A092443 (program): Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
  • A092444 (program): a(n+1) = 11*a(n) - a(n-1) - 3, a(0)=a(1)=1.
  • A092459 (program): Numbers that are not Catalan numbers (A000108).
  • A092460 (program): Numbers that are not Bell numbers (A000110).
  • A092462 (program): n times the coefficient of x^n in log[1 + sum(k>=0, x^2^k)].
  • A092463 (program): Greatest number in the n-th successive group of natural numbers containing exactly n prime powers.
  • A092464 (program): Numbers n congruent to 4 or 9 mod 13.
  • A092467 (program): a(n) = Sum_{i+j+k=n, 0<=i,j,k<=n} (n+2k)!/(i! * j! * (3*k)!).
  • A092471 (program): a(n) = Sum_{i+j+k=n, 0<=i<=n, 0<=j<=n, 0<=k<=n} (n+i+j)!/((i+j)! * j! * k!).
  • A092472 (program): a(n)=sum(i+j+k=n,(2n)!/(i+j)!/(j+k)!/(k+i)!) 0<=i<=n, 0<=j<=n, 0<=k<=n.
  • A092474 (program): a(n) is the first term in a sequence of primes such that a(n)+4m^2 is also prime for m = 1 to n.
  • A092476 (program): Numbers that are congruent to {1, 3, 9} mod 13.
  • A092477 (program): Triangle read by rows: T(n,k) = (2^k - 1)^n, 1<=k<=n.
  • A092482 (program): Sequence contains no 3-term arithmetic progression, other than its initial terms 1, 2, 3.
  • A092483 (program): a(n) is the floor of the average of the 1st moment of all previous entries.
  • A092486 (program): Take natural numbers, exchange first and third quadrisection.
  • A092489 (program): Arises in enumeration of 321-hexagon-avoiding permutations.
  • A092490 (program): a(n) = A058094(n) - 3*A058094(n-1) + A058094(n-2) for n >=4.
  • A092491 (program): a(n) = 2*A058094(n-2) - 5*A058094(n-3) + A058094(n-4) + a(n-1) for n >=4.
  • A092492 (program): Arises in enumeration of 321-hexagon-avoiding permutations.
  • A092493 (program): a(n) = 4a(n-1) - 4a(n-2) + 3a(n-3) + a(n-4) - a(n-5).
  • A092494 (program): a(n) = Sum_{p prime and p<=n} ceiling(n/p).
  • A092495 (program): Least factorial multiple of n.
  • A092498 (program): G.f.: (1+x+2x^2)/((1-x)^3*(1-x^3)).
  • A092499 (program): Chebyshev polynomials S(n-1,21) with Diophantine property.
  • A092502 (program): First derivative of cyclotomic(n,x) evaluated at x=1.
  • A092503 (program): a(n) = n^floor(n/2).
  • A092504 (program): a(n) = prime(n) + prime(n^2).
  • A092505 (program): a(n) = A002430(n) / A046990(n).
  • A092508 (program): G.f.: (1+x^18)/((1-x)*(1-x^4)*(1-x^8)*(1-x^12)).
  • A092509 (program): Möbius transform of sequence A008475.
  • A092510 (program): Difference between smallest semiperimeter (see A063655) and its integer log (see A001414).
  • A092514 (program): Decimal expansion of e^(1/5).
  • A092515 (program): Decimal expansion of e^(1/6).
  • A092516 (program): Decimal expansion of e^(1/7).
  • A092517 (program): Product of tau values for consecutive integers.
  • A092520 (program): Number of square divisors of n-th cube: a(n) = A046951(n^3).
  • A092521 (program): a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
  • A092523 (program): Number of distinct prime factors of n-th odd number.
  • A092525 (program): To binary representation of n, append as many ones as there are trailing zeros.
  • A092526 (program): Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.
  • A092530 (program): a(0) = 0; for n > 0, a(n) = T(n) + k where T(n) is the n-th triangular number (A000217) and k (see A026741) is the smallest positive number such that a(n) is divisible by n.
  • A092531 (program): Expansion of (1+x^10)/((1-x)*(1-x^4)^2*(1-x^8)).
  • A092532 (program): G.f.: 1/((1-x)*(1-x^4)*(1-x^8)).
  • A092533 (program): G.f.: (1+x^8)/((1-x)*(1-x^4)).
  • A092534 (program): Expansion of (1-x+x^2)*(1+x^4)/((1-x)^2*(1-x^2)).
  • A092535 (program): G.f.: (1+x^2)*(1+x^3)/((1-x)*(1-x^2)).
  • A092536 (program): Sorted numbers of edges in the Archimedean polyhedra.
  • A092537 (program): Sorted numbers of faces in the Archimedean polyhedra.
  • A092538 (program): Sorted numbers of vertices in the Archimedean polyhedra.
  • A092539 (program): Binary representation of a(n) equals first n+1 terms of A051023.
  • A092542 (program): Table below read by antidiagonals alternately upwards and downwards.
  • A092543 (program): Table below read by antidiagonals alternately upwards and downwards.
  • A092550 (program): Expansion of -x*(1+x+x^2+x^4)/(-1+2*x^3+x^6).
  • A092552 (program): Let X_{m,n}(q) be the chromatic polynomial of the complete bipartite graph K_{m,n}. Then a(n) is the negative of the coefficient of the linear term of X_{n,n}(q).
  • A092553 (program): Decimal expansion of 1/e^2.
  • A092554 (program): Decimal expansion of e^(-3).
  • A092555 (program): Decimal expansion of e^(-4).
  • A092557 (program): Triangle read by rows: T(1,1) = 1; for n>=2, write the first n^2 integers in an n X n array beginning with 1 in the upper left proceeding left to right and top to bottom; then T(n,k) is the last prime in the k-th row.
  • A092560 (program): Decimal expansion of e^(-5).
  • A092563 (program): Coefficients in asymptotic expansion of I_0(x)sqrt(2*Pi*x)/e^x in powers of 1/(16x).
  • A092569 (program): Permutation of integers a(a(n)) = n. In binary representation of n, transformation of inner bits, 1 <-> 0, gives binary representation of a(n).
  • A092572 (program): Numbers of the form x^2 + 3y^2 where x and y are positive integers.
  • A092573 (program): Number of solutions to x^2 + 3y^2 = n in positive integers x and y.
  • A092577 (program): Decimal expansion of e^(-6).
  • A092578 (program): Decimal expansion of e^(-7).
  • A092579 (program): A sieve using the Fibonacci sequence over the integers >=2. Any multiple of a Fibonacci number, F(n)*m, such that F(n)>=2 and m>=2 is excluded and what is left is included.
  • A092582 (program): Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.
  • A092590 (program): a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.
  • A092592 (program): a(n) = A001142(n)/A002944(n), i.e., the product of C(n,j) binomial coefficients (for j=0..n) is divided by the least common multiple of them.
  • A092594 (program): Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding both the 132- and the 231-pattern is equal to k.
  • A092596 (program): Natural numbers n for which sum of decimal digits is greater than n/2.
  • A092597 (program): Natural numbers n for which sum of decimal digits is not smaller than n/3.
  • A092598 (program): Natural numbers n for which sum of decimal digits is greater than n/4.
  • A092604 (program): Complement of A013939.
  • A092605 (program): Decimal expansion of e^(-1/2) or 1/sqrt(e).
  • A092606 (program): Fixed point of the morphism 0 -> 021, 1 -> 0, 2 -> 0; starting with a(1) = 0.
  • A092607 (program): Length of longest contiguous block of ones in binary representation of n!.
  • A092609 (program): Product of first n primes that end in 1.
  • A092610 (program): Product of first n primes that end in 3.
  • A092611 (program): Product of first n primes that end in 9.
  • A092612 (program): Product of first n primes that end in 7.
  • A092615 (program): Decimal expansion of e^(-1/3).
  • A092616 (program): Decimal expansion of e^(-1/4).
  • A092618 (program): Decimal expansion of e^(-1/5).
  • A092619 (program): Numbers with property that number of prime digits is prime.
  • A092620 (program): Numbers with exactly one prime digit.
  • A092624 (program): Numbers with exactly two prime digits.
  • A092625 (program): Numbers with exactly three prime digits.
  • A092629 (program): Number of prime digits is nonprime.
  • A092634 (program): a(n) = 1 - Sum_{k=2..n} k*k!.
  • A092668 (program): Bisection of A000011.
  • A092673 (program): a(n) = moebius(n) - moebius(n/2) where moebius(n) is zero if n is not an integer.
  • A092674 (program): Derived from a(n)=binomial(n+1,2) - sum{i=1,n-1,a(i)*floor(n/i)} (see A000010) - this is the value of the constant term.
  • A092684 (program): First column and main diagonal of triangle A092683, in which the convolution of each row with {1,1} produces a triangle that, when flattened, equals the flattened form of A092683.
  • A092690 (program): Row sums of triangle A092689, which is related to the central trinomial coefficients (A002426).
  • A092691 (program): a(n) = n! * Sum_{k=1..floor(n/2)} 1/(2k).
  • A092692 (program): Expansion of e.g.f. -log(1-x)/(1-x^2).
  • A092693 (program): Sum of iterated phi(n).
  • A092694 (program): Product of iterated phi(n).
  • A092695 (program): Number of positive integers less than or equal to n which are not divisible by the primes 2,3,5,7.
  • A092723 (program): Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.
  • A092724 (program): Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.
  • A092727 (program): Decimal expansion of e^(-1/6).
  • A092731 (program): Decimal expansion of Pi^5.
  • A092732 (program): Decimal expansion of Pi^6.
  • A092736 (program): Decimal expansion of Pi^8.
  • A092738 (program): Primes of the form prime(x)+prime(x+1)+1.
  • A092742 (program): Decimal expansion of 1/Pi^2.
  • A092749 (program): a(n) is the least k such that m^2 + m + k is prime for m = 0..n.
  • A092750 (program): Decimal expansion of e^(-1/7).
  • A092754 (program): a(1)=1, a(2n)=2a(n)+1, a(2n+1)=2a(n)+2.
  • A092755 (program): Partial sums of A000195 (floor(log(n))).
  • A092756 (program): Partial sums of round(exp(n)).
  • A092757 (program): Partial sums of round(log_2(n)).
  • A092759 (program): a(n) = prime(n)^7.
  • A092763 (program): a(n) = floor(3^n / n).
  • A092765 (program): Consider the 1-D random walk with jumps to next-nearest neighbors. Sequence gives number of paths of length n ending at origin.
  • A092769 (program): Squares of A006450: a(n) = prime(prime(n))^2.
  • A092770 (program): Cubes of A006450: a(n) = prime(prime(n))^3.
  • A092771 (program): Prime(prime(n))^2-1.
  • A092772 (program): (Prime(prime(n))^2-1)/24.
  • A092773 (program): Prime(prime(n))^2+1)/2.
  • A092774 (program): Prime(prime(n))^2+1
  • A092775 (program): (prime(prime(n))^4-1)/120.
  • A092778 (program): Concatenate pairs of successive Fibonacci numbers.
  • A092779 (program): Exponent of 2 in central fibonomial coefficient A003267.
  • A092780 (program): Sum(prime(k),k=1..n)^2-1.
  • A092782 (program): The ternary tribonacci word; also a Rauzy fractal sequence: fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 1, starting from a(1) = 1.
  • A092784 (program): a(n) = round(n*(Pi - 2)).
  • A092785 (program): a(n) = sum(sum(binomial(j-n-1,m),m=0..n),j=0..n).
  • A092787 (program): Primes in the sequence A005349 - 1.
  • A092788 (program): USUP perfect numbers.
  • A092790 (program): a(n) = (n+1)*phi(n-1)/2.
  • A092794 (program): Number of connected relations.
  • A092803 (program): Expansion of (1-5x)/((1-2x)(1-6x)).
  • A092804 (program): Expansion of (1+10x)/((1-1000x^3)).
  • A092805 (program): Expansion of (1+10x)/((1-x)(1-1000x^3)).
  • A092806 (program): Expansion of (1 + 8x - 9x^2)/(1 - 3x + 3x^2 - 1001x^3).
  • A092807 (program): Expansion of (1-6*x+4*x^2)/((1-2*x)*(1-6*x)).
  • A092808 (program): Pair reversal of Jacobsthal numbers.
  • A092809 (program): Expansion of (1+x-x^2) / ((1-x^2)*(1-4*x^2)).
  • A092810 (program): Binomial transform of a Jacobsthal trisection.
  • A092811 (program): Expansion of (1-4*x)/(1-8*x).
  • A092812 (program): Number of closed walks of length 2*n on the 4-cube.
  • A092813 (program): Schmidt’s problem sum for r = 3.
  • A092814 (program): Schmidt’s problem sum for r = 4.
  • A092815 (program): Schmidt’s problem sum for r = 5.
  • A092820 (program): a(0) = 1; for n>0, a(n) = 16 times sum of cubes of divisors of n.
  • A092822 (program): Row sums of A092821.
  • A092841 (program): Numerator of I(n) = Integral_{x=0..1/(4^n)} (1-sqrt(x))^2 dx; e.g., I(3) = 323/24576. The denominator is b(n) = 96*16^(n-1); e.g., b(3) = 24576.
  • A092843 (program): a(n) = Sum_{k|n, k>1} phi(k-1), where phi() is the Euler phi function.
  • A092844 (program): a(n) = Sum_{k=1..n} prime(k)*10^(k-1).
  • A092845 (program): A011545(n) reversed.
  • A092848 (program): Expansion of reciprocal of Hauptmodul for Gamma_0(18).
  • A092855 (program): Representation of sqrt(2) - 1 by an infinite sequence.
  • A092858 (program): “Sum” of the sequences of primes and the triangular numbers (A000217).
  • A092860 (program): “3 times the prime sequence”.
  • A092869 (program): Series expansion of the Ramanujan-Goellnitz-Gordon continued fraction.
  • A092870 (program): Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.
  • A092877 (program): Expansion of (eta(q^4) / eta(q))^8 in powers of q.
  • A092879 (program): Triangle of coefficients of the product of two consecutive Fibonacci polynomials.
  • A092880 (program): Number of ordered 2-multiantichains on an n-set.
  • A092886 (program): Expansion of x/(x^4-x^3-2x^2-x+1).
  • A092896 (program): Related to random walks on the 4-cube.
  • A092897 (program): Expansion of (1-x-x^2-3*x^3) / ((1+x)^2*(1-3*x)).
  • A092898 (program): Expansion of (1 - 4*x + 4*x^2 - 4*x^3)/(1 - 4*x).
  • A092899 (program): Expansion of (1+2x+3x^2+6x^3)/((1+x)(1-x)^2).
  • A092900 (program): A Jacobsthal sequence (A078008) to base 4.
  • A092904 (program): Number of decimal digits in the denominator of the Bernoulli number B(2n).
  • A092906 (program): Number of iterations of the sine function to be less than 1/n with an initial argument of Pi/2 radians.
  • A092908 (program): Primes in A051022.
  • A092909 (program): Interpolate 0’s between each pair of digits of n-th prime.
  • A092910 (program): a(n) is the (3n+2)-th component of the continued fraction for sum(k>=0,2^(-k!)).
  • A092915 (program): a(n) = largest k such that n divides (n-1)!/k!, or 0 if no such k exists (i.e., if n is prime).
  • A092919 (program): Partial sums of A000193 (round(log(n))).
  • A092923 (program): Number of permutations containing exactly one occurrence of the pattern #, with # one of {1-23, 3-21, 12-3, 32-1}.
  • A092929 (program): n-th partial arithmetic mean is equal to the n-th noncomposite number or to prime(n-1) for n>1.
  • A092930 (program): n-th partial arithmetic mean is equal to the n-th composite number.
  • A092935 (program): a(1) = 1; a(n) = floor {(n+1)(n+2)(n+3)…(n+k)}/{(n-1)(n-2)(n-3)…(n-k)} for the least value of k.
  • A092936 (program): Area of n-th triple of hexagons around a triangle.
  • A092940 (program): a(n) = largest prime p such that 2*prime(n) - p is prime.
  • A092942 (program): A Fibonacci sequence with “corrections” at every third step: -++-++-++-++-++…, i.e., at every 3rd step there is a subtraction instead of an addition.
  • A092949 (program): Numbers of the form prime(n+1) + prime(n) + 1.
  • A092956 (program): a(n) = (2*n+2)!/((n+2)*n!).
  • A092958 (program): a(1) = 1, a(2) = (1+2)*(2+3), a(3) = (1+2+3)*(2+3+4)*(3+4+5), … etc. Or a(n) = (T(n))*(T(n)+n)*(T(n)+2n)*(T(n)+3n)*… n terms. where T(n) = n(n+1)/2 given by A000217.
  • A092966 (program): Number of interior balls in a truncated tetrahedral arrangement.
  • A092968 (program): Numbers n such that 2n^2 + 11 is a prime.
  • A092975 (program): Consider all partitions of n into parts all of which are divisors of n; a(n) = maximal product of parts.
  • A092978 (program): (Product of first n even numbers)/(product of first k odd numbers) where k is chosen to give the least integer.
  • A092979 (program): Least k such that (n+1)*(n+2)*…*(k-1)*k >= n!.
  • A092983 (program): Least squarefree number > n!.
  • A092984 (program): a(n) = the least k >= 1 such that n! + k is squarefree.
  • A092985 (program): a(n) is the product of first n terms of an arithmetic progression with the first term 1 and common difference n.
  • A092996 (program): Least k such that 1 < p < n < c < k, where p is a prime and c is a composite number such that for every p there exists a distinct c.
  • A092998 (program): Least integer k > n such that the number of primes between 1 and n (exclusive) is the same as the number of primes between n and k (exclusive).
  • A093000 (program): Least k such that Sum_{r=n+1..k} r >= n!.
  • A093001 (program): Least k such that Sum_{r=n+1..k} r is greater than or equal to the sum of the first n positive integers (i.e., the n-th triangular number, A000217(n)). Or, least k such that (sum of first n positive integers) <= (sum of numbers from n+1 up to k).
  • A093003 (program): Number of composite numbers among next n numbers.
  • A093005 (program): a(n) = n * ceiling(n/2).
  • A093007 (program): First nonprime number reached when iterating n under x->2*x+1.
  • A093012 (program): Numbers k such that prime(k) == 3 (mod 16).
  • A093013 (program): Numbers n such that 10*n-3 and 10*n+3 are both primes.
  • A093030 (program): Largest palindromic divisor of n.
  • A093033 (program): Number of interior balls in a truncated octahedral arrangement.
  • A093039 (program): Sequence resulting from a sum of three repeated binomial(n+3,4) sequences.
  • A093040 (program): Expansion of (1+x)/((1+x+x^2)(1-x-x^2)).
  • A093041 (program): Expansion of (1-4x+6x^2-3x^3)/(1-5x+9x^2-8x^3+4x^4).
  • A093042 (program): Jacobsthal(n)*Fibonacci(n).
  • A093043 (program): Jacobsthal(n)*Fibonacci(n-1).
  • A093044 (program): A Jacobsthal Fibonacci product: a(n) = (2^n + 2*(-1)^n)*Fibonacci(n-1)/3.
  • A093045 (program): 2*Jacobsthal(n-1)*Fibonacci(n).
  • A093048 (program): a(n) = n minus exponent of 2 in n, with a(0) = 0.
  • A093049 (program): n-1 minus exponent of 2 in n, a(0) = 0.
  • A093050 (program): Exponent of 2 in (3^n-3)*2^(n-1).
  • A093051 (program): Exponent of 2 in (3^n-3)*2^n.
  • A093052 (program): Exponent of 2 in 6^n - 2^n.
  • A093057 (program): Triangle T(j,k) read by rows, where T(j,k) = number of matrix elements remaining at fixed position in the in-situ transposition of a rectangular j X k matrix (singleton cycles).
  • A093061 (program): 6 * Sum_{d|n} (d mod 3).
  • A093064 (program): Decimal expansion of (4 + 3*log(3))/20.
  • A093069 (program): a(n) = (2^n + 1)^2 - 2.
  • A093070 (program): Decimal expansion of 128/(45*Pi).
  • A093074 (program): Greatest prime factor of n and its direct neighbors.
  • A093083 (program): Partial sums of digits of decimal expansion of golden ratio phi.
  • A093085 (program): Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
  • A093101 (program): Cancellation factor in reducing Sum_{k=0…n} 1/k! to lowest terms.
  • A093103 (program): a(1)=1, a(2)=8, a(n+2) = 8*a(n+1) + 21*a(n).
  • A093112 (program): a(n) = (2^n-1)^2 - 2.
  • A093117 (program): a(1)=1, a(2)=15, a(n+2) = 8*a(n+1) + 21*a(n).
  • A093119 (program): Number of convex polyominoes with a 3 X n+1 minimal bounding rectangle.
  • A093121 (program): A Jacobsthal Fibonacci product.
  • A093122 (program): a(n) = Jacobsthal(n) * Fibonacci(n+1).
  • A093123 (program): Third binomial transform of Fib(3n-1) (A015448).
  • A093128 (program): Number of dissections of a polygon using strictly disjoint diagonals.
  • A093129 (program): Binomial transform of Fibonacci(2n-1) (A001519).
  • A093130 (program): Third binomial transform of Fibonacci(3n).
  • A093131 (program): Binomial transform of Fibonacci(2n).
  • A093132 (program): Third binomial transform of Fibonacci(3n+2).
  • A093133 (program): Third binomial transform of Fib(3n-3) divided by 2.
  • A093134 (program): A Jacobsthal trisection.
  • A093135 (program): Expansion of (1-8*x)/((1-x)*(1-10*x)).
  • A093136 (program): Expansion of (1 - 8*x)/(1 - 10*x).
  • A093137 (program): Expansion of (1-7*x)/((1-x)(1-10*x)).
  • A093138 (program): Expansion of (1-7x)/(1-10x).
  • A093140 (program): Expansion of (1-6*x)/((1-x)*(1-10*x)).
  • A093141 (program): Expansion of (1-6x)/(1-10x).
  • A093142 (program): Expansion of (1-5x)/((1-x)(1-10x)).
  • A093143 (program): Expansion of (1-5*x)/(1-10*x).
  • A093144 (program): Third binomial transform of Pell(3*n)/Pell(3).
  • A093145 (program): Third binomial transform of Fibonacci(3n)/Fibonacci(3).
  • A093146 (program): Third binomial transform of Pell(3n-1).
  • A093147 (program): Third binomial transform of Pell(3n+1).
  • A093148 (program): a(n) = gcd(Fibonacci(n+5), Fibonacci(n+1)).
  • A093149 (program): a(1) = 4; a(n) = (n^(n+1)+2*n-3)/(n-1) for n > 1.
  • A093150 (program): Absolute value of difference between (sum of digits of n if n odd, otherwise sum of digits of 2n) and (sum of digits of n if n even, otherwise sum of digits of 2n).
  • A093158 (program): Reduced numerators of the raw moments of the distribution of areas for triangles picked at random in a unit square.
  • A093160 (program): Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q.
  • A093175 (program): Tetranacci numbers starting with first four squares.
  • A093178 (program): If n is even then 1, otherwise n.
  • A093190 (program): Array t read by antidiagonals: number of {112,212}-avoiding words.
  • A093191 (program): Primes congruent to 4 mod 13.
  • A093194 (program): Minimal values of m=a^2+b^2=c^2+d^2 for each x=a+b+c+d=6*p (p=any odd prime).
  • A093197 (program): Number of labeled plane 2-trees on n triangles.
  • A093198 (program): Number of 4 X 4 symmetric magic squares with line sum 2n.
  • A093220 (program): a(n) is the largest number such that all of a(n)’s length-n substrings are distinct and divisible by 20.
  • A093230 (program): a(n) is the largest number such that all of a(n)’s length-n substrings are distinct and divisible by 30.
  • A093260 (program): a(n) is the largest number such that all of a(n)’s length-n substrings are distinct and divisible by 60.
  • A093270 (program): a(n) is the largest number such that all of a(n)’s length-n substrings are distinct and divisible by 70.
  • A093275 (program): a(n) is the largest number such that all of a(n)’s length-n substrings are distinct and divisible by 75.
  • A093280 (program): a(n) is the largest number such that all of a(n)’s length-n substrings are distinct and divisible by 80.
  • A093302 (program): a(n) = (a(n-1)+(2n-1))*(2n) with a(0) = 0.
  • A093303 (program): a(n) = a(n-1)*(2n-1) + 2n with a(0)=0.
  • A093304 (program): ((Cumulative sum A000045) + (A000079)) - A092176.
  • A093305 (program): Number of binary necklaces of length n with no subsequence 000.
  • A093308 (program): a(n) = Fibonacci(prime(prime(n))).
  • A093317 (program): Consider numbers n such that mu(n) = mu(n+1), A064148; sequence gives values of mu(A064148(n)).
  • A093318 (program): d(n) = number of positive divisors k of n where mu(k) = 1 and mu(n/k) = -1.
  • A093328 (program): a(n) = 2*n^2 + 3.
  • A093331 (program): Number of ternary necklaces of length n with no subsequence 00.
  • A093335 (program): a(0) = 0, a(1) = 1 and for n >= 0, a(n+2) = int(4 * a(n) * a(n+1) / (a(n) + a(n+1))).
  • A093336 (program): Second digit of prime(n).
  • A093337 (program): Penultimate digits of the primes.
  • A093338 (program): Scan primes, write down initial digit if it is a prime.
  • A093344 (program): a(n) = n! * Sum_{i=1..n} (1/i)*Sum_{j=0..i-1} 1/j!.
  • A093345 (program): a(n) = n! * {1 + Sum[i=1..n, 1/i*Sum(j=0..i-1, 1/j!)]}.
  • A093347 (program): A 3-fractal “castle” starting with 0.
  • A093348 (program): A 5-fractal “castle” starting with 0.
  • A093349 (program): A 7-fractal “castle” starting with 0.
  • A093350 (program): Primes congruent to 6 mod 13.
  • A093353 (program): a(n) = (n + n mod 2)*(n + 1)/2.
  • A093356 (program): Number of occurrences of pattern 1-2 after n iterations of morphism A007413.
  • A093357 (program): Number of occurrences of pattern 2-1 after n iterations of morphism A007413.
  • A093359 (program): Primes of the form 28n + 1.
  • A093360 (program): a(n) = prime(n)^(n-1).
  • A093361 (program): Add/multiply sequence, see example.
  • A093367 (program): Number of n-bead necklaces using exactly three colors with no adjacent beads having the same color.
  • A093374 (program): Number of 1-2-3-avoiding permutations with exactly thrice the 1-3-2 pattern.
  • A093375 (program): Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
  • A093379 (program): Expansion of x(1-2x-2x^2)/((1+x)(1-2x)(1-3x)).
  • A093380 (program): Expansion of (1+4x+x^2-10x^3)/((1-x)(1-x-2x^2)).
  • A093381 (program): Expansion of (1 - 2*x - 3*x^2 - 4*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
  • A093383 (program): One of the 16 sequences illustrating the fact that A093382(2) = 31.
  • A093384 (program): Another of the 16 sequences illustrating the fact that A093382(2) = 31.
  • A093387 (program): a(n) = 2^(n-1) - binomial(n, floor(n/2)).
  • A093390 (program): a(n) = floor(n/9) + floor((n+1)/9) + floor((n+2)/9).
  • A093391 (program): a(n) = floor(n/16) + floor((n+1)/16) + floor((n+2)/16) + floor((n+3)/16).
  • A093394 (program): a(n) is the GCD of n and the product of the anti-divisors of n.
  • A093395 (program): Numerators of n divided by the product of the anti-divisors of n.
  • A093396 (program): Denominators of n divided by the product of the anti-divisors of n.
  • A093406 (program): a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) + a(n-4).
  • A093411 (program): Divide n by the largest factorial that divides it and repeat until an odd number is reached, which will be the result; a(0) = 0.
  • A093412 (program): Triangle read by rows: a(n, k) is the numerator of (n + (n-1) + … + (n-k+1))/(1 + 2 + … + k), 0 < k <= n.
  • A093413 (program): Largest number in row n of A093412.
  • A093414 (program): Row sums of A093412.
  • A093415 (program): Triangle read by rows: a(n, k) is the denominator of (n + (n-1) + … + (n-k+1))/(1 + 2 + … + k), 0 < k <= n.
  • A093417 (program): Row sums of A093415.
  • A093418 (program): Numerator of -3*n + 2*(1+n)*HarmonicNumber(n).
  • A093419 (program): Denominators of row sums in triangle described in A093412.
  • A093421 (program): Triangle read by rows: T(n,k) is the denominator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.
  • A093434 (program): a(n) = Product_{i=1..n} (2*n-i)*(2*n+i).
  • A093445 (program): The triangular triangle.
  • A093446 (program): Largest member of the n-th row of the triangular triangle (A093445).
  • A093448 (program): Rows sums of the triangle A093447.
  • A093450 (program): Number of consecutive integers whose product = A093449(n).
  • A093451 (program): Number of distinct prime divisors of Product_{k=1+(n-1)n/2..n(n+1)/2)} k (i.e., of 1, 2*3, 4*5*6, 7*8*9*10, …).
  • A093453 (program): 1/1, 2*3/lcm(2,3), 4*5*6/lcm(4,5,6), 7*8*9*10/lcm(7,8,9,10), …
  • A093454 (program): a(n) = floor((LCM of next n numbers)/n!).
  • A093457 (program): Product of primes in the range [T(n-1} + 1, T(n - 1) + n], where T(n) is the n-th triangular number.
  • A093460 (program): a(n) = 2n^(n-1) - 1.
  • A093461 (program): a(1)=1, a(n) = 2*(n^(n-1)-1)/(n-1) for n >= 2.
  • A093462 (program): a(1)=1, a(n) = 2(n^(n-1)-1)/(n-1)^2.
  • A093463 (program): a(1) = 1; for n>1, a(n) = n*a(n-1) + 1 if n is a prime else a(n) = n*a(n-1) - 1.
  • A093464 (program): a(1) = 1; for n>1, a(n) = n*a(n-1) - 1 if n is a prime else a(n) = n*a(n-1) + 1.
  • A093467 (program): a(1) = 1, a(2) = 2; for n >= 2, a(n+1) = a(n) + Sum_{i = 1..n} (a(i) - a(1)).
  • A093468 (program): a(1) = 1, a(2) = 2; for n >= 2, a(n+1) = a(n) + Sum {a(n)-a(i), i = 1 to n}.
  • A093471 (program): a(n) = floor(n^(1/2)*10^n).
  • A093476 (program): Index of occurrence of the first 0 bit in binary representation of 3^n.
  • A093479 (program): Number of regular (infinite) apeirotopes of full rank in n-dimensional space.
  • A093480 (program): Numbers n such that n^3-(n-1)^2 is prime.
  • A093485 (program): a(n) = (27*n^2 + 9*n + 2)/2.
  • A093500 (program): a(n) = (15*n^2 + 5*n + 2)/2.
  • A093503 (program): a(0) = 2, a(n) = least prime >= a(n-1) + n.
  • A093505 (program): a(n) = floor(A001969(n)/2 + 1/2).
  • A093509 (program): Multiples of 5 or 6.
  • A093510 (program): Transform of the prime sequence by the Rule30 cellular automaton.
  • A093511 (program): Transform of the prime sequence by the Rule45 cellular automaton.
  • A093512 (program): Transform of the prime sequence by the Rule73 cellular automaton.
  • A093513 (program): Transform of the prime sequence by the Rule89 cellular automaton.
  • A093514 (program): Transform of the prime sequence by the Rule90 cellular automaton.
  • A093515 (program): Numbers k such that either k or k-1 is a prime.
  • A093516 (program): Transform of the prime sequence by the Rule137 cellular automaton.
  • A093517 (program): Transform of the prime sequence by the Rule225 cellular automaton.
  • A093518 (program): Number of ways of representing n as exactly 2 generalized pentagonal numbers.
  • A093519 (program): Numbers n with no representation as the sum of exactly 2 generalized pentagonal numbers.
  • A093521 (program): Runs of 1’s of lengths 1, prime(1), prime(2), prime(3), … separated by 0’s.
  • A093523 (program): Inverse binomial transform of A010054 (1 if triangular number else 0).
  • A093526 (program): Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.
  • A093527 (program): Denominators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.
  • A093528 (program): Numerators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.
  • A093529 (program): Pi*denominators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.
  • A093544 (program): Numerator of (4n-3)/A000265(n). Numerator of pairwise quotients of A004130.
  • A093545 (program): Sorted mapping of A093544 onto the integers.
  • A093555 (program): Number of non-prime-powers between consecutive prime-powers.
  • A093560 (program): (3,1) Pascal triangle.
  • A093561 (program): (4,1) Pascal triangle.
  • A093562 (program): (5,1) Pascal triangle.
  • A093563 (program): (6,1)-Pascal triangle.
  • A093564 (program): (7,1) Pascal triangle.
  • A093565 (program): (8,1) Pascal triangle.
  • A093566 (program): a(n) = n*(n-1)*(n-2)*(n-3)*(n^2-3*n-2)/48.
  • A093567 (program): Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).
  • A093570 (program): a(n) = Product_{k=1..n}(k + prime(k)).
  • A093572 (program): Greatest prime factor of Product(k+prime(k): 1<=k<=n).
  • A093577 (program): Decimal expansion of (3/4)*sqrt(2).
  • A093581 (program): Numerators of odd moments in the distribution of chord lengths for picked at random on the circumference of a unit circle.
  • A093582 (program): Decimal expansion of 3/(2*Pi).
  • A093585 (program): Numerators of even raw moments in the distribution of a triangle picked at random from points on the circumference of a unit circle.
  • A093586 (program): Denominators of even raw moments in the distribution of a triangle picked at random from points on the circumference of a unit circle.
  • A093593 (program): n! times sum of Farey fractions of order n.
  • A093599 (program): Composite numbers having an odd number of prime factors, all of which are distinct.
  • A093600 (program): Numerator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.
  • A093602 (program): Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)).
  • A093605 (program): Numerators of sqrt(2) term in expected number of complex eigenvalues in an n X n real matrix with entries chosen from a standard normal distribution.
  • A093609 (program): Upper Beatty sequence for e^G, G = Euler’s gamma constant.
  • A093610 (program): Lower Beatty sequence for e^G, G = Euler’s gamma constant.
  • A093611 (program): Numerators of convergents to 3/(1 + sqrt(10)).
  • A093613 (program): Table read by rows: row n is the n-th Primary Phyllotaxis Sequence (PPS), which has F(n+1) terms, where F(n) is the n-th Fibonacci number. For 1 <= k < F(n+1), a(n, k) = -k*F(n-1) mod F(n+1). a(n, F(n+1)) = F(n+1).
  • A093616 (program): a(n) = smallest k such that k*n has exactly as many divisors as n^2.
  • A093620 (program): Values of Laguerre polynomials: a(n) = 2^n*n!*LaguerreL(n,-1/2,-2).
  • A093627 (program): a(n) = lcm(1,2,3,…,n) * (sum of Farey fractions of order n).
  • A093641 (program): Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.
  • A093644 (program): (9,1) Pascal triangle.
  • A093645 (program): (10,1) Pascal triangle.
  • A093646 (program): Higher dimensional figurate numbers based on 12-gonal numbers A051624.
  • A093652 (program): Let a(1) = 1, a(2) = 2, a(3) = 7, a(4) = 15 and for n >= 5 set a(n) = (n*b(n) - b(n-2)) / 2, where b(n) = 4*b(n-2) - b(n-4) for n >= 5 and b(1) = 1, b(2) = 2, b(3) = 5, b(4) = 8.
  • A093653 (program): Total number of 1’s in binary expansion of all divisors of n.
  • A093658 (program): Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].
  • A093659 (program): First column of lower triangular matrix A093658; factorial of the number of 1’s in binary expansion of n.
  • A093660 (program): Row sums of lower triangular matrix A093658.
  • A093661 (program): Partial sums of A093660.
  • A093667 (program): a(n) = prime(n) - prime(n+1) + prime(n+2).
  • A093688 (program): Numbers m such that all divisors of m, excluding the divisor 1, have an even number of 1’s in their binary expansions.
  • A093695 (program): Number of one-element transitions among partitions of the integer n for unlabeled parts.
  • A093696 (program): Numbers n such that all divisors of n have an odd number of 1’s in their binary expansions.
  • A093697 (program): Least k so that n! >= primorial(k).
  • A093699 (program): Maximum number of odd 2 X 2 submatrices over all 2n X 2n (0,1) matrices.
  • A093701 (program): a(n) = smallest m>a(n-1) such that 1+m*n is prime, a(1) = 1.
  • A093702 (program): a(n) = smallest prime p>a(n-1) such that n divides (p-1) and (p-1)/n > (a(n-1)-1)/(n-1), a(1) = 2.
  • A093704 (program): Value of initial digit of n in Roman numeral representation.
  • A093707 (program): Numbers n such that n^3 + (n-1)^2 is a prime.
  • A093709 (program): Characteristic function of squares or twice squares.
  • A093712 (program): Repeatedly subtract largest prime from n until either a prime or 1 remains.
  • A093718 (program): a(n) = (n mod 3)^(n mod 2).
  • A093719 (program): a(n) = (n mod 2)^(n mod 3).
  • A093722 (program): Integers of the form (n^2 - 1) / 120.
  • A093762 (program): Numerators of 1-2*HarmonicNumber(n)/(n+1).
  • A093766 (program): Decimal expansion of Pi/(2*sqrt(3)).
  • A093768 (program): Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.
  • A093783 (program): Sum of digits of n in Roman numeral representation.
  • A093785 (program): Numbers that are divisible by every digit in their Roman numeral representation.
  • A093801 (program): a(n) = b(n)*Integral_{x=0..1/(4^n)} (1 - sqrt(x)) dx, where b(n) = 3*24^n.
  • A093803 (program): Greatest odd proper divisor of n; a(1)=1.
  • A093809 (program): a[n] =a[n-1] + 2*n*Prime[n]-n^2
  • A093811 (program): Sum of the digital products of the divisors of n.
  • A093818 (program): a(n) = gcd(A001008(n), n!).
  • A093819 (program): Algebraic degree of sin(2*Pi/n).
  • A093820 (program): a(n) = Sum_{k=1..n-1} gcd(n, a(k)) for n > 1; a(1) = 1.
  • A093821 (program): Decimal expansion of (2*(3 - sqrt(3)))/3.
  • A093825 (program): Decimal expansion of Pi/(3*sqrt(2)).
  • A093828 (program): Decimal expansion of (3*Pi)/8.
  • A093829 (program): Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.
  • A093833 (program): 3^n-Jacobsthal(n).
  • A093834 (program): Expansion of (1-2x)^2/((1-3x)(1-4x)).
  • A093835 (program): n*Jacobsthal(n).
  • A093836 (program): Numerator of A000328(n)/n^2, where A000328(n) is the number of lattice points (x,y) with x^2 + y^2 <= n^2.
  • A093837 (program): Denominator of N(n)/n^2, where N(n) is the number of lattice points (x,y) with x^2 + y^2 <= n^2.
  • A093838 (program): Primes of the form 36n + 1.
  • A093847 (program): First column of triangle A093846.
  • A093851 (program): a(n) = A002283(n-1) + floor(A052268(n)/(1+n)).
  • A093856 (program): a(0)=1; a(1)=2; a(n+1) = 2*n*a(n) - a(n-1) for n >= 1.
  • A093858 (program): a(0) = 1, a(1)= 2, a(n) = (a(n+1) - a(n-1))/n, or a(n+1) = n*a(n) + a(n-1).
  • A093859 (program): Numbers n such that sum of the digital products of the divisors is prime.
  • A093866 (program): a(0)=1, then a(n) is the least number such that there are exactly n numbers coprime to a(n-1) between a(n-1) and a(n) (excluded).
  • A093868 (program): Smallest prime that differs from a multiple of n by unity.
  • A093871 (program): a(n) is the n-th prime = -1 (mod n).
  • A093873 (program): Numerators in Kepler’s tree of harmonic fractions.
  • A093875 (program): Denominators in Kepler’s tree of harmonic fractions.
  • A093879 (program): First differences of A004001.
  • A093880 (program): a(n) = lcm(1, 2, …, 2n) / lcm(1, 2, …, n).
  • A093881 (program): Let n! = 2^a*3^b*5^c*7^d…. in canonical form, then a(n) = concatenation a,b,c,d,…
  • A093883 (program): Product of all possible sums of two distinct numbers taken from among first n natural numbers.
  • A093896 (program): Least positive k such that n^n divides k!.
  • A093898 (program): Triangle read by rows: T(n,r) = n^r + r^n (1 <= r <= n).
  • A093905 (program): Triangle read by rows: for 0 <= k < n, a(n, k) is the sum of the products of all subsets of {n-k, n-k+1, …, n} with k members.
  • A093907 (program): Number of elements in the n-th period of the periodic table as predicted by the Aufbau principle.
  • A093915 (program): Triangle with r-th row containing r consecutive integers that sum to the smallest possible proper multiple of A006003(r).
  • A093916 (program): a(2*k-1) = (2*k-1)^2 + 2 - k, a(2*k) = 6*k^2 + 2 - k: First column of the triangle A093915.
  • A093917 (program): a(n) = n^3+n for odd n, (n^3+n)*3/2 for even n: Row sums of A093915.
  • A093918 (program): a(2k-1)=(2k-1)^2+k, a(2k)=6k^2+k+1: Last term in rows of triangle A093915.
  • A093920 (program): Row sums of A093919.
  • A093935 (program): a(1) = 1, a(n+1) = a(n) + n*(a(1) + a(2) + … + a(n)).
  • A093951 (program): Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.
  • A093952 (program): Partition number A000041(n) mod n.
  • A093953 (program): a(n) = rightmost term in M^n * [1,1,1], where M = a 3 X 3 matrix composed of the first 3 rows of A050166 (fill in the matrix with zeros): = [1 0 0 / 1 2 0 / 1 4 5].
  • A093954 (program): Decimal expansion of Pi/(2*sqrt(2)).
  • A093957 (program): A091799(n) - 3.
  • A093958 (program): A091844(n) - 4.
  • A093960 (program): a(1) = 1, a(2) = 2, a(n+1) = n*a(1) + (n-1)*a(2) + … + (n-r)*a(r+1) + … + a(n).
  • A093963 (program): Antidiagonal sums of array in A093966.
  • A093964 (program): a(n) = Sum_{k=1..n} k*k!*C(n,k).
  • A093965 (program): Number of functions of [n] to [n] that simultaneously avoid the patterns 112 and 221.
  • A093966 (program): Array read by antidiagonals: number of {112,221}-avoiding words.
  • A093967 (program): a(n) = n * Pell(n).
  • A093968 (program): Inverse binomial transform of n*Pell(n).
  • A093969 (program): a(n) = n*Pell(n-2).
  • A093985 (program): a(1) = 1, a(2) = 2; a(n+1) = 2n*a(n) - a(n-1). Symmetrically, a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).
  • A093986 (program): a(1) = 1, a(2) = 1, a(n+1) = 2n*a(n) - a(n-1). Symmetrically a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).
  • A093995 (program): n^2 repeated n times, triangle read by rows.
  • A093996 (program): G.f.: Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.
  • A094002 (program): a(n+3) = 3*a(n+2) - 2*a(n+1) + 1 with a(0)=1, a(1)=5.
  • A094012 (program): Expansion of x*(1-6*x+10*x^2)/(1-4*x+2*x^2)^2.
  • A094013 (program): Expansion of (1-4*x)/(1-4*x-4*x^2).
  • A094014 (program): Expansion of (1-2*x)/(1-8*x^2).
  • A094015 (program): Expansion of (1+4*x)/(1-8*x^2).
  • A094021 (program): Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n).
  • A094024 (program): Alternating 1 with one less than the powers of 2.
  • A094025 (program): Expansion of (1+3x)/((1-x^2)(1-3x^2)).
  • A094026 (program): Expansion of x(1+10x)/((1-x^2)(1-10x^2)).
  • A094027 (program): Expansion of x(1+100x)/((1-x^2)(1-100x^2)).
  • A094028 (program): Expansion of 1/((1-x)*(1-100*x)).
  • A094029 (program): Number of n-crossing links with alternating braids of 3 strands.
  • A094031 (program): Number of n-crossing 2 component links with alternating braids of 3 strands.
  • A094033 (program): Number of connected 2-element antichains on a labeled n-set.
  • A094038 (program): Binomial transform of (Pell(-n)+Pell(n))/2.
  • A094039 (program): Binomial transform of (Jacobsthal(n) + 2^n*Jacobsthal(-n))/2.
  • A094040 (program): Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.
  • A094041 (program): Beatty sequence for e^Pi - Pi^e - i^i.
  • A094042 (program): Beatty sequence for e^Pi - Pi^e - i^i.
  • A094047 (program): Number of seating arrangements of n couples around a round table (up to rotations) so that each person sits between two people of the opposite sex and no couple is seated together.
  • A094051 (program): Phi(phi(p))/2 where p = prime(n).
  • A094052 (program): Number of walks of length n between two adjacent nodes in the cycle graph C_7.
  • A094053 (program): Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.
  • A094056 (program): Number of digits in A002860(n) (number of Latin squares).
  • A094058 (program): Positions of ones in binary expansion of e/4.
  • A094061 (program): Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard.
  • A094064 (program): Sequences has the properties shown in the Comments lines.
  • A094074 (program): Coefficients arising in combinatorial field theory.
  • A094075 (program): Denominator of I(n)=integral_{x=0 to 1/n}(x^2-1)^3 dx.
  • A094076 (program): Smallest k such that prime(n)+2^k is prime, or -1 if no such prime exists.
  • A094077 (program): a(1)=1 and, for n>1, a(n)=a(n-1)+n if n is even and a(n)=smallest positive integer which has not yet appeared in the sequence if n is odd.
  • A094081 (program): Smallest integral ladder whose ends slide over the respective distances 1 and m=2n+1 while slipping down along horizontal ground and vertical wall against which it leans.
  • A094083 (program): Numerators of ratio of sides of n-th triple of rectangles of unit area sum around a triangle.
  • A094084 (program): Denominators associated with A094083.
  • A094085 (program): Denominator of (3*2^(n-1) - 1)*integral_{x=0 to 1/(4^n)}1-sqrt x dx.
  • A094088 (program): E.g.f. 1/(2-cosh(x)) (even coefficients).
  • A094091 (program): a(1) = 0; for n>0, a(n) = smaller of 0 and 1 such that we avoid the property that, for some i and j in the range S = 2 <= i < j <= n/2, a(i) … a(2i) is a subsequence of a(j) … a(2j).
  • A094094 (program): Define x[1]…x[n] by the equations Sum_{j=1..n} x[j]^i = i, i=1..n; a(n) = n! * Sum_{j=1..n} x[j]^(n+1).
  • A094102 (program): Triangle read by rows in which row n contains Fib(1), …, Fib(n-1), Fib(n), Fib(n-1), …, Fib(1).
  • A094103 (program): a(n) = sum along n-th diagonal of A094102 (sloping downward to left).
  • A094112 (program): Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-, the 132- and the 231-pattern is equal to k.
  • A094113 (program): Total area of all 1-histograms of length n.
  • A094114 (program): a(n) = -Sum_{i=1..n-1} (-1)^i*2^valuation(i,3).
  • A094115 (program): Partial sums of A093347.
  • A094116 (program): Partial sums of A093348.
  • A094117 (program): Partial sums of A093349.
  • A094120 (program): a(n) = Sum_{k=1..n} Sum_{i=1..k} (-2)^valuation(i,2).
  • A094125 (program): a(n) = 3*2^n + 2*3^n.
  • A094151 (program): Remainder when concatenation 1,2,3,…up to (n-1) is divided by n.
  • A094159 (program): 3 times hexagonal numbers: a(n) = 3*n*(2*n-1).
  • A094160 (program): Column 4 of A048790.
  • A094175 (program): Round( n / sum of digits of n ).
  • A094178 (program): Numbers n such that 4n+1 is divisible only by primes of form 4m+1 (i.e., by the Pythagorean primes A002144).
  • A094179 (program): Numbers congruent to 3 mod 4 which are divisible only by primes congruent to 3 mod 4.
  • A094180 (program): Numbers k such that 4*k-1 is divisible only by primes of form 4*m-1 (i.e., by the Gaussian primes A002145).
  • A094181 (program): a(n) = (n - tau(n))*(n - phi(n)), where tau=A000005 and phi=A000010.
  • A094184 (program): Triangle read by rows in which each term equals the entry above minus the entry left plus twice the entry left-above.
  • A094186 (program): Taking a(1)=0 and a(2)=1, sequence (a(n))n>1 is defined as follows : letting w(k)=a(1)a(2)…a(k) and w(infinity)= limit k ->infinity a(1)a(2)w(1)w(2)…w(k) we have w(infinity)=a(1)a(2)a(3)a(4)…
  • A094188 (program): Number of levels in the compositions of n with odd summands.
  • A094189 (program): Number of primes between n^2-n and n^2 (inclusive).
  • A094192 (program): Values x of the generator pairs (x, y), x>y of primitive Pythagorean triples, sorted.
  • A094194 (program): Hypotenuses x^2 + y^2 of primitive Pythagorean triangles, sorted on values x of the generator pair (x, y), x>y.
  • A094195 (program): G.f.: (1-4*x)/((1-5*x)*(1-x)^2).
  • A094200 (program): a(n) = 16*n^4 + 32*n^3 + 36*n^2 + 20*n + 3.
  • A094201 (program): a(n) = 4*n^5 + 10*n^4 + 13*n^3 + 11*n^2 + 5*n + 1.
  • A094207 (program): a(n) = prime(4n-3) + prime(4n-2) + prime(4n-1) + prime(4n).
  • A094210 (program): Numbers k such that k^2 + 3k + 1 is a prime.
  • A094211 (program): a(n) = Sum_{k=0..n} binomial(7*n,7*k).
  • A094213 (program): a(n) = Sum_{k=0..n} binomial(9*n,9*k).
  • A094214 (program): Decimal expansion of 1/phi = phi - 1.
  • A094218 (program): Number of permutations of length n with exactly 2 occurrences of the pattern 2-13.
  • A094219 (program): Number of permutations of length n with exactly 3 occurrences of the pattern 2-13.
  • A094222 (program): a(n+1) = a(n) + (number of distinct prime factors of a(n)) for n>1; a(1)=1, a(2)=2.
  • A094233 (program): Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.
  • A094247 (program): Expansion of (phi(-q^5)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
  • A094248 (program): Consider 3 X 3 matrix M = [0 1 0 / 0 0 1 / 5 2 0]; a(n) = the center term in M^n * [1 1 1].
  • A094250 (program): Array read by antidiagonals: T(n,k) = (n+2)^(k+1)/(n+1)^2+k+1-(k+1)/(n+1)-1/(n+1)^2, n >= 0, k >= 0.
  • A094252 (program): a(n) = partition(n) mod prime(n).
  • A094253 (program): Let M be the 3 X 3 Matrix [ -4 4 8 / 1 0 0 / 0 1 0], a(n) = absolute value of the center term of M^n * [1 1 1].
  • A094254 (program): Let M be the 3 X 3 matrix [ 6 0 -8 / 1 0 0 / 0 1 0]. Then M^n * [1 1 1] = [a(n-1), a(n), a(n+1)].
  • A094256 (program): Expansion of x / ( (x-1)*(x^3 - 9*x^2 + 6*x - 1) ).
  • A094258 (program): a(1) = 1, a(n+1) = n*n! for n >= 1.
  • A094259 (program): G.f.: (1-5*x)/((1-6*x)*(1-x)^2).
  • A094260 (program): Sum of next n numbers/n if n divides the sum else n times the sum of next n numbers.
  • A094261 (program): a(n) = n(n-1)(n-3)(n-6)…(n-t), where t is the largest triangular number less than n; number of factors in the product is ceiling((sqrt(1+8*n)-1)/2).
  • A094263 (program): Triangle read by rows: for 1 <= k <= n, a(n, k) = n^k mod k.
  • A094264 (program): a(n) = Sum_{r = 1 .. n} (n^r mod r).
  • A094265 (program): Largest number in n-th row of triangle A094263.
  • A094266 (program): LQTL Lean Quaternary Temporal Logic: a terse form of temporal logic created by assigning four descriptors such that false, becoming true, true and becoming false are represented and become a linear sequence. In a branching tree two alternative are open, change or no change. The integer sequence above is the count of the row possibilities of the four states over successive iterations.
  • A094267 (program): First differences of A001511.
  • A094283 (program): Row sums of triangle A094280.
  • A094284 (program): a(n) = A094283(n+1)/A094283(n).
  • A094286 (program): Number of (s(0), s(1), …, s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,…,n, s(0) = 1, s(n) = 1.
  • A094287 (program): Number of (s(0), s(1), …, s(n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| <= 1 for i = 1,2,…,n, s(0) = 1, s(n) = 1.
  • A094290 (program): a(n) = prime(A001511(n)), where A001511 is one more than the 2-adic valuation of n.
  • A094292 (program): Number of (s(0), s(1), …, s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,…,n, s(0) = 2, s(n) = 4.
  • A094294 (program): a(n) = n*a(n-1) - n + 2 for n > 1; a(1)=1.
  • A094297 (program): Number of (s(0), s(1), …, s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,…,n, s(0) = 2, s(n) = 2.
  • A094301 (program): a(0)=0, a(1)=1, a(2)=2; for n>2, a(n) = a(n-1) - a(n-2) + a(n-3)^2.
  • A094303 (program): a(1) = 1, a(2) = 2, and a(n+1) = a(n) * sum of all previous terms up to a(n-1) for n >= 2.
  • A094304 (program): Sum of all possible sums formed from all but one of the previous terms, starting 1.
  • A094305 (program): Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
  • A094306 (program): Number of (s(0), s(1), …, s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,…,n, s(0) = 2, s(n) = 4.
  • A094309 (program): Number of (s(0), s(1), …, s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,…,n, s(0) = 2, s(n) = 5.
  • A094310 (program): Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.
  • A094311 (program): n(1+n^2)((2+n^2)^2-n^2)/16.
  • A094323 (program): a(n) = n*(n+1)*(2n+1)*(3n+1)*(4n+1)/30.
  • A094328 (program): Iterate the map in A006369 starting at 4.
  • A094329 (program): Iterate the map in A006369 starting at 16.
  • A094330 (program): Product of next n numbers divided by n.
  • A094331 (program): Least k such that n! < (n+1)(n+2)(n+3)…(n+k).
  • A094332 (program): Iterate the map in A006368 starting at 12.
  • A094337 (program): a(n) = floor((product of composites among next n numbers)/(product of primes among next n numbers)).
  • A094343 (program): List of pairs of primes (p, q) with q - p = 4.
  • A094345 (program): Sum of all digits in ternary expansions of 0, …, n.
  • A094347 (program): a(n) = 14*a(n-1) - a(n-2); a(0) = a(1) = 2.
  • A094350 (program): Numbers n such that A094291(n) is not a square.
  • A094359 (program): Pair reversal of a Jacobsthal sequence.
  • A094360 (program): Pair reversal of Jacobsthal-Lucas numbers.
  • A094361 (program): Pair-reversal of 1,4,4,16,16…
  • A094364 (program): Expansion of (1-5x)/(1-10x-100x^2).
  • A094366 (program): a(n) is the number of two-generated numerical semigroups whose Frobenius number is 2n-1.
  • A094373 (program): Expansion of (1-x-x^2)/((1-x)*(1-2*x)).
  • A094374 (program): a(n) = (3^n-1)/2 + 2^n.
  • A094375 (program): a(n)=(4^n-2^n)/2+3^n.
  • A094384 (program): Determinant of n X n partial Hadamard matrix with coefficient m(i,j) 1<=i,j<=n (see comment).
  • A094385 (program): Another version of triangular array in A062991 unsigned and transposed : triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 1, 1, 1, 1, 1, …] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, …] where DELTA is the operator defined in A084938.
  • A094386 (program): a(n) = floor(sqrt(3)*2^(n-1)).
  • A094387 (program): Numbers k such that gcd(k, A000120(k)) = 1.
  • A094388 (program): Expansion of (1- 2x - x^2)/((1-x)*(1-3x)).
  • A094389 (program): Last decimal digit of the odd Catalan number A038003(n).
  • A094390 (program): A Beatty sequence using exp(Pi/4).
  • A094391 (program): A Beatty sequence using exp(Pi/4)/(exp(Pi/4)-1).
  • A094403 (program): a(1) = 1; a(n) = (sum of previous terms)^n mod n.
  • A094404 (program): Numerators of low-water marks of mu(n)/n, where mu(n) is A002034.
  • A094405 (program): a(1) = 1; a(n) = (sum of previous terms) mod n.
  • A094407 (program): Primes of the form 16n+1.
  • A094414 (program): Triangle T read by rows: dot product <1,2,…,r> * <s+1,s+2,…,r,1,2,…,s>.
  • A094415 (program): Triangle T read by rows: dot product <r,r-1,…,1> * <s+1,s+2,…,r,1,2,…,s>.
  • A094417 (program): Generalized ordered Bell numbers Bo(4,n).
  • A094418 (program): Generalized ordered Bell numbers Bo(5,n).
  • A094419 (program): Generalized ordered Bell numbers Bo(6,n).
  • A094420 (program): Generalized ordered Bell numbers Bo(n,n).
  • A094421 (program): a(n) = n * (6*n^2 + 6*n + 1).
  • A094423 (program): A045873(n)^2.
  • A094429 (program): Given the 3 X 3 matrix M = [0 1 0 / 0 0 1 / 7 -14 7], a(n) = (-) rightmost term of M^n * [1 1 1].
  • A094430 (program): a(n) is the rightmost term of M^n * [1 0 0], where M is the 3 X 3 matrix [0 1 0 / 0 0 1 / 7 -14 7].
  • A094431 (program): a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [1 -1 0 / -1 4 -3 / 0 -3 3].
  • A094432 (program): a(n) = rightmost term in M^n * [1 0 0]. M = the 3 X 3 stiffness matrix [1 -1 0 / -1 4 -3 / 0 -3 3].
  • A094433 (program): a(n) is the left term in M^n * [1 0 0], M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].
  • A094434 (program): a(n) = rightmost term of M^n * [1 0 0], with M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].
  • A094435 (program): Triangular array read by rows: T(n,k) = Fibonacci(k)*C(n,k), k = 1…n; n>=1.
  • A094436 (program): Triangular array T(n,k) = Fibonacci(k+1)*binomial(n,k) for k = 0..n; n >= 0.
  • A094437 (program): Triangular array T(n,k) = Fibonacci(k+2)*C(n,k), k=0..n, n>=0.
  • A094438 (program): Triangular array T(n,k) = Fibonacci(k+3)*C(n,k), k=0..n, n>=0.
  • A094439 (program): Triangular array T(n,k) = Fibonacci(k+4)*C(n,k), k=0..n, n>=0.
  • A094440 (program): Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k-1), k = 1..n; n >= 1.
  • A094441 (program): Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.
  • A094442 (program): Triangular array T(n,k) = Fibonacci(n+2-k)*C(n,k), 0 <= k <= n.
  • A094443 (program): Triangular array T(n,k) = Fibonacci(n+3-k)*C(n,k), k=0..n, n>=0.
  • A094444 (program): Triangular array T(n,k) = Fibonacci(n+4-k)*C(n,k), k=0..n, n>=0.
  • A094447 (program): Numbers which are the sum of two positive cubes and divisible by 13.
  • A094451 (program): a(n) = A033485(n) modulo 3.
  • A094453 (program): Numbers n with property that binomial (2n, n) / (n+2) is not an integer.
  • A094460 (program): a(n) is the third term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.
  • A094471 (program): a(n) = n*tau(n) - sigma(n) = n*A000005(n) - A000203(n).
  • A094472 (program): a(n) = n*tau(n) - sigma(n) - phi(n), where tau(n) is the number of divisors of n.
  • A094473 (program): Smallest prime factor of 2^n+3^n.
  • A094495 (program): Table of binomial coefficients mod m^2, read by rows: T(m, n) = binomial(m, n) mod m^2.
  • A094497 (program): Triangular table A(n,j) = C(n,j) - C(n,j) mod n^3, difference of binomial coefficient and its residue mod n^3, read by rows.
  • A094500 (program): Least number k such that (n+1)^k / n^k >= 2.
  • A094506 (program): Numerator of I(n) = (-1) * Integral_{x=0..4^n} (1-x^(3/2)) dx.
  • A094517 (program): Primes p such that 6p+11 is not a prime.
  • A094524 (program): Primes of form 3*prime(m) + 2.
  • A094527 (program): Triangle T(n,k), read by rows, defined by T(n,k) = binomial(2*n,n-k).
  • A094531 (program): Array read by rows: right-hand side of triangle A027907 of trinomial coefficients.
  • A094536 (program): Number of binary words of length n that are not “bifix-free”.
  • A094537 (program): A094536/2.
  • A094538 (program): Number of ternary words of length n that are not “bifix-free”.
  • A094539 (program): a(n) = A094538(n)/3.
  • A094540 (program): Last digit of the n-th perfect number.
  • A094545 (program): Number of minimal T_0-covers of an n-set.
  • A094547 (program): A019309(n)/4 for n >= 1.
  • A094550 (program): Numbers n such that there are integers a < b with a+(a+1)+…+(n-1) = (n+1)+(n+2)+…+b.
  • A094551 (program): Numbers n such that there are integers a < b with a+(a+1)+…+(n-1) = n+(n+1)+…+b.
  • A094554 (program): Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).
  • A094555 (program): Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).
  • A094556 (program): Number of walks of length n between opposite vertices on a triangular prism.
  • A094559 (program): Number of words of length n over an alphabet of size 4 that are not “bifix-free”.
  • A094560 (program): Initial decimal digit of Pi*n, Pi=3.1415…
  • A094561 (program): Final decimal digit of floor(Pi*n), Pi=3.1415…
  • A094562 (program): Initial decimal digit of fractional part of Pi*n, Pi=3.1415…
  • A094565 (program): Triangle read by rows: binary products of Fibonacci numbers.
  • A094566 (program): Triangle of binary products of Fibonacci numbers.
  • A094567 (program): Associated with alternating row sums of array in A094566.
  • A094568 (program): Triangle of binary products of Fibonacci numbers.
  • A094569 (program): Associated with alternating row sums of array in A094568.
  • A094570 (program): Triangle T(n,k) read by rows: T(n,k) = F(k) + F(n-k) where F(n) is the n-th Fibonacci number.
  • A094572 (program): Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.
  • A094575 (program): Numbers n with property that binomial (2n, n) / (n-1) is an integer.
  • A094576 (program): Numbers n with property that binomial (2n, n) / (n-2) is an integer.
  • A094577 (program): Central Peirce numbers. Number of set partitions of {1,2,..,2n+1} in which n+1 is the smallest of its block.
  • A094578 (program): A094559/4.
  • A094584 (program): Dot product of (1,2,…,n) and first n distinct Fibonacci numbers.
  • A094585 (program): Triangle T of all positive differences of distinct Fibonacci numbers; also, triangle of all sums of consecutive distinct Fibonacci numbers.
  • A094586 (program): Central numbers of the triangle T of all positive differences of distinct Fibonacci numbers.
  • A094587 (program): Triangle of permutation coefficients arranged with 1’s on the diagonal. Also, triangle of permutations on n letters with exactly k+1 cycles and with the first k+1 letters in separate cycles.
  • A094588 (program): a(n) = n*F(n-1) + F(n), where F = A000045.
  • A094590 (program): a(1) = 1; a(n+1) = a(n) * k(n), where k(n) is the number of elements of {a(j)}, 1<=j<=n, which are <= n.
  • A094591 (program): a(0) = 1; a(n) = n + (largest element of {a} <= n).
  • A094603 (program): a(n) is the length of the maximal sequence of rightmost black cells in the n-th row of Rule 30 (begun from an initial black cell).
  • A094615 (program): Triangular array T of numbers generated by these rules: 1 is in T; and if x is in T, then 2x+1 and 3x+2 are in T.
  • A094616 (program): Row sums of A094615.
  • A094617 (program): Triangular array T of numbers generated by these rules: 2 is in T; and if x is in T, then 2x-1 and 3x-2 are in T.
  • A094618 (program): a(n) = 3^(n+1) - 2^(n+1) + n + 1.
  • A094620 (program): Expansion of x*(11 + 22*x + 20*x^2)/((1-x)*(1+x)*(1 - 10*x^2)).
  • A094621 (program): Expansion of x*(11+13*x+20*x^2) / ( (x-1)*(1+x)*(10*x^2-1) ).
  • A094622 (program): Expansion of x*(11+20*x)/((1-x)*(1-10*x^2)).
  • A094623 (program): Expansion of x*(1+10*x)/((1-x)*(1-10*x^2)).
  • A094624 (program): Expansion of x*(1+11*x+x^2)/((1-x)*(1+x)*(1-10*x^2)).
  • A094625 (program): Expansion of x*(2+22*x+11*x^2)/((x-1)*(1+x)*(10*x^2-1)).
  • A094626 (program): Expansion of x*(1+x)/((1-x)*(1-10*x^2)).
  • A094627 (program): Expansion of (1+x)^2/((1-x)*(1-10*x^2)).
  • A094628 (program): Erroneous version of A052218.
  • A094632 (program): A trace sequence for a Napoleon graph.
  • A094633 (program): A Lucas Jacobsthal product.
  • A094635 (program): Smallest digit of n in Roman numeral representation.
  • A094639 (program): Partial sums of squares of Catalan numbers (A000108).
  • A094647 (program): a(n) = n^(2n) - (2n)^n.
  • A094648 (program): An accelerator sequence for Catalan’s constant.
  • A094649 (program): An accelerator sequence for Catalan’s constant.
  • A094657 (program): Primes congruent to 4 mod 17.
  • A094659 (program): Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.
  • A094667 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 1, s(2n+1) = 4.
  • A094669 (program): Number of halving and tripling steps for 10^n to reach 1 in ‘3x+1’ problem.
  • A094677 (program): Sum of digits is divisible by 10.
  • A094684 (program): Records in A094683.
  • A094686 (program): A Fibonacci convolution.
  • A094687 (program): Convolution of Fibonacci and Jacobsthal numbers.
  • A094688 (program): Convolution of Fibonacci(n) and 3^n.
  • A094693 (program): Records in A094685.
  • A094695 (program): Numbers having in binary representation fewer ones than in their squares.
  • A094699 (program): Number of prime partition numbers <= n-th partition number.
  • A094703 (program): a(1)=1, a(2)=11, a(n+2) = 8*a(n+1) + 21*a(n).
  • A094704 (program): Convolution of Fibonacci(n) and 10^n.
  • A094705 (program): Convolution of Jacobsthal(n) and 3^n.
  • A094706 (program): Convolution of Pell(n) and 2^n.
  • A094707 (program): Partial sums of repeated Fibonacci sequence.
  • A094715 (program): a(n) = sum(2i+3j=n, 0<=i<=n, 0<=j<=n, n!/((2i)!(3j)!)).
  • A094717 (program): a(n) = n!*Sum_(i+2j+3k=n, 0<=i<=n, 0<=j<=n, 0<=k<=n, 1/i!/(2j)!/(3k)!).
  • A094723 (program): a(n) = Pell(n+2) - 2^n.
  • A094726 (program): Let M = the 2 X 2 matrix [ 0 3 / 3 2]. Take (M^n * [1 1])/3 = [p q]; then a(n) = p.
  • A094727 (program): Triangle read by rows: T(n,k) = n + k, 0 <= k < n.
  • A094728 (program): Triangle read by rows: T(n,k) = n^2 - k^2, 0<=k<n.
  • A094729 (program): Number of connected ordered 2-element multiantichains on a labeled n-set.
  • A094734 (program): Number of connected 2-element multiantichains on a labeled n-set.
  • A094741 (program): Number of primes of the form k+1 where k is coprime to n, k < n.
  • A094743 (program): Beginning with 2, increasing primes such that the sum of successive differences is also prime.
  • A094761 (program): a(n) = n + (square excess of n).
  • A094762 (program): a(n) = Bell(n+1) - 2^n + 1 + n, where Bell(i) is the i-th Bell number A000110(i).
  • A094763 (program): Trajectory of 2 under repeated application of the map n -> n + square excess of n.
  • A094764 (program): Trajectory of 7 under repeated application of the map n –> n + square excess of n.
  • A094765 (program): a(n) = n + 2 * square excess of n.
  • A094766 (program): Trajectory of 11 under repeated application of the map n -> n + 2*square excess of n (see A094765).
  • A094779 (program): Let 2^k = smallest power of 2 >= binomial(n,[n/2]); a(n) = 2^k - binomial(n,[n/2]).
  • A094780 (program): Let 2^k = smallest power of 2 >= binomial(2n,n); a(n) = 2^k - binomial(2n,n).
  • A094784 (program): Numbers that are neither squares nor cubes.
  • A094788 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 1, s(2n+1) = 6.
  • A094789 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 1, s(2n+1) = 4.
  • A094790 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 1, s(2n) = 3.
  • A094792 (program): a(n) = (1/n!)*A001565(n).
  • A094793 (program): a(n) = (1/n!)*A001688(n).
  • A094794 (program): a(n) = (1/n!)*A001689(n).
  • A094795 (program): a(n) = (1/n!)*A023043(n).
  • A094797 (program): Number of times 1 is used in writing out all numbers 1 through 10^n.
  • A094798 (program): Number of times 1 is used in writing out all the numbers 1 through n.
  • A094802 (program): a(n) = smallest k such that all of 1 through n divides k!.
  • A094803 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 1, s(2n) = 3.
  • A094806 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 1, s(2n) = 5.
  • A094811 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 1, s(2n+1) = 6.
  • A094815 (program): (prime(prime(n)))^n.
  • A094817 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 3, s(2n) = 3.
  • A094820 (program): Partial sums of A038548.
  • A094821 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 3, s(2n) = 5.
  • A094822 (program): E.g.f.: exp(3x)/(1-3x)^(1/3).
  • A094823 (program): If n = c0 + c1*10 + c2*10^2 + …cn*10^n then a(n) = c0 + c1*13 + c2*13^2 + …cn*13^k.
  • A094825 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 1, s(2n) = 7.
  • A094826 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 1, s(2n) = 3.
  • A094827 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 1, s(2n+1) = 4.
  • A094828 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 1, s(2n) = 5.
  • A094829 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 1, s(2n+1) = 6.
  • A094831 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 3, s(2n) = 3.
  • A094832 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 3, s(2n+1) = 4.
  • A094833 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 3, s(2n) = 5.
  • A094834 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 3, s(2n+1) = 6.
  • A094835 (program): a(n) = 2702*a(n-1) - a(n-2); a(-1) = a(0) = 26.
  • A094836 (program): a(n) = 2702*a(n-1) - a(n-2); a(-1) = -15; a(0) = 15.
  • A094838 (program): The longest subsequence length that provides an example for A094837.
  • A094854 (program): Number of (s(0), s(1), …, s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n, s(0) = 4, s(2n) = 4.
  • A094855 (program): Number of (s(0), s(1), …, s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,…,2n+1, s(0) = 4, s(2n+1) = 5.
  • A094856 (program): E.g.f.: exp(4x)/(1-4x)^(1/4).
  • A094861 (program): Same as A094858, except that we fix X = 123123123…
  • A094864 (program): a(0)=1, a(1)=2, a(2)=6, a(3)=18; for n >= 4, a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
  • A094865 (program): Expansion of x^3/((1-3*x+x^2)*(1-5*x+5*x^2)).
  • A094869 (program): E.g.f.: exp(5x)/(1-5x)^(1/5).
  • A094874 (program): Decimal expansion of (5-sqrt(5))/2.
  • A094875 (program): a(n)=1 if floor(Pi*10^n) is prime, otherwise a(n)=0.
  • A094876 (program): a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; then a(n) = sum( a(2*i)*a(3*j)) where 1< = i< = n, 1< = j< = n and 2*i+3*j = n.
  • A094879 (program): Where 1’s occur in A094830 (nonprimes that reach a prime in one step under iteration of “x -> x + sum of squares of digits of x”).
  • A094883 (program): Decimal expansion of sqrt(2)/phi, where phi = (1+sqrt(5))/2.
  • A094884 (program): Decimal expansion of phi/sqrt(2), where phi = (1+sqrt(5))/2.
  • A094887 (program): Decimal expansion of phi*sqrt(2), where phi = (1+sqrt(5))/2.
  • A094892 (program): a(n) is the number of primes between n*210 and (n+1)*210.
  • A094896 (program): If 4*n+1 is prime and 4*n+3 is not prime then a(n)=4*n+1, else a(n)=0.
  • A094897 (program): If 4*n+1 is not prime and 4*n+3 is prime then a(n)=4*n+3, else a(n)=0.
  • A094905 (program): Expansion of e.g.f.: exp(6*x)/(1-6*x)^(1/6).
  • A094909 (program): Let p_k(n) = number of partitions of n into exactly k parts; sequence gives p_3(n-3) + p_2(n-2) + 1.
  • A094911 (program): E.g.f.: exp(7x)/(1-7x)^(1/7).
  • A094912 (program): Output from a certain finite automaton when fed binary representation of n read from right to left.
  • A094913 (program): Maximal number of distinct nonempty substrings of any binary string of length n.
  • A094914 (program): Absolute value of n^2 + n - 1354363.
  • A094918 (program): a(n) = (3^n-1)/2 mod n.
  • A094919 (program): (4^n-1)/3 mod n.
  • A094920 (program): a(n) = (5^n-1)/4 mod n.
  • A094921 (program): (6^n-1)/5 mod n.
  • A094922 (program): (7^n-1)/6 mod n.
  • A094923 (program): (8^n-1)/7 mod n.
  • A094924 (program): a(n) = (9^n-1)/8 mod n.
  • A094930 (program): Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).
  • A094935 (program): E.g.f.: exp(8x)/(1-8x)^(1/8).
  • A094938 (program): a(n)=(-36^n/18)*B(2n,1/6)/B(2n,1/3) where B(n,x) is the n-th Bernoulli polynomial.
  • A094941 (program): a(n) is n! times the coefficient of Pi^floor(n/2) in the volume of an n-dimensional unit ball.
  • A094943 (program): A sequence generated from a semi-magic square.
  • A094945 (program): Initial n terms of A023532(n) taken as digits.
  • A094949 (program): Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.
  • A094951 (program): a(n) = A081038(n) + A077616(n).
  • A094952 (program): A sequence derived from pentagonal numbers, or a Stirling number of the first kind matrix.
  • A094953 (program): Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.
  • A094954 (program): Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1.
  • A094955 (program): Main diagonal of array A094954.
  • A094956 (program): Diagonal T(n+2,n) of array A094954.
  • A094957 (program): Antidiagonal sums of array A094954.
  • A094958 (program): Numbers of the form 2^k or 5*2^k.
  • A094960 (program): Positive integers k such that the derivative of k-th Bernoulli polynomial B(k,x) contains only integer coefficients.
  • A094966 (program): Left-hand neighbors of Fibonacci numbers in Stern’s diatomic series.
  • A094967 (program): Right-hand neighbors of Fibonacci numbers in Stern’s diatomic series.
  • A094968 (program): Indices of Fibonacci numbers in Stern’s diatomic series A049456 regarded as a single linear sequence.
  • A094969 (program): a(n) = floor(5^n/2^n).
  • A094970 (program): a(n) = floor(7^n/2^n).
  • A094971 (program): a(n) = floor(9^n/2^n).
  • A094972 (program): a(n) = floor(11^n/2^n).
  • A094974 (program): a(n) = floor(5^n/3^n).
  • A094975 (program): a(n) = floor(7^n/3^n).
  • A094976 (program): a(n) = floor(8^n/3^n).
  • A094977 (program): a(n) = floor(10^n/3^n).
  • A094978 (program): a(n) = floor(11^n/3^n).
  • A094980 (program): a(n) = floor(7^n/4^n).
  • A094981 (program): a(n) = floor(9^n/4^n).
  • A094982 (program): a(n) = floor(11^n/4^n).
  • A094983 (program): a(n) = floor(6^n/5^n).
  • A094984 (program): a(n) = floor(7^n/5^n).
  • A094985 (program): a(n) = floor(8^n/5^n).
  • A094986 (program): a(n) = floor(9^n/5^n).
  • A094987 (program): a(n) = floor(11^n/5^n).
  • A094988 (program): a(n) = floor(7^n/6^n).
  • A094989 (program): Floor(11^n/6^n).
  • A094990 (program): a(n) = floor(8^n/7^n).
  • A094991 (program): a(n) = floor(9^n/7^n).
  • A094992 (program): a(n) = floor(10^n/7^n).
  • A094993 (program): a(n) = floor(11^n/7^n).
  • A094994 (program): a(n) = floor(9^n/8^n).
  • A094995 (program): a(n) = floor(11^n/8^n).
  • A094996 (program): a(n) = floor(10^n/9^n).
  • A094997 (program): a(n) = floor(11^n/9^n).
  • A094999 (program): a(n) = floor(12^n/11^n).
  • A095000 (program): E.g.f.: exp(x)/(1-x)^4.
  • A095002 (program): a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 3, a(3) = 19.
  • A095003 (program): a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3).
  • A095004 (program): a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.
  • A095037 (program): An example of a (v,k,lambda)=(23,11,5) cyclic difference set.
  • A095039 (program): An example of a (v,k,lambda)=(40,13,4) cyclic difference set.
  • A095041 (program): One of two (v,k,lambda)=(31,15,7) cyclic difference sets. The other one is A095042.
  • A095042 (program): One of two (v,k,lambda)=(31,15,7) cyclic difference sets. The other one is A095041.
  • A095047 (program): An example of a (v,k,lambda)=(107,53,26) cyclic difference set.
  • A095051 (program): E.g.f.: exp(-x)/eta(x), where eta(x) is the Dedekind eta function.
  • A095070 (program): One-bit dominant primes, i.e., primes whose binary expansion contains more 1’s than 0’s.
  • A095071 (program): Zero-bit dominant primes, i.e., primes whose binary expansion contains more 0’s than 1’s.
  • A095074 (program): Primes in whose binary expansion the number of 0-bits is less than or equal to number of 1-bits.
  • A095075 (program): Primes in whose binary expansion the number of 1-bits is less than or equal to number of 0-bits.
  • A095076 (program): Parity of 1-fibits in Zeckendorf expansion A014417(n).
  • A095078 (program): Primes with a single 0 bit in their binary expansion.
  • A095080 (program): Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.
  • A095081 (program): Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.
  • A095082 (program): Fib00 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros.
  • A095083 (program): Fibodious primes, i.e., primes p whose Zeckendorf-expansion A014417(p) contains an odd number of 1-fibits.
  • A095084 (program): Fibevil primes, i.e., primes p whose Zeckendorf-expansion A014417(p) contains an even number of 1-fibits.
  • A095085 (program): Fib000 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with three zeros.
  • A095086 (program): Fib001 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros and final 1.
  • A095087 (program): Fib010 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero, one and zero.
  • A095088 (program): Fib100 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one and two final zeros.
  • A095089 (program): Fib101 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends as one, zero, one.
  • A095096 (program): Fibevil numbers: those n for which the parity of 1-fibits in Zeckendorf expansion A014417(n) is even, i.e., for which A095076(n) = 0.
  • A095097 (program): Erroneous version of A101345.
  • A095098 (program): Fib001 numbers: those k for which the Zeckendorf expansion A014417(k) ends with two zeros and a final one.
  • A095111 (program): One minus the parity of 1-fibits in Zeckendorf expansion A014417(n).
  • A095112 (program): a(n) is the sum of n/k over all prime powers k > 1 which divide n.
  • A095114 (program): a(1)=1. a(n) = a(n-1) + (number of elements of {a(1),…,a(n-1)} that are <= n-1).
  • A095116 (program): a(n) = prime(n) + n - 1.
  • A095117 (program): a(n) = pi(n) + n, where pi(n) = A000720(n) is the number of primes <= n.
  • A095118 (program): a(n) is the sum of the squares of the divisors of n which are <= sqrt(n).
  • A095121 (program): Expansion of (1-x+2x^2)/((1-x)*(1-2x)).
  • A095122 (program): Fib(n)(2Fib(n)-1).
  • A095125 (program): Expansion of -x*(-1-x+x^2) / ( 1-2*x-3*x^2+x^3 ).
  • A095126 (program): Expansion of x*(4+5*x-x^2)/ (1-2*x-3*x^2+x^3).
  • A095127 (program): a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 1, a(2) = 4, a(3) = 10.
  • A095128 (program): a(n+3) = 3*a(n+2) + 2*a(n+1) - a(n).
  • A095129 (program): Jac(n)(2Jac(n)-1).
  • A095130 (program): Expansion of (x+x^2)/(1-x^6); period 6: repeat [0, 1, 1, 0, 0, 0].
  • A095134 (program): Sum of the product of the first floor(n/2) even-indexed primes and the product of the first floor(n/2) odd-indexed primes.
  • A095137 (program): Absolute difference between the product of the first floor(n/2) even-indexed primes and the product of the first floor(n/2) odd-indexed primes.
  • A095140 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 5.
  • A095141 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 6.
  • A095142 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 7.
  • A095143 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 9.
  • A095144 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 11.
  • A095145 (program): Triangle, read by rows, formed by reading Pascal’s triangle (A007318) mod 12.
  • A095149 (program): Triangle read by rows: Aitken’s array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, …
  • A095151 (program): a(n+3) = 3*a(n+2) - 2*a(n+1) + 1 with a(0)=0, a(1)=2.
  • A095152 (program): Number of 3-block covers of a labeled n-set.
  • A095153 (program): Number of 4-block covers of a labeled n-set.
  • A095166 (program): Group the natural numbers >= 1 so that the n-th group contains n(n+1)/2 numbers and obtain the group sum.
  • A095175 (program): Denominator of b(n) given by b(1) = 1, b(2) = 2; for n >= 3, b(n) = (-1)^n (2n-1) ((n-2)!!)^2/((n-1)!!)^2, where n!! is the double factorial A006882.
  • A095176 (program): E.g.f.: exp(9x)/(1-9x)^(1/9).
  • A095177 (program): E.g.f.: exp(x)/(1-x)^5.
  • A095179 (program): Numbers whose reversed digit representation is prime.
  • A095180 (program): Reverse digits of primes, append to sequence if result is a prime.
  • A095187 (program): Least significant digit of (n mod 10)^floor(n/10).
  • A095190 (program): Doubled Thue-Morse sequence: the A010060 sequence replacing 0 with 0,0 and 1 with 1,1.
  • A095195 (program): T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k<n, triangle read by rows.
  • A095197 (program): a(1) = 1, a(n+1) = 2 * digit reversal of a(n).
  • A095199 (program): Least integer multiple of f(1/n) where f(1/n) is the number obtained by retaining only n digits after decimal and deleting the rest.
  • A095200 (program): Greatest multiple of n of the form (n-1) + (n-2) + … + (n-k), or 0 if no such number exists.
  • A095201 (program): A095200(n)/n.
  • A095202 (program): Value of largest k such that (n-1) + (n-2) + (n-3) + … + (n-k) is a multiple of n, or 0 if no such k exists.
  • A095208 (program): n if n is composite else 10*n.
  • A095209 (program): a(0) = 1, and for n > 0, a(n) = the least multiple of prime(n) such that the geometric mean of a(0) to a(n) is an integer.
  • A095219 (program): First differences of A053067.
  • A095221 (program): (Concatenation 1,2,3,…,n) mod n.
  • A095222 (program): (Concatenation of T(n)+1..T(n+1)) mod (concatenation of T(n-1)+1..T(n)), where T(k) is the k-th triangular number, A000217(k).
  • A095227 (program): Numbers which can be partitioned into distinct parts all using the same digit.
  • A095228 (program): n-th decimal digit of 1/n!.
  • A095232 (program): a(1) = 1, a(n+1) = floor(a(n)*10^n/n).
  • A095233 (program): a(n) = a(n-1) + Sum(floor(n/p): p prime), a(1) = 1.
  • A095237 (program): a(1)=1; then for n even, a(n)=(sum of previous terms times n) plus 1, for n odd, a(n)=(sum of previous terms times n) minus 1.
  • A095238 (program): a(1) = 1, a(n) = n*(sum of all previous terms mod n).
  • A095243 (program): a(n) = concatenation of 1,2,3,…,n mod prime(n).
  • A095244 (program): a(n) = concatenation n,n-1,n-2,…,3,2,1 mod (prime(n)).
  • A095248 (program): a(n) = least k > 0 such that n-th partial sum is divisible by n if and only if n is not prime.
  • A095249 (program): Reverse concatenation of first n positive integers modulo forward concatenation of first n positive integers.
  • A095250 (program): a(n) = 11111111… (n times) = (10^n-1)/9 reduced mod n.
  • A095252 (program): a(n) = floor(sqrt{concatenation n,(n-1),…,3,2,1}).
  • A095262 (program): A sequence derived from a truncated Pascal’s Triangle matrix.
  • A095263 (program): a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n).
  • A095264 (program): a(n) = 2^(n+2) - 3*n - 4.
  • A095265 (program): A sequence generated from a 4th degree Pascal’s Triangle polynomial.
  • A095266 (program): A sequence generated from the Narayana triangle considered as a matrix, or from Pascal’s triangle.
  • A095267 (program): Least perimeter common to 2^n primitive arithmetic triangles, i.e., Heronian triangles whose sides are in arithmetic progression.
  • A095276 (program): Length of n-th run of identical symbols in A095076 and A095111.
  • A095277 (program): Numbers k such that 4k + 3 is composite.
  • A095278 (program): Numbers k such that 4k + 3 is prime.
  • A095279 (program): Partial sums of A095276.
  • A095280 (program): Lower Wythoff primes, i.e., primes in A000201.
  • A095281 (program): Upper Wythoff primes, i.e., primes in A001950.
  • A095282 (program): Primes whose binary-expansion ends with an even number of 1’s.
  • A095283 (program): Primes whose binary-expansion ends with an odd number of 1’s.
  • A095285 (program): Primes in whose binary expansion the number of 1 bits is <= 5 + number of 0 bits.
  • A095286 (program): Primes in whose binary expansion the number of 1 bits is > 1 + number of 0 bits.
  • A095287 (program): Primes in whose binary expansion the number of 1-bits is <= 1 + number of 0-bits.
  • A095288 (program): Least middle side of 2^n primitive arithmetic triangles, i.e., primitive Heronian triangles whose sides are in arithmetic progression.
  • A095289 (program): a(n) = the smallest number (in base 10) such that the product of its digits is >= n.
  • A095305 (program): Numbers n such that A094020(n) < n.
  • A095307 (program): Number of walks of length n between two nodes at distance 2 in the cycle graph C_7.
  • A095308 (program): Number of walks of length n between two nodes at distance 3 in the cycle graph C_7.
  • A095310 (program): a(n+3) = 2*a(n+2) + 3*(n+1) - a(n).
  • A095311 (program): 47-gonal numbers.
  • A095313 (program): Primes in whose binary expansion the number of 1-bits is <= 6 + number of 0-bits.
  • A095314 (program): Primes in whose binary expansion the number of 1 bits is > 2 + number of 0 bits.
  • A095315 (program): Primes in whose binary expansion the number of 1 bits is <= 2 + number of 0 bits.
  • A095316 (program): Primes in whose binary expansion the number of 1-bits is > number of 0-bits minus 2.
  • A095318 (program): Primes in whose binary expansion the number of 1 bits is > 3 + number of 0 bits.
  • A095319 (program): Primes in whose binary expansion the number of 1 bits is <= 3 + number of 0 bits.
  • A095320 (program): Primes in whose binary expansion the number of 1-bits is > number of 0-bits minus 3.
  • A095322 (program): Primes in whose binary expansion the number of 1 bits is > 4 + number of 0 bits.
  • A095323 (program): Primes in whose binary expansion the number of 1 bits is <= 4 + number of 0 bits.
  • A095338 (program): Total number of leaves in the labeled graphs of order n.
  • A095340 (program): Total number of nodes in all labeled graphs on n nodes.
  • A095342 (program): Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1.
  • A095343 (program): Length of n-th string generated by a Kolakoski(7,1) rule starting with a(1)=1.
  • A095344 (program): Length of n-th string generated by a Kolakoski(9,1) rule starting with a(1)=1.
  • A095345 (program): a(n) is the length of the n-th run in A095346.
  • A095346 (program): a(n) is the length of the n-th run of A095345.
  • A095351 (program): Total number of edges in all labeled graphs on n nodes.
  • A095364 (program): Number of walks of length n between two adjacent nodes in the cycle graph C_9.
  • A095366 (program): Least k > 1 such that k divides 1^n + 2^n +…+ (k-1)^n.
  • A095367 (program): Number of walks of length n between two nodes at distance 2 in the cycle graph C_9.
  • A095368 (program): Number of walks of length n between two nodes at distance 3 in the cycle graph C_9.
  • A095369 (program): Number of walks of length n between two nodes at distance 4 in the cycle graph C_9.
  • A095372 (program): 1+integers repeating “90” decimal digit pattern:.
  • A095374 (program): One less than the number of divisors of 2*n + 1.
  • A095375 (program): Total number of 1’s in the binary expansions of the first n primes: summatory A014499.
  • A095386 (program): Largest prime factor of peak values of 3x+1 trajectory started at n.
  • A095394 (program): a(n) = Floor[n^((n)/(n+1))], integer part of n^x where x = n/(n+1) < 1.
  • A095402 (program): Sum of digits of all distinct prime factors of n.
  • A095407 (program): Total number of decimal digits of all distinct prime factors of n.
  • A095408 (program): Total number of decimal digits in all distinct prime factors of n minus number of digits in n.
  • A095409 (program): Numbers n such that total number of decimal digits of all distinct prime factors of n is smaller than number of digits of n.
  • A095410 (program): Numbers n such that total number of decimal digits of all distinct prime factors of n equals the number of digits of n.
  • A095660 (program): Pascal (1,3) triangle.
  • A095661 (program): Fifth column (m=4) of (1,3)-Pascal triangle A095660.
  • A095662 (program): Seventh column (m=6) of (1,3)-Pascal triangle A095660.
  • A095663 (program): Eighth column (m=7) of (1,3)-Pascal triangle A095660.
  • A095664 (program): Ninth column (m=8) of (1,3)-Pascal triangle A095660.
  • A095665 (program): Tenth column (m=9) of (1,3)-Pascal triangle A095660.
  • A095666 (program): Pascal (1,4) triangle.
  • A095667 (program): Fifth column (m=4) of (1,4)-Pascal triangle A095666.
  • A095668 (program): Sixth column (m=5) of (1,4)-Pascal triangle A095666.
  • A095669 (program): Seventh column (m=6) of (1,4)-Pascal triangle A095666.
  • A095670 (program): Eighth column (m=7) of (1,4)-Pascal triangle A095666.
  • A095671 (program): Ninth column (m=8) of (1,4)-Pascal triangle A095666.
  • A095676 (program): Row sums of A095675.
  • A095681 (program): G.f.: 1/((1-x)(1-x-x^2)(1-x-x^2-x^3)).
  • A095685 (program): Expansion of (1+x)^4/(1-11*x+11*x^2-x^3).
  • A095686 (program): Half the number of divisors of nonsquares (A000037).
  • A095687 (program): Numbers n such that n-th Pisano number = 6*n.
  • A095691 (program): Multiplicative with a(p^e) = A000720(e)+1.
  • A095692 (program): Primes of the form n^3 + n + 1.
  • A095694 (program): T(n,3) diagonal of triangle in A095693.
  • A095697 (program): Primes of the form x^2 + y^2 + z, where x, y and z are three successive numbers.
  • A095698 (program): Number of permutations of {1,2,3,…,n} where, for 1 < i <= n, the i-th number has maximized sum of the i-1 absolute differences from all previous numbers of the permutation.
  • A095701 (program): Define string S_0 to be the null sequence; string S_n is derived from string S_{n-1} by inserting n’s in the rightmost n gaps; sequence gives limit S_n as n -> infinity.
  • A095704 (program): Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).
  • A095708 (program): Tau-functions of the q-discrete Painlevé I equation, f(n+1) = (A*q^n*f(n) + B)/(f(n)^2*f(n-1)), for q=2 and A=B=1, with f(n) = a(n+1)*a(n-1)/a(n)^2.
  • A095709 (program): Triangle of numbers obtained by reversing the first n digits of Pi and juxtaposing.
  • A095711 (program): Triangle of numbers obtained by reversing the first n digits of e and juxtaposing.
  • A095713 (program): Triangle of numbers obtained by reversing the first n digits of golden ratio phi and juxtaposing.
  • A095715 (program): Triangle of numbers obtained by reversing the first n digits of 1/phi and juxtaposing (phi denotes the golden ratio: A001622).
  • A095718 (program): a(n) = Sum_{k=0..n} floor(binomial(n,k)/(k+1)).
  • A095719 (program): Sum(floor(C(n-k,k)/(k+1)),k=0..n/2).
  • A095720 (program): a(1)=1 and, for n>1, a(n)=a(n-1)+Floor(3n/4) if n is even, else a(n)=smallest positive integer which has not yet appeared in the sequence.
  • A095722 (program): E.g.f.: exp(x)/(1-x)^8.
  • A095729 (program): A002260 squared, as an infinite lower triangular matrix, read by rows.
  • A095736 (program): Numbers with binary weight (A000120) <= 3.
  • A095740 (program): E.g.f.: exp(x)/(1-x)^9.
  • A095743 (program): Primes p for which A037888(p) = 1, i.e., primes whose binary expansion is almost symmetric, needing just a one-bit flip to become palindrome.
  • A095750 (program): “Degree” of the Sophie Germain primes (A005384).
  • A095752 (program): a(n) = (1/9)*(-1+10^n)*(4*n+3)
  • A095761 (program): a(n) = A014824(2*n-1).
  • A095764 (program): Numbers whose name in English contains an “l”.
  • A095767 (program): a(n) = valuation(A004001(n),2).
  • A095768 (program): a(n) = 2^(n+1) - n.
  • A095773 (program): a(1)=1, a(n) = 1 + a(n - a(a(a(n-1)))).
  • A095774 (program): a(n)=2*A003160(n)-n.
  • A095775 (program): n/2 when 2*A003160(n) = n.
  • A095790 (program): Numbers whose name in English contains an “r”.
  • A095791 (program): Number of digits in lazy-Fibonacci-binary representation of n.
  • A095792 (program): a(n) = Z(n) - L(n), where Z=A072649 and L=A095791 are lengths of Zeckendorf and lazy Fibonacci representations in binary notation.
  • A095794 (program): a(n) = A005449(n) - 1, where A005449 = second pentagonal numbers.
  • A095795 (program): a(0)=2, a(1)=5, a(n+2) = a(n+1) + (-1)^n a(n).
  • A095796 (program): 1 + (26*n+17+7*n^2)*n/2.
  • A095800 (program): Triangle T(n,k) = abs( k *( (2*n+1)*(-1)^(n+k)+2*k-1) /4 ) read by rows, 1<=k<=n.
  • A095803 (program): Values of r in Wolfram’s iteration for sqrt(2).
  • A095804 (program): Values of s in Wolfram’s iteration for sqrt(2).
  • A095805 (program): Reduced numerators in Wolfram’s iteration for sqrt(2).
  • A095806 (program): Reduced denominators in Wolfram’s iteration for sqrt(2).
  • A095807 (program): Number of integers from 0 to 10^n-1 which contain at least one decimal digit = 0.
  • A095808 (program): Number of ways to write n in the form m + (m+1) + … + (m+k-1) + (m+k) + (m+k-1) + … + (m+1) + m with integers m>= 1, k>=1. Or, number of divisors a of 4n-1 with 0 < (a-1)^2 < 4n.
  • A095814 (program): Number of nonisomorphic partitions of n on the Ferrers diagram.
  • A095815 (program): n + largest digit of n.
  • A095819 (program): Tenth column (m=9) of (1,4)-Pascal triangle A095666.
  • A095821 (program): Denominators of some (trivial) upper bounds for Euler’s Zeta-function Zeta(n).
  • A095822 (program): Numerators of certain upper bounds for Euler’s number e.
  • A095823 (program): Denominators of certain upper bounds for Euler’s number e.
  • A095824 (program): Numbers n such that n + largest digit of n is prime.
  • A095827 (program): a(n) is the smallest k such that ((A007953)^k)(9n)=9.
  • A095831 (program): Triangle read by rows: T(n,k) = (n-k)^2, n>=1, 1<=k<=n.
  • A095832 (program): Triangle read by rows: T(n,k) = (n-k+1)*(n-k), n>=1, 1<=k<=n.
  • A095833 (program): Triangle read by rows: T(n,k) = (n-k+1)*n, n>=1, 1<=k<=n.
  • A095834 (program): Triangle read by rows: T(n,k) = (n-k)*n, n>=1, 1<=k<=n.
  • A095835 (program): Triangle read by rows: T(n,k) = n^((n-k)^2), n>=1, 1<=k<=n.
  • A095836 (program): Triangle read by rows: T(n,k) = k^((n-k)^2), n>=1, 1<=k<=n.
  • A095837 (program): Triangle read by rows: T(n,k) = (n-k+1)^((n-k)^2), n>=1, 1<=k<=n.
  • A095838 (program): Triangle read by rows: T(n,k) = n^((n-k+1)^2), n>=1, 1<=k<=n.
  • A095839 (program): a(n) = ((2*n)!/(n!*2^(n-1)))*integral_{x=1/2..1} (Sqrt(1-x^2)/x)^(2*n) dx.
  • A095843 (program): Triangle read by rows: T(n,k) = (n-k)^((n-k+1)^2), n>=1, 1<=k<=n.
  • A095850 (program): Triangle read by rows: T(n,k) = k^((n-k+1)^2), n>=1, 1<=k<=n.
  • A095851 (program): Triangle read by rows: T(n,k) = (n-k+1)^((n-k+1)^2), n>=1, 1<=k<=n.
  • A095852 (program): Triangle read by rows: T(n,k) = (n-k+1)^(k^2), n>=1, 1<=k<=n.
  • A095859 (program): Triangle read by rows: T(n,k) = (n-k)^(k^2), n>=1, 1<=k<=n.
  • A095860 (program): Triangle read by rows: T(n,k) = n^(k^2), n>=1, 1<=k<=n.
  • A095861 (program): Number of primitive Pythagorean triangles of form (X,Y,Y+1) with hypotenuse Y+1 less than or equal to n.
  • A095863 (program): Index of largest triangular number with n digits.
  • A095864 (program): Largest n-digit triangular number.
  • A095865 (program): Smallest n-digit triangular number - smallest n-digit number.
  • A095866 (program): Largest n-digit number - largest n-digit triangular number.
  • A095871 (program): Triangle read by rows: T(n,k)=(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2
  • A095873 (program): Triangle T(n,k) = (2*k-1)*(n+k-1)*(n-k+1) read by rows, 1<=k<=n.
  • A095874 (program): a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.
  • A095875 (program): Number of lattice points on graph of parabola y >= x^2 with y <= n.
  • A095876 (program): Triangle read by rows: T(n,k) = k^(n^2), n>=1, 1<=k<=n.
  • A095879 (program): Numbers whose lazy Fibonacci representation has an odd number of summands.
  • A095880 (program): Numbers whose lazy Fibonacci representation has an even number of summands.
  • A095881 (program): Triangle read by rows: T(n,k) = (n-k+1)^(n^2), n>=1, 1<=k<=n.
  • A095882 (program): Triangle read by rows: T(n,k) = (n-k)^(n^2), n>=1, 1<=k<=n.
  • A095884 (program): Triangle read by rows: T(n,k) = (n-k)^k, n>=1, 1<=k<=n.
  • A095886 (program): Triangle read by rows: T(n,k) = (n-k)^n, n>=1, 1<=k<=n.
  • A095887 (program): Triangle read by rows: T(n,k) = (n-k+1)^n, n>=1, 1<=k<=n.
  • A095888 (program): Triangle read by rows: T(n,k) = n^(n-k), n>=1, 1<=k<=n.
  • A095890 (program): Triangle read by rows: T(n,k) = (n-k+1)^(n-k), n>=1, 1<=k<=n.
  • A095891 (program): Triangle read by rows: T(n,k) = (n-k+1)^(n-k+1), n>=1, 1<=k<=n.
  • A095893 (program): Triangle read by rows: T(n,k) = (n-k)^(n-k+1), n>=1, 1<=k<=n.
  • A095894 (program): a(2n) = 6*n^2 + 7*n + 1; a(2n+1) = 6*n^2 + 13*n + 7.
  • A095896 (program): Triangle read by rows: T(n,k) = n^(n-k+1), n>=1, 1<=k<=n.
  • A095898 (program): The (1,1)-term of the 3 X 3 matrix M^n, where M = [1,2,3 / 4,7,11 / 6,10,16].
  • A095901 (program): A004001 (mod 2).
  • A095903 (program): Lexical ordering of the lazy Fibonacci representations.
  • A095905 (program): Sequence generated from Golomb’s proof of de Bruijn’s theorem on a torus considered as a matrix.
  • A095907 (program): Digits in the concatenation of strings formed from a previous string by substituting “01” for “0” and “011” for “1” simultaneously at each occurrence. Start with [0].
  • A095914 (program): Binary numbers with 2 replacing 1 in odd positions.
  • A095915 (program): Each number is twice times the product of the digits of the previous number.
  • A095916 (program): Differences between adjacent digits of Pi.
  • A095917 (program): Unreduced numerator of Sum[k=1..n, -(-1)^k/(F(k)*F(k+1))], with F(i) = A000045(i) the Fibonacci numbers.
  • A095922 (program): Dimension of invariants of n-th tensor power of 5-dimensional irreducible representation of B_2.
  • A095925 (program): Smallest m such that prime(n) mod m > 1.
  • A095929 (program): Number of closed walks of length 2n at a vertex of the cyclic graph on 10 nodes C_10.
  • A095930 (program): Number of walks of length 2n between two nodes at distance 2 in the cycle graph C_10.
  • A095931 (program): Number of walks of length 2n between two nodes at distance 4 in the cycle graph C_10.
  • A095932 (program): Number of walks of length 2n+1 between two nodes at distance 3 in the cycle graph C_10.
  • A095933 (program): Number of walks of length 2n+1 between two nodes at distance 5 in the cycle graph C_10.
  • A095934 (program): Expansion of (1-x)^2/(1-5*x+3*x^2).
  • A095937 (program): a(n) = Sum_{k=0..n} (k-1)^k.
  • A095939 (program): a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 1, a(1) = 2, a(2) = 9.
  • A095940 (program): a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 0, a(1) = 1, a(2) = 4.
  • A095942 (program): Differences between adjacent digits of e.
  • A095943 (program): Differences between adjacent digits of golden ratio phi.
  • A095944 (program): Number of subsets S of {1,2,…,n} which contain a number that is greater than the sum of the other numbers in S.
  • A095949 (program): Position of consonants in English alphabet.
  • A095958 (program): Twin prime pairs concatenated in decimal representation.
  • A095959 (program): Primes modulo 30.
  • A095960 (program): Number of divisors of n that are less than the squarefree kernel of n.
  • A095968 (program): Number of tilings of an n X n section of the square lattice with “ribbon tiles”. A ribbon tile is a polyomino which has at most one square on each diagonal running from northwest to southeast.
  • A095973 (program): Yard markers on a U.S.A. football field.
  • A095977 (program): Expansion of 2*x / ((1+x)^2*(1-2*x)^2).
  • A095982 (program): a(n) = a(n-1) + a(n-2) + a(n-4) with a(0) = 2, a(1) = 3, a(2) = 6, a(3) = 9.
  • A095987 (program): a(n) = gcd(n!!, (n-1)!!) where n!! = A006882.
  • A095989 (program): INVERTi transform applied to the ordered Bell numbers.
  • A095992 (program): a(1) = 30; for n > 1, a(n+1) = a(n) + {product of nonzero digits of a(n)}.
  • A095995 (program): Primes of the form 100n - 1.
  • A095996 (program): a(n) = largest divisor of n! that is coprime to n.
  • A095997 (program): Number of divisors of n! that are coprime to n.
  • A095998 (program): n! * (fractional part of n-th harmonic number).
  • A096000 (program): Cupolar numbers: a(n) = (n+1)*(5*n^2+7*n+3)/3.
  • A096006 (program): Scan Pascal’s triangle (A007318) from left to right, record largest prime factor of each entry.
  • A096007 (program): Scan Pascal’s triangle (A007318) from left to right, record smallest prime factor of each entry.
  • A096010 (program): Number of different cycles computed with the generalized 3x+1 problem using C=2, B=Cn+m, A=C^m.
  • A096014 (program): a(n) = (smallest prime factor of n) * (least prime that is not a factor of n), with a(1)=2.
  • A096015 (program): A096014(n) / 2.
  • A096019 (program): a(0)=3, a(n) = 3*a(n-1) + 2*(-1)^n.
  • A096022 (program): Numbers that are congruent to {15, 27, 39, 51} mod 60.
  • A096023 (program): Numbers congruent to {63, 123, 183, 243, 303, 363} mod 420.
  • A096024 (program): Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 5 and (n+6) mod 8 <> 1.
  • A096025 (program): Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 6 and (n+7) mod 9 <> 1.
  • A096026 (program): Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 8 and (n+9) mod 11 <> 1.
  • A096027 (program): Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 10 and (n+11) mod 13 <> 1.
  • A096033 (program): Difference between leg and hypotenuse in primitive Pythagorean triangles.
  • A096034 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^3-M)/2, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096035 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^4-M)/3, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096037 (program): Triangle T(n,m) = (3*n+3*m-2)*(n+1-m)/2 read by rows.
  • A096038 (program): Triangle T(n,m) = (3*n^2-3*m^2+5*m-4+n)/2 read by rows.
  • A096039 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^5-M)/4, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096040 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^6-M)/5, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096041 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^7-M)/6, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096042 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^8-M)/7, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096043 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^9-M)/8, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096044 (program): Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^10-M)/9, where M is the infinite lower Pascal’s triangle matrix, 1<=k<=n.
  • A096045 (program): a(n) = B(2*n,2)/B(2*n) (see comment).
  • A096046 (program): a(n) = B(2n,3)/B(2n) (see comment).
  • A096047 (program): a(n)=B(2n,4)/B(2n) (see comment).
  • A096051 (program): Decimal expansion of lim_{n->infty} B(2n,8)/(B(2n)*64^n) ( see comment for B(n,k) definition ).
  • A096052 (program): Decimal expansion of lim_{n–>infty} B(2n,5)/(B(2n)*25^n) ( see comment for B(n,k) definition ).
  • A096053 (program): a(n) = (3*9^n - 1)/2.
  • A096054 (program): a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.
  • A096055 (program): Let {s(i)}, i=0,1,2,… be a sequence of finite sequences with terms s(i)(j), j=1,2,3,… Start with s(0)={1}. Then, for k>0, let s(k)=s(k-1)Us(k-1) if s(k-1)(k)=0, s(k)=s(k-1)U{0}Us(k-1) if s(k-1)(k)=1, where s(i)(j) is the j-th element of s(i) and U denotes concatenation of the terms of the two operands. {a(n)} is the limit of s(k) as k goes to infinity.
  • A096060 (program): (2^(p-1)-1)/(3*p) where p = prime(n).
  • A096061 (program): a(n) = floor((Sum of the first n natural numbers)/(Sum of the first n terms of the harmonic series)).
  • A096065 (program): Let p(k) = k-th prime; sequence gives primes q of the form q = k*p(k) - 1 for some k.
  • A096080 (program): a(n) = floor((a(n-2)*a(n-1))/(a(n-1)+a(n-2))) + a(n-1) + a(n-2), a(0) = 0, a(1) = 1, a(2) = 1, …
  • A096081 (program): a(n) = floor(a(n-2)^2/a(n-1)) + a(n-1) + a(n-2), a(0) = 0, a(1) = 1, a(2) = 1, …
  • A096089 (program): Let f(n) = largest number formed using digits of n, g(n) = smallest number formed using digits of n; then a(n) = floor[f(n)/g(n)].
  • A096094 (program): Analog of A094091 for S=3.
  • A096104 (program): Digit reversal of A096299(n).
  • A096111 (program): If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).
  • A096115 (program): If n = (2^k)-1, a(n) = a((n+1)/2) = k, if n = 2^k, a(n) = a(n-1)+1 = k+1, otherwise a(n) = (A000523(n)+1)*a(A035327(n-1)).
  • A096121 (program): Number of full spectrum rook’s walks on a (2 X n) board.
  • A096123 (program): Least product n*(n-1)*(n-2)*…*(n-k+1) divisible by (n-k)!.
  • A096125 (program): Least value of k such that n!/((n-k)!)^2 is an integer.
  • A096129 (program): Middle term of a triple of consecutive numbers which are sums of two squares.
  • A096130 (program): Triangle read by rows: T(n,k) = binomial(k*n,n), 1 <= k <= n.
  • A096131 (program): Sum of the terms of the n-th row of triangle pertaining to A096130.
  • A096132 (program): Triangle read by rows in which the r-th term of the n-th row is C(n^r,r*n), where r = 1 to n.
  • A096133 (program): Triangle T(j,k) = (j^k) mod (j*k) for 1 <= k <= j, read by rows.
  • A096138 (program): a(1) = 1, a(n) = digit reversal of n*a(n-1).
  • A096139 (program): Number of ways to write 2*n as an ordered sum of two numbers which are prime or 1.
  • A096140 (program): a(n) = sum of n Fibonacci numbers starting from F(n).
  • A096141 (program): a(n) = sum of n n-th powers starting from n^n.
  • A096143 (program): a(n) = ceiling(Sum_{i=1..n} 1/i).
  • A096145 (program): Sum of digits in Pascal’s triangle (A007318) in decimal representation, triangle read by rows, 0<=k<=n.
  • A096146 (program): Prime numerators of the rational convergents to sqrt(3).
  • A096156 (program): Numbers with ordered prime signature (2,1).
  • A096157 (program): Numbers whose numbers of odd and even proper divisors differ at most by 1.
  • A096163 (program): Primes p of the form qrs + 1 where q, r and s are distinct primes.
  • A096170 (program): Primes of the form (n^4 + 1)/2.
  • A096172 (program): Largest prime factor of n^4 + 1.
  • A096173 (program): Numbers k such that k^3+1 is an odd semiprime.
  • A096174 (program): Even numbers k such that (k^3+1)/(k+1) is prime.
  • A096175 (program): Numbers k such that k^3-1 is an odd semiprime.
  • A096176 (program): Numbers k such that (k^3-1)/(k-1) is prime.
  • A096182 (program): Index of first occurrence of n in A095773.
  • A096191 (program): a(n) = Sum_{k=1..n} C(n,k)^3 where C(n,k) is binomial(n,k).
  • A096192 (program): a(n) = Sum_{k=1..n} C(n,k)^4 where C(n,k) is binomial(n,k).
  • A096196 (program): a(n) = (1 + 2^n) mod n.
  • A096197 (program): a(n) = (1+prime(n)) mod n.
  • A096198 (program): Triangle read by rows: T(m,n)=A029837(m)+A029837(n), where (m,n)=(1,1); (2,1), (1,2); (3,1), (2,2), (1,3); …
  • A096199 (program): Numbers such that in binary representation the length is a multiple of the number of ones.
  • A096200 (program): n*(n-1)*(n-2)*(3*n-2)/6.
  • A096215 (program): Greatest primes not greater than the sum of two succeeding primes.
  • A096216 (program): a(n) = number of terms among {a(1), a(2), a(3), …, a(n-1)} that are coprime to n; a(1)=1.
  • A096217 (program): a(n) = sum of terms of {a(1),a(2),a(3),…a(n-1)} which are coprime to n.
  • A096222 (program): Number of different rectangles when a piece of paper is folded n times in alternate directions.
  • A096223 (program): Let p(k) be the number of partitions of k (A000041); a(n) = Sum_{1<=k<=n, gcd(k,n)=1} p(k).
  • A096226 (program): a(n) is the least exponent k > 1 such that m^k is congruent to m modulo n for all natural numbers m, or a(n) = 1 if no such k exists.
  • A096228 (program): a(n) = floor(n^2*((n-1)/n)^(n-1/2)).
  • A096230 (program): Period 5: repeat [9, 7, 5, 3, 1].
  • A096231 (program): Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {Pi/2, Pi/3, 0}.
  • A096252 (program): Array read by rows, starting with n=0: row n lists A057077(n+1)*8^(n+1)/2, A057077(n+2)*8^(n+1)/2, A057077(n+1)*8^(n+1).
  • A096268 (program): Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.
  • A096269 (program): a(n) = number of distinct palindromes of length n that occur in A096268.
  • A096271 (program): Ternary sequence that is a fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 00.
  • A096273 (program): a(0)=0, then a(n)=a(n-1)+(n-1) if n is odd, a(n)=a(n/2)+n/2 otherwise.
  • A096277 (program): Sum of successive sums of successive primes: a(n) = s(n) + s(n+1) where s(n) = prime(n) + prime(n+1) (A001043).
  • A096278 (program): Sums of successive sums of successive sums of successive primes.
  • A096279 (program): Sums of successive sums of successive sums of successive sums of successive primes.
  • A096281 (program): Sums of successive twin primes of order 1.
  • A096282 (program): Sums of successive twin primes of order 2.
  • A096283 (program): First sums of successive twin primes of order n.
  • A096284 (program): Numerator of the ratio of the preceding two terms.
  • A096285 (program): Denominator of the ratio of the preceding two terms.
  • A096288 (program): Sum of digits of n in bases 2 and 3.
  • A096289 (program): Let n=Sum(c(k)*2^k), c(k)=0,1, be the binary form of n, n=Sum(d(k)*5^k), d(k)=0,1,2,3,4 the base 5 form; then a(n)=Sum(c(k)+d(k)).
  • A096292 (program): Primes p such that p!-1 is divisible by the next prime larger than p.
  • A096299 (program): List of strings in lexicographic order with property that for the 2^(m-1) strings of length m, the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.
  • A096301 (program): (Product of nonzero digits(sum of digits(n^n)).
  • A096302 (program): Number of combinations of two natural numbers that together have n digits.
  • A096304 (program): Numbers k such that 3k does not divide (6k-4)!/((3k-2)!*(3k-1)!).
  • A096307 (program): E.g.f.: exp(x)/(1-x)^6.
  • A096309 (program): a(1)=1; for n > 1, a(n) is the number of levels in the “stacked” prime number factorization of n (prime number factorization of the exponents if necessary and so on …).
  • A096313 (program): a(n) = determinant of n X n matrix m(i,j) = Product_{k=1..i} k+j.
  • A096316 (program): Given the number wheel 0,1,2,3,4,5,6,7,8,9 then starting with 2, the next number is a prime p number of positions from the previous number found, for p=2,3,…
  • A096319 (program): Given the number wheel 0,1,2,3,4,5,6,7,8,9 then starting with 0, the next number is a prime p number of positions from the previous number found, for p=2,3,…
  • A096320 (program): a(n) = (n^2+n+4)/2, modulo 10.
  • A096334 (program): Triangle read by rows: T(n,k) = prime(n)#/prime(k)#, 0<=k<=n.
  • A096338 (program): a(n) = (2/(n-1))*a(n-1) + ((n+5)/(n-1))*a(n-2) with a(0)=0 and a(1)=1.
  • A096341 (program): E.g.f.: exp(x)/(1-x)^7.
  • A096344 (program): Number of 1’s in binary expansion(sum of digits(n^n)).
  • A096346 (program): Complement of A004128.
  • A096363 (program): Length of repeating cycle of the final n digits in the Fibonacci sequence.
  • A096364 (program): Number of ways to generate a Coxeter element of the reflection group of the root system B_n with certain restrictions on generators: (3n-4)*(n-2)^(n-2) - (n-1)^(n-1).
  • A096365 (program): Maximum number of iterations of the RUNS transform needed to reduce any binary sequence of length n to a sequence of length 1.
  • A096367 (program): Number of winning paths of length n+1 across an n X n Hex board.
  • A096373 (program): Number of partitions of n such that the least part occurs exactly twice.
  • A096376 (program): a(n) = n + (n-1)^2 + (n+1)^2.
  • A096379 (program): a(n) = prime(n) + prime(n+1) - prime(n+2).
  • A096380 (program): Differences between the sum of the first three primes and the fourth prime in consecutive prime quadruples.
  • A096382 (program): Consider a Pythagorean triangle with sides a=u^2-v^2, b=2uv, c=u^2+v^2. The sequence is the area of the triangle when v=2, u=3,4,5,…
  • A096383 (program): Area of the Pythagorean triangle a = u^2 - v^2 (cf. A096382) when u=3, v=4,4,5,…
  • A096386 (program): Number of numbers <= n which are divisible by 2 or 3.
  • A096398 (program): Numbers n such that 0= #{ 1<=i<=n : k(n,i)=-1 } where k(n,i) is the Kronecker symbol.
  • A096400 (program): Number of equivalence classes of triangles having equal angles of integral degrees and smallest angle = n.
  • A096406 (program): a(1) = 4; then alternately add -4 and multiply by -2.
  • A096422 (program): a(1)=a(2)=a(3)=1, a(n) = 2*a(n-1)*a(n-3) + a(n-2)^2 for n > 3.
  • A096423 (program): a(1)=a(2)=1, a(n) = (a(n-1)+1)*(a(n-2)+1) for n > 2.
  • A096424 (program): a(1)=a(2)=1, a(n) = (a(n-2)+F(n-2)) * (a(n-1)+F(n-1)) for n > 2 where F(i) is the i-th Fibonacci no.
  • A096430 (program): Numerator of (9(n^4 - 2n^3 + 2n^2 - n) + 2)/(2(2n-1)).
  • A096431 (program): Denominator of (9*(n^4 - 2*n^3 + 2*n^2 - n) + 2)/(2*(2*n - 1)).
  • A096432 (program): Let n = 2^e_2 * 3^e_3 * 5^e_5 * … be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, …} is a prime.
  • A096441 (program): Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.
  • A096442 (program): a(n) = Max { k>0 : denominator(S(k,2n+1)) } where S(k,s)=sum(i=1,k,i^s*H(i,2)) - H(k,2)*H(k,-s) and H(k,r)=sum(i=1,k,1/i^r) are the generalized harmonic numbers.
  • A096444 (program): Decimal expansion of (Pi - 1)/2.
  • A096446 (program): Number of reduced primitive positive definite binary quadratic forms of determinant n.
  • A096457 (program): If n is prime replace n with the next prime.
  • A096458 (program): If n is prime, the next prime after the next prime after n. otherwise n.
  • A096459 (program): Triangle read by rows: T(n,k) = n^2 mod prime(k), 1<=k<=n.
  • A096461 (program): a(1)=1, a(2)=2; for n >= 2, a(n+1) = a(n) + sum of prime factors of a(n).
  • A096465 (program): Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).
  • A096470 (program): Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column and in the main diagonal ((n,n) entries) to 1 and defining the rest of the entries by the recursion T(n,k) = T(n-1,k) - T(n,k-1).
  • A096471 (program): Number of degree-n permutations with exactly one even cycle.
  • A096472 (program): Numbers containing squares of Pythagorean triples in their divisor set.
  • A096474 (program): Difference prime(q+2) - prime(q) as q runs through the lesser-twin-primes.
  • A096483 (program): Integer part of the square root of n-th decimal repunit.
  • A096484 (program): Integer part of the square root of [2n-1]-th decimal repunit.
  • A096487 (program): Largest term in periodic part of continued fraction expansion of square root of n-th repunit.
  • A096489 (program): Noncomposite numbers n such that number of decimal digits of n = number of divisors of n.
  • A096491 (program): a(n) = sqrt(n) of n if n is a perfect square, otherwise a(n) = largest term in period of continued fraction expansion of square root of n.
  • A096494 (program): Largest value in the periodic part of the continued fraction of sqrt(prime(n)).
  • A096500 (program): Let f(n) = smallest prime > n; a(n) = f(n+1) - f(n).
  • A096501 (program): Difference between primes preceding n+1 and n.
  • A096532 (program): Number of composite numbers not greater than the n-th composite number that do not divide any number not greater than the n-th composite number.
  • A096533 (program): Number of composite numbers not greater than the n-th composite number that divide at least one other number not greater than the n-th composite number.
  • A096534 (program): a(1) = 0; a(2) = 1; a(n) = (a(n-1) + a(n-2)) mod n.
  • A096535 (program): a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.
  • A096569 (program): Number of compositions of n with first part 1 and no equal adjacent parts; this is column 1 of the array in A096568.
  • A096577 (program): Number of fixed points of solid partitions under ‘time-lapse’ operation.
  • A096579 (program): Number of partitions of an n-set with exactly one even block.
  • A096582 (program): From the “100 Green Bottles” song.
  • A096603 (program): Numbers occurring twice in A096607.
  • A096604 (program): Numbers that appear at most once in A096607.
  • A096605 (program): a(n) = floor((floor(n/2) + a(floor(n/2)))/2), a(1) = 1.
  • A096606 (program): a(n) = A002264(n-1) - A096605(n).
  • A096607 (program): a(n) = A096605(2*n).
  • A096615 (program): Decimal expansion of 5 Pi^2/96.
  • A096617 (program): Numerator of n*HarmonicNumber(n).
  • A096620 (program): Denominator of -3*n + 2*(1+n)*HarmonicNumber(n).
  • A096627 (program): Decimal expansion of golden angle in degrees, 360*(2-phi).
  • A096630 (program): Numbers n for which the set of first n primes (mod 3) is biased towards 1.
  • A096646 (program): Triangle (read by rows) where the number of row entries increases by steps of 2 and the entries are stacked in a rectangular fashion. The end entries = 1. Rest of entries in the n-th row are the sum of the entries directly above and to the left and right in all previous rows (total of 3*(n-1) entries).
  • A096647 (program): Number of partitions of an n-set with even number of even blocks.
  • A096648 (program): Number of partitions of an n-set with odd number of even blocks.
  • A096654 (program): Denominators of self-convergents to 1/(e-2).
  • A096655 (program): a(n) = F(n+1)*a(n-1) + F(n)*a(n-2), where F = A000045 (Fibonacci numbers), a(0)=1, a(1)=1.
  • A096656 (program): a(n) = F(n+2)*a(n-1) + F(n+1)*a(n-2), where F = A000045 (Fibonacci numbers), a(0)=1, a(1)=2.
  • A096657 (program): a(n) = (2^n)*a(n-1) + (2^(n-1))*a(n-2), a(0)=1, a(1)=3.
  • A096658 (program): a(n) = (2^n)*a(n-1) + (2^(n-1))*a(n-2), a(0)=1, a(1)=2.
  • A096661 (program): Fine’s numbers J(n).
  • A096675 (program): a(n) = A096786(n)/2.
  • A096676 (program): a(n) = (A096788(n)-1)/2.
  • A096677 (program): A060254 indexed by A000040.
  • A096678 (program): A096785 indexed by A000040.
  • A096679 (program): A096787 indexed by A000040.
  • A096689 (program): Numbers n such that 2n^2 + 3n + 3 is prime.
  • A096691 (program): Numbers n such that 8n^2 + 6n + 3 is prime.
  • A096713 (program): Irregular triangle T(n,k) of nonzero coefficients of the modified Hermite polynomials (n >= 0 and 0 <= k <= floor(n/2)).
  • A096726 (program): Expansion of eta(q^3)^10 / (eta(q) * eta(q^9))^3 in powers of q.
  • A096727 (program): Expansion of eta(q)^8 / eta(q^2)^4 in powers of q.
  • A096736 (program): a(1) = 2; for n>1: a(n) = integer part of x-value when y=0 in (y-tau(n))/(x-1)=(1-tau(n))/(n-1), tau=A000005.
  • A096737 (program): a(1) = 2; for n>1: a(n) = integer part of y-value when x=0 in (y-tau(n))/(x-1)=(1-tau(n))/(n-1), tau=A000005.
  • A096738 (program): Numbers n such that A096736(n) = (n*tau(n)-1)/(tau(n)-1).
  • A096748 (program): Expansion of (1+x)^2/(1-x^2-x^4).
  • A096750 (program): Expansion of (1-x+x^2)/(1-2x+2x^2-x^3-x^4).
  • A096765 (program): Number of partitions of n into distinct parts, the least being 1.
  • A096769 (program): a(n)=Max{ (i+j)!/i!^2 | 0<=i,j<=n }.
  • A096773 (program): a(n+2) = 4*a(n) + 1; a(1) = 0, a(2) = 3.
  • A096777 (program): a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.
  • A096778 (program): Number of partitions of n with at most two even parts.
  • A096784 (program): Numbers n such that both n and n+1 are composite numbers that sum up to a prime.
  • A096785 (program): Primes of form 4n+1 which are the sum of two consecutive composite numbers.
  • A096786 (program): Numbers n such that both n and n+1 are composite numbers that sum up to a Pythagorean prime (i.e., of the form 4k+1).
  • A096787 (program): Primes of form 4n+3 that are the sum of two consecutive composite numbers.
  • A096788 (program): Numbers n such that both n and n+1 are composite numbers that add up to a prime of the form 4k+3.
  • A096789 (program): Decimal expansion of BesselI(1,2).
  • A096791 (program): Number of partitions of n into distinct parts with even number of even parts.
  • A096792 (program): Number of partitions of n into distinct parts with odd number of even parts.
  • A096794 (program): Triangle read by rows: a(n,k) = number of Dyck n-paths such that number of DUs at level 1 plus number of UDs at level 2 is k, 0<=k<=n-1.
  • A096796 (program): a(n) = n for n <= 2; for n > 2, a(n) = 2a(n-1) - a(n - floor( 1/2 + sqrt(2n) )).
  • A096824 (program): a(n) = n for n <= 2; for n > 2, a(n) = 2a(n-1) - a(n - floor(1/2 + sqrt(2(n-1)))).
  • A096825 (program): Maximal size of an antichain in divisor lattice D(n).
  • A096828 (program): Numbers that must appear in any variation of A097390.
  • A096829 (program): Numbers that can appear an infinite number of times in a variation of A097390.
  • A096881 (program): Expansion of (1+4*x)/(1-17*x^2).
  • A096882 (program): Expansion of (1+7x)/(1-50x^2).
  • A096883 (program): Expansion of (1+10x)/(1-101x^2).
  • A096884 (program): a(n) = 101^n.
  • A096885 (program): Related to diagonals of Pascal’s triangle.
  • A096886 (program): Expansion of (1+3*x)/(1-8*x^2).
  • A096891 (program): Least hypotenuse of primitive Pythagorean triangles with odd leg 2n+1.
  • A096892 (program): Least semiperimeter of primitive Pythagorean triangles with odd leg 2n+1.
  • A096893 (program): Least area/6 of primitive Pythagorean triangles with odd leg 2n+1.
  • A096894 (program): Least inradius of primitive Pythagorean triangles with odd leg 2n+1.
  • A096895 (program): a(n) = A088557(n)/4.
  • A096896 (program): Least hypotenuse of primitive Pythagorean triangles with even leg 4n.
  • A096897 (program): Least semiperimeter of primitive Pythagorean triangles with even leg 4n.
  • A096898 (program): Least area/6 of primitive Pythagorean triangles with even leg 4n.
  • A096899 (program): Least inradius of primitive Pythagorean triangles with even leg 4n.
  • A096900 (program): a(n) = (A088558(n)-1)/2.
  • A096911 (program): Number of partitions of n into distinct parts with exactly one even part.
  • A096914 (program): Number of partitions of 2*n into distinct parts with exactly two odd parts.
  • A096916 (program): Lesser prime factor of n-th product of two distinct primes.
  • A096917 (program): Smallest prime factor of n-th product of 3 distinct primes.
  • A096918 (program): Intermediate prime factor of n-th product of 3 distinct primes.
  • A096919 (program): Greatest prime factor of n-th product of 3 distinct primes.
  • A096920 (program): Expansion of q^(-1/12) * eta(q^2)^4 / (eta(q)^2 * eta(q^4)) in powers of q.
  • A096921 (program): Triangle array of binomial coefficients.
  • A096922 (program): Numbers n for which there is a unique k such that n = k + (product of nonzero digits of k).
  • A096923 (program): Numbers n for which there are exactly two k such that n = k + (product of nonzero digits of k).
  • A096924 (program): Numbers n for which there are exactly three k such that n = k + (product of nonzero digits of k).
  • A096932 (program): Smallest number having exactly s divisors, where s is the n-th semiprime (A001358).
  • A096936 (program): Half of number of integer solutions to the equation x^2 + 3y^2 = n.
  • A096939 (program): Number of sets of odd number of even lists, cf. A000262.
  • A096940 (program): Pascal (1,5) triangle.
  • A096941 (program): Fourth column of (1,5)-Pascal triangle A096940.
  • A096942 (program): Fifth column of (1,5)-Pascal triangle A096940.
  • A096943 (program): Sixth column of (1,5)-Pascal triangle A096940.
  • A096944 (program): Seventh column of (1,5)-Pascal triangle A096940.
  • A096945 (program): Eighth column of (1,5)-Pascal triangle A096940.
  • A096946 (program): Ninth column of (1,5)-Pascal triangle A096940.
  • A096947 (program): Tenth column of (1,5)-Pascal triangle A096940.
  • A096948 (program): Triangular table read by rows: T(n,m) = number of rectangles found in an n X m rectangle built from 1 X 1 squares, 1 <= m <= n.
  • A096949 (program): Numerators of partial sums of series for 3*arctanh(1/3) = (3/2)*log(2).
  • A096950 (program): Denominators of partial sums of series for 3*arctanh(1/3) = (3/2)*log(2).
  • A096951 (program): Sum of odd powers of 2 and of 3 divided by 5.
  • A096952 (program): Numerators of upper bounds for Lagrange remainder in Taylor’s expansion of log((1+x)/(1-x)) for x=1/3, multiplied by 6/5.
  • A096956 (program): Pascal (1,6) triangle.
  • A096957 (program): Fourth column (m=3) of (1,6)-Pascal triangle A096956.
  • A096958 (program): Fifth column (m=4) of (1,6)-Pascal triangle A096956.
  • A096959 (program): Sixth column (m=5) of (1,6)-Pascal triangle A096956.
  • A096960 (program): a(n) = Sum_{0<d|n, n/d odd} d^5.
  • A096961 (program): a(n) = Sum_{0<d|n, n/d odd} d^7.
  • A096962 (program): a(n) = Sum_{0<d|n, n/d odd} d^9.
  • A096963 (program): a(n) = Sum {0<d|n, n/d odd} d^11.
  • A096965 (program): Number of sets of even number of even lists, cf. A000262.
  • A096968 (program): a(n) = (prime(n)*prime(n+2))^4.
  • A096972 (program): Number of preimages of n (or immediate predecessors) under map f(k) = k + (product of nonzero digits of k).
  • A096975 (program): Trace sequence of a path graph plus loop.
  • A096976 (program): Number of walks of length n on P_3 plus a loop at the end.
  • A096977 (program): a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).
  • A096978 (program): Sum of the areas of the first n Jacobsthal rectangles.
  • A096979 (program): Sum of the areas of the first n+1 Pell triangles.
  • A096980 (program): Expansion of (1+3x)/(1-2x-7x^2).
  • A096982 (program): Define a(1)=0. Then define b(1)=1, b(2)=1 and for k > 2, b(k) = (b(k-2) + b(k-1)) (mod n). This gives for each n a cyclic repetitive sequence; a(n) is the least k > 1 for the first 1 of the sequence 1,1,….
  • A097009 (program): a(n) = gcd(prime(2*n)-1, prime(n)-1).
  • A097011 (program): Remainder of sigma(n) modulo 30.
  • A097017 (program): a(n) = sigma(5*n) modulo 6.
  • A097022 (program): a(n) = (sigma(2n^2)-3)/6.
  • A097038 (program): A Jacobsthal variant.
  • A097039 (program): a(n) = Sum_{i=0..n} i*L(i), where L = A000032.
  • A097040 (program): a(n) = 2*sum(C(n,2k+1)*F(2k), k=0..floor((n-1)/2)), where F(n) are Fibonacci numbers A000045.
  • A097041 (program): Expansion of (1+x)/(1-x^2-9*x^3).
  • A097043 (program): a(n) = n - a(floor(sqrt(n))) for n > 1; a(1) = 1.
  • A097044 (program): a(n) = n + a(floor(sqrt(n))) for n > 1; a(1) = 1.
  • A097050 (program): Smallest prime > n(n+1)/2.
  • A097053 (program): First occurrence of n in A097051.
  • A097057 (program): Number of integer solutions to a^2 + b^2 + 2*c^2 + 2*d^2 = n.
  • A097058 (program): Numbers of the form p^2 + 2^p for p prime.
  • A097059 (program): Numbers of the form p^3 + 3^p for p prime.
  • A097062 (program): Interleave 2*n+1 and 2*n-1.
  • A097063 (program): Expansion of (1-2*x+3*x^2)/((1+x)*(1-x)^3).
  • A097064 (program): Expansion of (1-4x+6x^2)/(1-2x)^2.
  • A097065 (program): Interleave n+1 and n-1.
  • A097066 (program): Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).
  • A097067 (program): Expansion of (1-4*x+5*x^2)/(1-2*x)^2.
  • A097068 (program): a(n)=Sum(C(n,2k+1)5^k 3^(2k+1) 7^(n-2k-1), k=0,..,Floor[(n-1)/2]).
  • A097069 (program): Positive integers n such that 2n - 9 is prime.
  • A097070 (program): Consider all compositions (ordered partitions) of n into n parts, allowing zeros. E.g., for n = 3 we get 300, 030, 003, 210, 120, 201, 102, 021, 012, 111. Then a(n) is the total number of 1’s.
  • A097071 (program): Number of Shubnikov compounds.
  • A097072 (program): Expansion of (1 - 2*x + 2*x^2)/((1 - x^2)*(1 - 2*x)).
  • A097073 (program): Expansion of (1-x+2*x^2)/((1+x)*(1-2*x)).
  • A097074 (program): Expansion of (1-x+2*x^2)/((1-x)*(1-x-2*x^2)).
  • A097075 (program): Expansion of (1-x-x^2)/(1-x-3*x^2-x^3).
  • A097076 (program): Expansion of x/(1 - x - 3*x^2 - x^3).
  • A097080 (program): a(n) = 2*n^2 - 2*n + 3.
  • A097081 (program): a(n) = Sum_{k=0..n} C(n,4k)*2^k.
  • A097083 (program): Values of k such that there is exactly one permutation p of (1,2,3,…,k) such that i+p(i) is a Fibonacci number for 1<=i<=k.
  • A097105 (program): Gregorian years containing two Islamic New Year Days.
  • A097108 (program): If a geodesic dome is made by dividing each triangle of an icosahedron into n^2 identical equilateral triangles and the vertices of those newly created triangles are pushed out from the center to lie on the surface of the sphere in which the icosahedron is inscribed, then this sequence gives the number of different strut lengths that are required to build the dome.
  • A097109 (program): G.f.: s(2)^2*s(3)^3/(s(1)*s(6)^2), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind’s function, cf. A010815.
  • A097110 (program): Expansion of (1 + 2x - 2x^3) / (1 - 3x^2 + 2x^4).
  • A097111 (program): Expansion of (1 + 3x - 2x^2 - 12x^3)/(1 - 9x^2 + 20x^4).
  • A097112 (program): Expansion of (1+4x-6x^2-36x^3)/(1-19x^2+90x^4).
  • A097113 (program): Expansion of (1 + 5*x - 12*x^2 - 80*x^3)/(1 - 33*x^2 + 272*x^4).
  • A097114 (program): Expansion of (1 + 8x - 42x^2 - 392x^3)/(1 - 99x^2 + 2450x^4).
  • A097115 (program): Expansion of (1 + 11*x - 90*x^2 - 1100*x^3)/(1 - 201*x^2 + 10100*x^4).
  • A097116 (program): Expansion of (1-x)/((1-x)^2-3x^3).
  • A097117 (program): Expansion of (1-x)/((1-x)^2 - 4*x^3).
  • A097118 (program): Expansion of (1+x)/(1-x)^2-5x^3).
  • A097119 (program): Expansion of (1-x)^2/((1-x)^3-2x^4).
  • A097120 (program): Expansion of (1-x)^2/((1-x)^3-3x^4).
  • A097121 (program): Expansion of (1-x)^2/((1-x)^3-4x^4).
  • A097122 (program): Expansion of (1-x)^2/((1-x)^3 - 3*x^3).
  • A097123 (program): Expansion of (1-x)^2/((1-x)^3 - 4*x^3).
  • A097124 (program): Expansion of (1-x)^2/((1-x)^3-5x^3).
  • A097131 (program): F(n)+(-1)^n*F(n-1).
  • A097132 (program): a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).
  • A097133 (program): 3*Fibonacci(n)+(-1)^n.
  • A097134 (program): a(n) = 3*Fibonacci(2*n) + 0^n.
  • A097135 (program): a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).
  • A097136 (program): a(n) = 3*Fibonacci(2*n) + 1.
  • A097137 (program): Convolution of 3^n and floor(n/2).
  • A097138 (program): Convolution of 4^n and floor(n/2).
  • A097139 (program): Convolution of 5^n and floor(n/2).
  • A097140 (program): Interleave n and 1-n.
  • A097141 (program): Expansion of x*(1+2*x)/(1+x)^2.
  • A097162 (program): Sum( k=0..n, C(floor((n+1)/2),floor((k+1)/2))*2^k ).
  • A097163 (program): Expansion of (1+x-x^2)/((1-x)*(1-4*x^2)).
  • A097164 (program): Expansion of (1+3x)/((1-x)(1-4x^2)).
  • A097165 (program): Expansion of (1-3x)/((1-x)(1-4x)(1-5x)).
  • A097166 (program): Expansion of (1+2*x)/((1-x)*(1-10*x)).
  • A097167 (program): 3*10^n-2*9^n.
  • A097168 (program): Expansion of (1-7x)/((1-x)(1-9x)(1-10x).
  • A097169 (program): a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2)) * 3^k.
  • A097173 (program): Total number of green nodes among tricolored labeled trees on n nodes.
  • A097174 (program): Total number of red nodes among tricolored labeled trees on n nodes.
  • A097175 (program): a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 4^k.
  • A097176 (program): a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 5^k.
  • A097177 (program): a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 10^k.
  • A097178 (program): Expansion of (1+10*x-101*x^2-900*x^3)/((1-100*x^2)*(1-101*x^2)).
  • A097179 (program): Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).
  • A097180 (program): Row sums of triangle A097179, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A077860(y)^(n+1), where R_n(1/2) = 4^n for all n>=0.
  • A097183 (program): Main diagonal of triangle A097181, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097182(y)^(n+1), where R_n(1/2) = 8^n for all n>=0.
  • A097184 (program): G.f. A(x) satisfies A097182(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097181.
  • A097186 (program): Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 3^n, where R_n(y) forms the initial (n+1) terms of g.f. A057083(y)^(n+1).
  • A097188 (program): G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.
  • A097189 (program): Row sums of triangle A097186, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A057083(y)^(n+1), where R_n(1/3) = 3^n for all n >= 0.
  • A097192 (program): Main diagonal of triangle A097190, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A097191(y)^(n+1), where R_n(1/3) = 9^n for all n>=0.
  • A097193 (program): G.f. A(x) satisfies A097191(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097190.
  • A097195 (program): Expansion of s(12)^3*s(18)^2/(s(6)^2*s(36)), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind’s function, cf. A010815. Then replace q^6 with q.
  • A097196 (program): Expansion of psi(x^3)^2 / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions.
  • A097197 (program): Expansion of q^(-1/3) * eta(q^6)^2 / (eta(q) * eta(q^3)) in powers of q.
  • A097199 (program): Numbers of the form p^4 + 4^p for p prime.
  • A097200 (program): Numbers of the form p^5 + 5^p for p prime.
  • A097201 (program): Numbers of the form 4^p - p^4 for p prime.
  • A097202 (program): Numbers of the form 5^p - p^5 for p prime.
  • A097204 (program): Binomial transform of A033312.
  • A097205 (program): Numbers of the form 3^p * p^3 for p prime.
  • A097206 (program): Numbers of the form 5^p * p^5 for p prime.
  • A097207 (program): Triangle read by rows: T(n,k) = binomial(n,k) + 2*binomial(n,k-1).
  • A097219 (program): Numbers n that are the hypotenuse of exactly 6 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 6 ways.
  • A097230 (program): Triangle read by rows: number of binary sequences with no isolated 1’s.
  • A097240 (program): a(n) = (n+1)*prime(n) + n*prime(n+1).
  • A097241 (program): Primes of the form (k+1)*prime(k) + k*prime(k+1).
  • A097242 (program): Expansion of q-series 1 / (q^2, q^3, q^9, q^10; q^12)_infinity.
  • A097243 (program): Expansion of 1 + 32 * (eta(q^4) / eta(q))^8 in powers of q.
  • A097250 (program): Smallest m such that A097249(m) = n; from n=1 onwards, twice the primorials, 2*A002110(n).
  • A097251 (program): Numbers whose set of base 5 digits is {0,4}.
  • A097252 (program): Numbers whose set of base 6 digits is {0,5}.
  • A097253 (program): Numbers whose set of base 7 digits is {0,6}.
  • A097254 (program): Numbers whose set of base 8 digits is {0,7}.
  • A097255 (program): Numbers whose set of base 9 digits is {0,8}.
  • A097256 (program): Numbers whose set of base 10 digits is {0,9}.
  • A097257 (program): Numbers whose set of base 11 digits is {0,A}, where A base 11 = 10 base 10.
  • A097258 (program): Numbers whose set of base 12 digits is {0,B}, where B base 12 = 11 base 10.
  • A097259 (program): Numbers whose set of base 13 digits is {0,C}, where C base 13 = 12 base 10.
  • A097260 (program): Numbers whose set of base 14 digits is {0,D}, where D base 14 = 13 base 10.
  • A097261 (program): Numbers whose set of base 15 digits is {0,E}, where E base 15 = 14 base 10.
  • A097262 (program): Numbers whose set of base 16 digits is {0,F}, where F base 16 = 15 base 10.
  • A097268 (program): Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.
  • A097269 (program): Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.
  • A097272 (program): Least integer with same “mod 2 prime signature” as n.
  • A097273 (program): Least integer with each “mod 2 prime signature”.
  • A097280 (program): Perimeter of integer triangle (A001611(n), A001611(n+1), A001611(n+2)).
  • A097283 (program): Contains exactly once every pair (i,j) satisfying 0 < i < j.
  • A097285 (program): Contains exactly once every pair (i,j) of distinct positive integers.
  • A097291 (program): Contains exactly once every pair (i,j) of positive integers.
  • A097297 (program): Seventh column (m=6) of (1,6)-Pascal triangle A096956.
  • A097298 (program): Eighth column (m=7) of (1,6)-Pascal triangle A096956.
  • A097299 (program): Ninth column (m=8) of (1,6)-Pascal triangle A096956.
  • A097300 (program): Tenth column (m=9) of (1,6)-Pascal triangle A096956.
  • A097302 (program): Denominators of rationals used in the Euler-Maclaurin type derivation of Stirling’s formula for N!.
  • A097308 (program): Chebyshev T-polynomials T(n,13) with Diophantine property.
  • A097309 (program): Chebyshev polynomials of the second kind, U(n,x), evaluated at x=13.
  • A097310 (program): Chebyshev T-polynomials T(n,14) with Diophantine property.
  • A097311 (program): Chebyshev polynomials of the second kind, U(n,x), evaluated at x=14.
  • A097313 (program): Chebyshev polynomials of the second kind, U(n,x), evaluated at x=15.
  • A097314 (program): Pell equation solutions (3*a(n))^2 - 10*b(n)^2 = -1 with b(n) = A097315(n), n >= 0.
  • A097315 (program): Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n) = A097314(n), n >= 0.
  • A097316 (program): Chebyshev U(n,x) polynomial evaluated at x=33.
  • A097317 (program): Numbers of the form 7^p + p^7 for p prime.
  • A097321 (program): a(n) = (3*n-1) * 3*n * (3*n+1).
  • A097322 (program): (2n)! divided by denominator of Taylor expansion of exp(cos(x)-1).
  • A097325 (program): Period 6: repeat [0, 1, 1, 1, 1, 1].
  • A097326 (program): Largest integer m such that m*n has the same decimal digit length as n.
  • A097327 (program): Least positive integer m such that m*n has greater decimal digit length than n.
  • A097328 (program): Denominator of 1 + 1/5 + 1/9 +…+ 1/(4n+1).
  • A097329 (program): Least common multiple of {3,7,11,…,4n+3}.
  • A097330 (program): In the sequence of prime numbers replace each term p with floor(p/2) and ceiling(p/2).
  • A097331 (program): Expansion of 1 + 2x/(1 + sqrt(1 - 4x^2)).
  • A097332 (program): Expansion of (1/(1-x))(1+2x/(1-x+sqrt(1-2x-3x^2))).
  • A097333 (program): a(n) = Sum_{k=0..n} C(n-k, floor(k/2)).
  • A097334 (program): Sum k=0..n, C(n-k, floor(k/2))2^k.
  • A097335 (program): Sum k=0..n, C(n-k, floor(k/2))3^k.
  • A097336 (program): Sum k=0..n, C(n-k, floor(k/2))4^k.
  • A097337 (program): Integer part of the edge of a cube that has space-diagonal n.
  • A097338 (program): Positive integers n such that 2n-11 is prime.
  • A097339 (program): 2^n+n^3.
  • A097340 (program): McKay-Thompson series of class 4A for the Monster group with a(0) = 24.
  • A097344 (program): Numerators in binomial transform of 1/(n+1)^2.
  • A097345 (program): Numerators of the partial sums of the binomial transform of 1/(n+1).
  • A097357 (program): For definition see Comments lines.
  • A097358 (program): Primes of the form 2(p+q) + 1, p and q prime.
  • A097361 (program): Primes of the form 2(p+q) - 1, where p and q are consecutive primes.
  • A097362 (program): a(n) = (n+1)/2 if n is odd, n+2 otherwise.
  • A097363 (program): Positive integers n such that 2n-13 is prime.
  • A097367 (program): Fibonacci regression array: For n>=2 and 1<=k<=n-1, T(n,k) is the last term before the first nonpositive term in the sequence n, k, n-k, 2k-n, 2n-3k, 5k-3n, …
  • A097369 (program): Position in row n of Fibonacci regression array (A097367) where the least term first occurs.
  • A097377 (program): CubeFreeKernel(n) + 1.
  • A097378 (program): SquareFreeKernel(n)*CubeFreeKernel(n) + 1.
  • A097379 (program): Numbers m such that 1+SquareFreeKernel(m) is prime.
  • A097380 (program): Numbers m such that 1+CubeFreeKernel(m) is prime.
  • A097381 (program): Numbers m such that 1+SquareFreeKernel(m)*CubeFreeKernel(m) is prime.
  • A097382 (program): a(h) = d(h,j) = lcm( f(h,j,1) … f(h,j,h) ), when j=2.
  • A097383 (program): Minimum total number of comparisons to find each of the values 1 through n using a binary search with 3-way comparisons (less than, equal and greater than).
  • A097384 (program): Total number of comparisons to find each of the values 1 through n using a binary search with 3-way comparisons (less than, equal and greater than), always choosing the mid-most value to compare to.
  • A097388 (program): 2n-th derivative of the Gaussian exp(-x^2) evaluated at x=0.
  • A097389 (program): Numbers that appear in A097390.
  • A097391 (program): The number of hierarchies with at least one subhierarchy composed of exactly 2 levels and no subhierarchy with more than 2 levels.
  • A097401 (program): Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct nonnegative integers chosen from the range 0..n.
  • A097403 (program): Minimum wind speed in knots for Beaufort Number n.
  • A097408 (program): Initial decimal digit of n^4.
  • A097409 (program): Initial decimal digit of n^5.
  • A097410 (program): Initial decimal digit of n^6.
  • A097411 (program): Initial decimal digit of n^7.
  • A097412 (program): Initial decimal digit of n^8.
  • A097413 (program): Initial decimal digit of n^9.
  • A097414 (program): Initial decimal digit of n^10.
  • A097415 (program): Values of k such that the first digit of 2^k is 9.
  • A097422 (program): Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j.
  • A097426 (program): Integer part of the circumference of circles with prime diameters.
  • A097427 (program): Integer part of the area of circles with prime radii.
  • A097428 (program): Integer part of the diameters of circles with prime circumferences.
  • A097429 (program): Integer part of the radii of circles with prime areas.
  • A097430 (program): Integer part of the radii of circles with area n.
  • A097431 (program): Integer part of the hypotenuse of right triangles with consecutive prime legs.
  • A097432 (program): Integer part of the hypotenuse of right triangles with consecutive integer legs.
  • A097433 (program): Integer part of the hypotenuse of prime leg isosceles right triangles.
  • A097446 (program): Concatenate consecutive prime-sided isosceles triangles.
  • A097447 (program): Primes in the concatenation of consecutive prime-sided isosceles triangles.
  • A097448 (program): If n is square, replace with sqrt(n).
  • A097449 (program): If n is a cube, replace it with the cube root of n.
  • A097451 (program): Number of partitions of n into parts congruent to {2, 3, 4} mod 6.
  • A097453 (program): Primes p of the form p = prime(k) - composite(k) for some k.
  • A097454 (program): a(n) = (number of nonprimes <= n) - (number of primes <= n).
  • A097455 (program): a(n) = gcd(prime(n)+1, composite(n)).
  • A097456 (program): Integer part of the ratio (number of composites <=n) / (number of primes <=n).
  • A097462 (program): Multiplication table for numbers 0 through 10 read by rows.
  • A097468 (program): Number of 1’s in the decimal expansion of the lesser of twin primes.
  • A097470 (program): Number of 0’s in the decimal expansion of the lesser of twin primes.
  • A097471 (program): First differences of A076678.
  • A097472 (program): Number of different candle trees having a total of m edges.
  • A097480 (program): Positive integers n such that 2n-15 is prime.
  • A097482 (program): a(1) = 1, a(2) = 1, a(n) = floor(sqrt(a(n-2)*a(n-1))) + 3 for n > 2.
  • A097489 (program): a(n) = product of first n terms of A001359.
  • A097492 (program): a(n) = product of first n terms of A006512.
  • A097494 (program): a(n) = floor( prime(n+1)*prime(n+2)/prime(n) ).
  • A097495 (program): Subsequence of terms of even index in the Somos-5 sequence.
  • A097496 (program): Subsequence of terms of odd index of the Somos-5 sequence.
  • A097497 (program): Floor( prime(n)*(prime(n)+prime(n+1))/prime(n+1)).
  • A097502 (program): Least integer m such that there are at least n composite numbers between m and 2*m.
  • A097503 (program): Numbers k such that A000203(k) = sigma(k) is not divisible by 6.
  • A097505 (program): Triangle read by rows: T(n,k) = Sum_{j=1..k} Prime(n+j-1).
  • A097508 (program): Differences between floor(n*sqrt(2)) and n.
  • A097509 (program): a(n) is the number of times that n occurs as floor(k * sqrt(2)) - k.
  • A097512 (program): a(n) = 6*Lucas(2n) - Fibonacci(2n+2).
  • A097514 (program): Number of partitions of an n-set without blocks of size 2.
  • A097520 (program): Numbers of the form p^11 + 11^p for p prime.
  • A097535 (program): Dimensions of spaces of cusp forms.
  • A097537 (program): -Sum_{k=1..2*q-1} J(k,q)*J(-4,k)*k/4 as q runs through primes == 3 (mod 4), where J(i,j) is the Jacobi symbol.
  • A097538 (program): Subtract 2 from primes == 3 (mod 4).
  • A097539 (program): Subtract 4 from primes == 1 (mod 4).
  • A097550 (program): Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.
  • A097551 (program): Number of positive words of length n in the monoid Br_4 of positive braids on 5 strands.
  • A097557 (program): a(n+1) = a(n) + number of nonprimes so far; a(1) = 1.
  • A097559 (program): a(n) = number of terms among {a(1), a(2), a(3), …, a(n-1)} that are coprime to n; a(1)=2.
  • A097560 (program): a(n) = number of terms among {a(1), a(2), a(3), …, a(n-1)} that are coprime to n; a(1)=3.
  • A097562 (program): a(n) = number of terms among {a(1), a(2), a(3), …, a(n-1)} that are coprime to n; a(1)=5.
  • A097564 (program): a(n) = (a(n-1) mod 2)*a(n-1) + a(n-2) with a(0)=0, a(1)=1.
  • A097575 (program): A 2 X 2 matrix Fibonacci sequence.
  • A097578 (program): a(n) = (2*n+1)*2^floor((n+1)/2).
  • A097579 (program): Triangular numbers k such that 2*k-1 is also a triangular number.
  • A097580 (program): Base 3 representation of the concatenation of the first n numbers with the most significant digits first.
  • A097581 (program): a(n) = 3*2^floor((n-1)/2) + (-1)^n.
  • A097582 (program): Base 7 representation of the concatenation of the first n numbers with the most significant digits first.
  • A097583 (program): Octal representation of the concatenation of the first n decimal numbers with the most significant digits first.
  • A097593 (program): Number of increasing runs of even length in all permutations of [n].
  • A097594 (program): a(0) = 3, a(1) = 2, a(n) = Mod[a(n-1),a(n-2)] + a(n-2) for n > 1.
  • A097595 (program): a(0) = 2, a(1) = 5, a(n) = a(n-2)^2 + a(n-1)a(n-2).
  • A097596 (program): An A001644 Binet like function for a Bonacci 3 type sequence using two negative roots instead of all positive.
  • A097599 (program): Differences between A097598 and A047842.
  • A097602 (program): a(n+1) = a(n) + number of squares so far; a(1) = 1.
  • A097603 (program): Multiples of perfect numbers.
  • A097606 (program): a(n) = number of terms among {a(1), a(2), a(3), …, a(n-1)} that are coprime to n; a(1)=6.
  • A097609 (program): Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k horizontal steps at level 0.
  • A097610 (program): Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.
  • A097613 (program): a(n) = binomial(2n-3,n-1) + binomial(2n-2,n-2).
  • A097625 (program): a(n) = Sum_{k=0..n} (-2)^k * binomial(2n-k,k) * (n-k)!.
  • A097627 (program): Number of rooted directed trees on n nodes with a red root.
  • A097628 (program): Number of rooted directed trees on n nodes with a green root.
  • A097629 (program): a(n) = 2*(2n)^(n-2).
  • A097630 (program): Number of unrooted directed trees on n nodes with a red root.
  • A097631 (program): Number of unrooted directed trees on n nodes with a green root.
  • A097632 (program): 2^n * Lucas(n).
  • A097648 (program): a(n) is the least non-palindromic number m such that phi(m)=phi(reversal(m))=4*10^(n+2), or 0 if no such number exists.
  • A097656 (program): Binomial transform of A038507.
  • A097657 (program): Fibonacci sequence with first two terms 11 and 23.
  • A097659 (program): a(n) = 1001^n.
  • A097660 (program): Number of zeros in decimal representation of 1001^n.
  • A097662 (program): a(n) = A002720(n) - 1.
  • A097677 (program): E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+1)/(3*i+1) ) for an order-3 linear recurrence with varying coefficients.
  • A097678 (program): E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+2)/(3*i+2) ) for an order-3 linear recurrence with varying coefficients.
  • A097679 (program): E.g.f.: (1/(1-x^4))*exp( 4*Sum_{i>=0} x^(4*i+1)/(4*i+1) ) for an order-4 linear recurrence with varying coefficients.
  • A097690 (program): Numerators of the continued fraction n-1/(n-1/…) [n times].
  • A097691 (program): Denominators of the continued fraction n-1/(n-1/…) [n times].
  • A097692 (program): Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs.
  • A097693 (program): Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct integers chosen from the range -n…n.
  • A097697 (program): Numbers k such that 4*k^2 + 3 is prime.
  • A097700 (program): Numbers not of form x^2 + 2y^2.
  • A097701 (program): Expansion of 1/((1-x)^2*(1-x^2)^2*(1-x^3)).
  • A097702 (program): a(n) = (A063880(n) - 108)/216.
  • A097703 (program): Numbers n such that m = 216n + 108 satisfies sigma(m) != 2*usigma(m).
  • A097704 (program): Elements of A097703 not of form 3k + 1.
  • A097705 (program): a(n) = 4*a(n-1) + 17*a(n-2), a(1)=1, a(2)=4.
  • A097706 (program): Part of n composed of prime factors of form 4k+3.
  • A097707 (program): Part of n! composed of prime factors of form 4k+3.
  • A097708 (program): Sum of prime-length repunits: Sum_{k=1..n} r(prime(k)), where r()=A002275.
  • A097714 (program): Repeatedly convert from sexagesimal to centesimal, starting with 60.
  • A097715 (program): Decimal expansion of 7*sqrt(3)/2.
  • A097718 (program): E.g.f. A(x) satisfies A(x) = exp(x(A(x)-2)).
  • A097723 (program): One fourth of sum of divisors of 4n+3.
  • A097725 (program): Chebyshev U(n,x) polynomial evaluated at x=51.
  • A097726 (program): Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.
  • A097727 (program): Pell equation solutions (5*b(n))^2 - 26*a(n)^2 = -1 with b(n)=A097726(n), n >= 0.
  • A097728 (program): Chebyshev U(n,x) polynomial evaluated at x=73 = 2*6^2+1.
  • A097729 (program): Pell equation solutions (6*a(n))^2 - 37*b(n)^2 = -1 with b(n):=A097730(n), n >= 0.
  • A097730 (program): Pell equation solutions (6*b(n))^2 - 37*a(n)^2 = -1 with b(n)=A097729(n), n >= 0.
  • A097731 (program): Chebyshev U(n,x) polynomial evaluated at x=99 = 2*7^2+1.
  • A097732 (program): Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=2*5^2 is not squarefree.
  • A097733 (program): Pell equation solutions (7*b(n))^2 - 2*(5*a(n))^2 = -1 with b(n)=A097732(n), n >= 0. Note that D=50=2*5^2 is not squarefree.
  • A097734 (program): Chebyshev U(n,x) polynomial evaluated at x=129 = 3*43.
  • A097735 (program): Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n >= 0.
  • A097736 (program): Pell equation solutions (8*b(n))^2 - 65*a(n)^2 = -1 with b(n):=A097735(n), n >= 0.
  • A097737 (program): Chebyshev U(n,x) polynomial evaluated at x=163.
  • A097738 (program): Pell equation solutions (9*a(n))^2 - 82*b(n)^2 = -1 with b(n):=A097739(n), n >= 0.
  • A097739 (program): Pell equation solutions (9*b(n))^2 - 82*a(n)^2 = -1 with b(n):=A097738(n), n >= 0.
  • A097740 (program): Chebyshev U(n,x) polynomial evaluated at x=201.
  • A097741 (program): Pell equation solutions (10*a(n))^2 - 101*b(n)^2 = -1 with b(n) = A097742(n), n >= 0.
  • A097742 (program): Pell equation solutions (10*b(n))^2 - 101*a(n)^2 = -1 with b(n)=A097741(n), n >= 0.
  • A097743 (program): Numbers of the form 3*2^(p - 1) - 1 where p is prime.
  • A097750 (program): Reversal of the binomial transform of the Whitney triangle A004070 (see A131250), triangle read by rows, T(n,k) for 0 <= k <= n.
  • A097761 (program): Inverse of binomial transform of Whitney triangle.
  • A097765 (program): Chebyshev U(n,x) polynomial evaluated at x=243=2*11^2+1.
  • A097766 (program): Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n >= 0.
  • A097767 (program): Pell equation solutions (11*b(n))^2 - 122*a(n)^2 = -1 with b(n):=A097766(n), n >= 0.
  • A097768 (program): Chebyshev U(n,x) polynomial evaluated at x=289=2*12^2+1.
  • A097769 (program): Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n >= 0.
  • A097770 (program): Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n)=A097769(n), n >= 0.
  • A097771 (program): Chebyshev U(n,x) polynomial evaluated at x=339=2*13^2+1.
  • A097772 (program): Pell equation solutions (13*a(n))^2 - 170*b(n)^2 = -1 with b(n):=A097771(n), n >= 0.
  • A097773 (program): Pell equation solutions (13*b(n))^2 - 170*a(n)^2 = -1 with b(n):=A097772(n), n >= 0.
  • A097774 (program): Chebyshev U(n,x) polynomial evaluated at x=393=2*14^2+1.
  • A097775 (program): Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n) = A097776(n), n >= 0.
  • A097776 (program): Pell equation solutions (14*b(n))^2 - 197*a(n)^2 = -1 with b(n)=A097775(n), n >= 0.
  • A097778 (program): Chebyshev polynomials S(n,23) with Diophantine property.
  • A097779 (program): Number of Motzkin paths of length n, starting with an up step, ending with a down step and having no peaks (can be easily expressed using RNA secondary structure terminology).
  • A097780 (program): Chebyshev polynomials S(n,25) with Diophantine property.
  • A097781 (program): Chebyshev polynomials S(n,27) with Diophantine property.
  • A097782 (program): Chebyshev polynomials S(n,29) with Diophantine property.
  • A097783 (program): Chebyshev polynomials S(n,11) + S(n-1,11) with Diophantine property.
  • A097784 (program): Partial sums of Chebyshev sequence S(n,10) = U(n,5) = A004189(n+1).
  • A097786 (program): a(n)=3a(n-1)+C(n+3,3),n>0, a(0)=1.
  • A097787 (program): a(n)=3a(n-1)+C(n+4,4),n>0, a(0)=1.
  • A097788 (program): a(n)=4a(n-1)+C(n+3,3),n>0, a(0)=1.
  • A097789 (program): a(n)=4a(n-1)+C(n+4,4),n>0, a(0)=1.
  • A097790 (program): a(n)=5a(n-1)+C(n+3,3),n>0, a(0)=1.
  • A097791 (program): a(n)=5a(n-1)+C(n+4,4),n>0, a(0)=1.
  • A097795 (program): Number of partitions of 2*n into perfect numbers.
  • A097801 (program): a(n) = (2*n)!/(n!*2^(n-1)).
  • A097802 (program): a(n) = 3*(25*n + 1).
  • A097803 (program): a(n) = 3*(2*n^2 + 1).
  • A097804 (program): a(n) = 3*(2*5^n + 1).
  • A097805 (program): Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.
  • A097806 (program): Riordan array (1+x, 1) read by rows.
  • A097807 (program): Riordan array (1/(1+x),1) read by rows.
  • A097808 (program): Riordan array ((1+2x)/(1+x)^2, 1/(1+x)) read by rows.
  • A097809 (program): a(n) = 5*2^n - 2*n - 4.
  • A097810 (program): a(n) = 7*2^n - 3n - 6.
  • A097813 (program): a(n) = 3*2^n - 2*n - 2.
  • A097814 (program): E.g.f. exp(3x)/(1-3x).
  • A097815 (program): E.g.f. exp(4x)/(1-4x).
  • A097816 (program): E.g.f. exp(5x)/(1-5x).
  • A097817 (program): E.g.f. exp(2x)/(1-3x).
  • A097818 (program): Expansion of (8+10x)/((1-x)(1-100x)).
  • A097819 (program): E.g.f. exp(3x)/(1-4x).
  • A097820 (program): Expansion of e.g.f. exp(2*x)/(1-4*x).
  • A097821 (program): E.g.f. exp(2x)/(1-5x).
  • A097826 (program): Partial sums of Chebyshev sequence S(n,11) = U(n,11/2) = A004190(n).
  • A097827 (program): Partial sums of Chebyshev sequence S(n,12)= U(n,6)=A004191(n).
  • A097828 (program): Partial sums of Chebyshev sequence S(n,13)= U(n,13/2)=A078362(n).
  • A097829 (program): Partial sums of Chebyshev sequence S(n,15)= U(n,15/2)=A078364(n).
  • A097830 (program): Partial sums of Chebyshev sequence S(n,16) = U(n,16/2) = A077412(n).
  • A097831 (program): Partial sums of Chebyshev sequence S(n,17)= U(n,17/2)=A078366(n).
  • A097832 (program): Partial sums of Chebyshev sequence S(n,19)= U(n,19/2)=A078368(n).
  • A097833 (program): Partial sums of Chebyshev sequence S(n,20)= U(n,10)=A075843(n+1).
  • A097834 (program): Chebyshev polynomials S(n,27) + S(n-1,27) with Diophantine property.
  • A097835 (program): First differences of Chebyshev polynomials S(n,27) = A097781(n) with Diophantine property.
  • A097836 (program): Chebyshev polynomials S(n,51).
  • A097837 (program): Chebyshev polynomials S(n,51) + S(n-1,51) with Diophantine property.
  • A097838 (program): First differences of Chebyshev polynomials S(n,51) = A097836(n) with Diophantine property.
  • A097839 (program): Chebyshev polynomials S(n,83).
  • A097840 (program): Chebyshev polynomials S(n,83) + S(n-1,83) with Diophantine property.
  • A097841 (program): First differences of Chebyshev polynomials S(n,83) = A097839(n) with Diophantine property.
  • A097842 (program): Chebyshev polynomials S(n,123) + S(n-1,123) with Diophantine property.
  • A097843 (program): First differences of Chebyshev polynomials S(n,123) = A049670(n+1) with Diophantine property.
  • A097844 (program): Chebyshev polynomials S(n,171).
  • A097845 (program): Chebyshev polynomials S(n,171) + S(n-1,171) with Diophantine property.
  • A097846 (program): Differences between A097598 and A045918.
  • A097852 (program): Expansion of (1+x^15)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^10)).
  • A097861 (program): Number of humps in all Motzkin paths of length n. (A hump is an upstep followed by 0 or more flatsteps followed by a downstep.)
  • A097869 (program): Expansion of g.f.: (1+x^4+x^5+x^9)/((1-x)*(1-x^2)*(1-x^4)^2).
  • A097870 (program): Molien series for group of order 4608 acting on joint weight enumerators of a pair of binary doubly-even self-dual codes.
  • A097882 (program): a(n) = floor( n^2/prime(n) ).
  • A097893 (program): Partial sums of the central trinomial coefficients (A002426).
  • A097894 (program): Partial sums of A014531.
  • A097895 (program): Number of compositions of n with at least 1 odd and 1 even part.
  • A097896 (program): Number of compositions of n with either all parts odd or all parts even.
  • A097900 (program): Number of runs of length 1 in all permutations of [n]. (The permutation 3574162 has two runs of length 1: 357/4/16/2.)
  • A097913 (program): G.f.: (1+x^18)/((1-x)*(1-x^8)*(1-x^12)*(1-x^24)).
  • A097920 (program): G.f.: (1+x^10)/((1-x)*(1-x^3)*(1-x^5)).
  • A097921 (program): G.f.: (1-x^6)*(1-x^8)/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)).
  • A097922 (program): G.f.: (1-x^4)*(1-x^10)/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^5)).
  • A097924 (program): a(n) = 4*a(n-1) + a(n-2), n>=2, a(0) = 2, a(1) = 7.
  • A097925 (program): Number of (n,3) Freiman-Wyner sequences.
  • A097926 (program): Number of (n,4) Freiman-Wyner sequences.
  • A097932 (program): Positive integers n such that 2n-19 is prime.
  • A097933 (program): Primes such that p divides 3^((p-1)/2) - 1.
  • A097934 (program): Primes p that divide 3^((p-1)/2) - 2^((p-1)/2).
  • A097939 (program): Sum of the smallest parts of all compositions of n.
  • A097944 (program): Number of digits in n-th prime.
  • A097945 (program): a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010).
  • A097947 (program): G.f.: (2+7*x+2*x^2)/((x^2-1)*(1+4*x+x^2)).
  • A097948 (program): G.f.: -(1-3*x^2-x^3)/(1+4*x-4*x^3-x^4).
  • A097949 (program): G.f.: -(2+7*x-x^3)/(1+4*x-4*x^3-x^4).
  • A097950 (program): G.f.: (1+x^5+x^10)/((1-x)*(1-x^3)).
  • A097953 (program): Boustrophedon transform of ((-2)^n+0^n)/2.
  • A097957 (program): Primes p such that p divides 5^((p-1)/2) + 4^((p-1)/2).
  • A097958 (program): Primes p such that p divides 6^((p-1)/2) - 3^((p-1)/2).
  • A097967 (program): a(n) = Sum_{k=1..n} (P(n,k) + C(n,k)).
  • A097971 (program): Number of alternating runs in all permutations of [n] (the permutation 732569148 has four alternating runs: 732, 2569, 91 and 148).
  • A097972 (program): Least m such that both p|m and p+2|m+2 for twin prime pairs (p,p+2) (p=A001359).
  • A097973 (program): Least m>p such that p|m, p+1|m+1 and p+2|m+2, for twin prime pairs (p, p+2), p in A001359.
  • A097974 (program): Sum of distinct prime divisors of n which are <= sqrt(n).
  • A097975 (program): a(n) is the prime divisor of n which is >= sqrt(n), or 0 if there is no such prime divisor.
  • A097986 (program): Number of strict integer partitions of n with a part dividing all the other parts.
  • A097987 (program): Numbers k such that 4 does not divide phi(k), where phi is Euler’s totient function (A000010).
  • A097988 (program): a(n) = Sum_{d dividing n} tau(d)^3 = (Sum_{d dividing n} tau(d))^2.
  • A097990 (program): A puzzle: reverse digits of n^2 + 10.
  • A097991 (program): A puzzle: reverse digits of n^2 + 10.
  • A097992 (program): G.f.: 1/((1-x)*(1-x^6)) = 1/ ( (1+x)*(x^2-x+1)*(1+x+x^2)*(x-1)^2 ).
  • A097999 (program): Number of 2-connected outerplanar graphs on n labeled nodes.
  • A098003 (program): Start with positive integers. On the m-th step, shift terms a(m+1) to a(2m-1) one position to the left, then move a(m) to position (2m-1). Sequence is limit of reordering.
  • A098005 (program): Beatty sequence for 1/(3 - e): a(n) = floor(n/(3-e)).
  • A098006 (program): (p-1)/2 - phi(p-1) as p runs through the odd primes.
  • A098011 (program): 10^a(n) + 1 = A088773(n).
  • A098012 (program): Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).
  • A098013 (program): Differences between consecutive primes that are twice primes.
  • A098014 (program): A098013/2.
  • A098015 (program): Indices x such that (1/2)(prime(x+1) - prime(x)) is prime.
  • A098018 (program): a(n) = Sum_{k|n, k>=2} mu(k-1), where mu() is the Moebius function.
  • A098019 (program): Irrational rotation of e as an implicit sequence with an uneven Cantor cartoon.
  • A098020 (program): Let f[n] = fractional part of n*Pi and let g[x] = -1 for the range 0<=x<=1/3, g[x] = 0 for the range 1/3<x<=2/3, g[x] = 11 for range 2/3<x<1. Sequence gives all positive integers n such that f[n+2]-2*f[n+1]+f[n]]-g[f[n+1]] = 0.
  • A098021 (program): Positions of 0’s in the zero-one sequence [nr+2r]-[nr]-[2r], where r=sqrt(2) and [ ]=floor; see A187967.
  • A098022 (program): Irrational rotation of Log(3)/Log(2) as an implicit sequence with an uneven Cantor cartoon.
  • A098025 (program): p and 2p-1 are both Pythagorean primes, i.e., congruent to 1 (mod 4).
  • A098033 (program): Parity of p*(p+1)/2 for n-th prime p.
  • A098035 (program): a(n) = Sum_{k|n} mu(k+1), where mu() is the Moebius function.
  • A098037 (program): Number of prime divisors, counted with multiplicity, of the sum of two consecutive primes.
  • A098046 (program): Inverse permutation to A098003.
  • A098058 (program): Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).
  • A098059 (program): Primes preceding gaps divisible by 4.
  • A098061 (program): Primes p such that p - 6 is a product of two consecutive primes.
  • A098062 (program): Primes of the form n^2 + 4n + 8.
  • A098065 (program): Minimal span for an absolute difference triangle of distinct entries whose base consists of a sequence of n positive integers.
  • A098075 (program): Threefold convolution of A004148 (the RNA secondary structure numbers) with itself.
  • A098077 (program): a(n) = n^2*(n+1)*(2*n+1)/3.
  • A098080 (program): Nontrivial slowest increasing sequence whose succession of digits is that of the nonnegative integers.
  • A098084 (program): a(n) satisfies P(n) + P(n+1) + a(n) = least prime >= P(n) + P(n+1), where P(i)=i-th prime.
  • A098085 (program): Primes P(n)+P(n+1)+b(n) = least prime >= P(n)+P(n+1), P(i)=i-th prime, b(n) given in A098084.
  • A098090 (program): Numbers k such that 2k-3 is prime.
  • A098096 (program): Numbers of the form p^2 * 2^p for p prime.
  • A098098 (program): a(n) = sigma(6*n+5)/6.
  • A098102 (program): Numbers of the form 2^(p - 1) - 1 where p is prime.
  • A098104 (program): Numbers of the form 7^p - p^7 where p is prime.
  • A098105 (program): a(n) = 2^p - p^2 where p is the n-th prime.
  • A098106 (program): Hankel transform of sequence (b(n)) where b(n) = Sum_{i=0..n} binomial(2*i,i).
  • A098107 (program): Sum of all matrix elements M(i,j) = n!*(i/j), (i,j = 1..n).
  • A098108 (program): a(n) = 1 if n is an odd square, otherwise 0.
  • A098111 (program): Inverse binomial transform of A098149.
  • A098116 (program): a(n) = 3^(p-1) + (3^p - 1) where p is the n-th prime.
  • A098117 (program): a(n) = 5^(prime(n) - 1) + 5^prime(n) - 1.
  • A098118 (program): a(n) = n!*[x^n] (log(x+1) * Sum_{j=0..n} C(2*n,j)*x^j).
  • A098123 (program): Number of compositions of n with equal number of even and odd parts.
  • A098127 (program): Fibonacci sequence with a(1) = 7 and a(2) = 26.
  • A098131 (program): Number of compositions of n where the smallest part is greater than or equal to the number of parts.
  • A098132 (program): Number of compositions of n where the smallest part is greater than the number of parts.
  • A098135 (program): a(h) = d(h,j) = lcm( f(h,j,1) … f(h,j,h) ), when j=3.
  • A098137 (program): a(n) = n^n + n^prime(n).
  • A098139 (program): a(n) = 2^p + 3^p + 5^p + 7^p where p = prime(n).
  • A098140 (program): 63-gonal numbers: a(n) = n*(61*n - 59)/2.
  • A098141 (program): a(1)=1, then if n not a square a(n)=a(n-1) + n, if n a square then a(n)= a(n-1) - n.
  • A098145 (program): p + P(p) where p is the n-th prime and P(p) is the unrestricted partition number of p.
  • A098149 (program): a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n>1.
  • A098150 (program): a(n) = 2(a(n-2) - a(n-1)) + a(n-3) where a(0)=-3, a(1)=11 & a(2)=-30.
  • A098151 (program): Number of partitions of 2n prime to 3 with all odd parts occurring with even multiplicities. There is no restriction on the even parts.
  • A098152 (program): a(n) = a(n-1)^2 + n, with a(0)=0.
  • A098156 (program): Interleave n+1 and 2n+1 and take binomial transform.
  • A098157 (program): Triangle T(n,k) with diagonals T(n,n-k)=binomial(n+1,2k).
  • A098158 (program): Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).
  • A098159 (program): Numbers n with property that when writing down all the nonnegative numbers from 0 to n one uses n odd digits.
  • A098160 (program): Numbers n with property that when writing down all the natural numbers from 0 to n one uses the same number of even and odd digits.
  • A098162 (program): a(n+1) = smallest number greater than a(n) having with a(n) a common divisor which is used before as such a common divisor at most once; a(1) = 1.
  • A098172 (program): Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
  • A098173 (program): Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 4k).
  • A098177 (program): Start with the first n, which reads: “Prolong the sequence with n numbers having their parity opposed to n”. Then read and obey the second n, then the third n, etc. This sequence is the slowest increasing one with such rule.
  • A098178 (program): Expansion of (1+x)(1-x+x^2)/((1-x)(1+x^2)).
  • A098179 (program): Expansion of (1-3*x+3*x^2)/(1-5*x+10*x^2-10*x^3+4*x^4).
  • A098180 (program): Odd numbers with twice the odd numbers repeated in order between them.
  • A098181 (program): Two consecutive odd numbers separated by multiples of four, repeated twice, between them, written in increasing order.
  • A098182 (program): a(n) = 3*a(n-1) - a(n-2) + a(n-3), a(0)=1,a(1)=1,a(2)=3.
  • A098183 (program): a(n) = 3*a(n-1) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 4.
  • A098184 (program): a(n) = 3a(n-1)+a(n-2)+a(n-3), a(0)=1, a(1)=1, a(2)=5.
  • A098189 (program): Sum of unitary divisors minus Euler phi: a(n) = A034448(n) - A000010(n).
  • A098198 (program): Decimal expansion of Pi^4/36 = zeta(2)^2.
  • A098200 (program): Number of distinct terms in iteration-list when cototient-function[=A051953] is iterated and the initial value is even number.
  • A098201 (program): Number of distinct terms in iteration-list when cototient-function[=A051953] is iterated and the initial value is odd number.
  • A098205 (program): A first order iteration: n-th term is obtained from (n-1)-th by adding n-th prime and then multiplying by the n-th prime; initial value is 0.
  • A098206 (program): A first order iteration: n-th term is obtained from (n-1)-th by adding n-th prime and then multiplying by the n-th prime; initial value is 1.
  • A098207 (program): a(n) is the square of near-repdigit number A033175(n).
  • A098209 (program): a(n) = A067275(n+1)^2 - 1.
  • A098210 (program): a(n) = -1 + A093137(n)^2.
  • A098212 (program): Expansion of (5-x^2)/((1+x)*(1-6*x+x^2)).
  • A098219 (program): a(n) = floor(sigma(sigma(n))/n) = floor(A051027(n)/n).
  • A098220 (program): a(n) = floor(sigma(sigma(p))/p) = floor(sigma(p+1)/p) = floor(A008333(n)/p), where p is the n-th prime number.
  • A098228 (program): a(n) = floor(n/(n-phi(n)) = floor(n/cototient(n)).
  • A098229 (program): a(n)=6*c(n,1) where n runs through the 3-smooth numbers (see comment).
  • A098230 (program): 75-gonal numbers: a(n) = n*(73*n-71)/2.
  • A098231 (program): 2^p - 11 for p prime.
  • A098232 (program): Largest power of 2 <= 3^n.
  • A098235 (program): Number of ways to write n as a sum of two ordered positive squarefree numbers.
  • A098236 (program): Number of ways to write n as the sum of two positive distinct squarefree numbers.
  • A098244 (program): First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.
  • A098245 (program): Chebyshev polynomials S(n,227).
  • A098246 (program): Chebyshev polynomials S(n,227) + S(n-1,227) with Diophantine property.
  • A098247 (program): First differences of Chebyshev polynomials S(n,227)=A098245(n) with Diophantine property.
  • A098248 (program): Chebyshev polynomials S(n,291).
  • A098249 (program): Chebyshev polynomials S(n,291) + S(n-1,291) with Diophantine property.
  • A098250 (program): First differences of Chebyshev polynomials S(n,291)=A098248(n) with Diophantine property.
  • A098251 (program): Chebyshev polynomials S(n,363).
  • A098252 (program): Chebyshev polynomials S(n,363) + S(n-1,363) with Diophantine property.
  • A098253 (program): First differences of Chebyshev polynomials S(n,363) = A098251(n) with Diophantine property.
  • A098254 (program): Chebyshev polynomials S(n,443).
  • A098255 (program): Chebyshev polynomials S(n,443) + S(n-1,443) with Diophantine property.
  • A098256 (program): First differences of Chebyshev polynomials S(n,443)=A098254(n) with Diophantine property.
  • A098257 (program): Chebyshev polynomials S(n,531).
  • A098258 (program): Chebyshev polynomials S(n,531) + S(n-1,531) with Diophantine property.
  • A098259 (program): First differences of Chebyshev polynomials S(n,531)=A098257(n) with Diophantine property.
  • A098260 (program): Chebyshev polynomials S(n,627).
  • A098261 (program): Chebyshev polynomials S(n,627) + S(n-1,627) with Diophantine property.
  • A098262 (program): First differences of Chebyshev polynomials S(n,627)=A098260(n) with Diophantine property.
  • A098263 (program): Chebyshev polynomials S(n,731).
  • A098264 (program): G.f.: 1/(1-2x-19x^2)^(1/2).
  • A098265 (program): G.f. : 1/(1-2x-23x^2)^(1/2).
  • A098269 (program): a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.
  • A098270 (program): a(n) = 2^n*P_n(5), 2^n times the Legendre polynomial of order n at 5.
  • A098272 (program): a(n) = 2^(2n+1) * binomial(3n,n)/(2n+1).
  • A098274 (program): Sum_{k = 0..n} C(n, k)^2*C(n+k, n)*C(n+2*k, n).
  • A098291 (program): Chebyshev polynomials S(n,731) + S(n-1,731) with Diophantine property.
  • A098292 (program): First differences of Chebyshev polynomials S(n,731)=A098263(n) with Diophantine property.
  • A098293 (program): Powers of 2 alternating with powers of 3.
  • A098294 (program): a(n) = ceiling(n*log_2(3/2)).
  • A098295 (program): ((3/2)^n)/2^a(n) lies in the half-open interval [1,2).
  • A098296 (program): Member r=11 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098297 (program): Member r=12 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098298 (program): Member r=13 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098299 (program): Member r=14 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098300 (program): Member r=15 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098301 (program): Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098302 (program): Member r=17 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098303 (program): Member r=18 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098304 (program): Member r=19 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098305 (program): Unsigned member r=-5 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098306 (program): Unsigned member r=-6 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098307 (program): Unsigned member r=-7 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098308 (program): Unsigned member r=-8 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098309 (program): Unsigned member r = -10 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098310 (program): Unsigned member r=-11 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A098316 (program): Decimal expansion of [3, 3, …] = (3 + sqrt(13))/2.
  • A098317 (program): Decimal expansion of phi^3 = 2 + sqrt(5).
  • A098318 (program): Decimal expansion of [5, 5, …] = (5 + sqrt(29))/2.
  • A098329 (program): Expansion of 1/(1-2x-31x^2)^(1/2).
  • A098331 (program): Expansion of 1/sqrt(1 - 2*x + 5*x^2).
  • A098332 (program): Expansion of 1/sqrt(1 - 2*x + 9*x^2).
  • A098333 (program): Expansion of 1/sqrt(1 - 2x + 13x^2).
  • A098334 (program): Expansion of 1/sqrt(1-2x+17x^2).
  • A098335 (program): Expansion of 1/sqrt(1-4x+8x^2).
  • A098336 (program): Expansion of 1/sqrt(1 - 4*x + 12*x^2).
  • A098337 (program): Expansion of 1/sqrt(1-4x+20x^2).
  • A098338 (program): Expansion of 1/sqrt(1-6x+13x^2).
  • A098339 (program): Expansion of 1/sqrt(1 - 6x + 17x^2).
  • A098340 (program): Expansion of 1/sqrt(1 - 6x + 21x^2).
  • A098341 (program): Expansion of 1/sqrt(1 - 6*x + 25*x^2).
  • A098347 (program): A sequence derived from a Ferrers graph partition of 16.
  • A098350 (program): Multiplication table of the primes read by antidiagonals.
  • A098352 (program): Multiplication table of the even numbers read by antidiagonals.
  • A098353 (program): Multiplication table of the odd numbers read by antidiagonals.
  • A098354 (program): Multiplication table of the powers of 2 read by antidiagonals.
  • A098355 (program): Multiplication table of the powers of three read by antidiagonals.
  • A098356 (program): Multiplication table of the Fibonacci numbers read by antidiagonals.
  • A098358 (program): Multiplication table of the triangular numbers read by antidiagonals.
  • A098359 (program): Multiplication table of the square numbers read by antidiagonals.
  • A098360 (program): Multiplication table of the cube numbers read by antidiagonals.
  • A098361 (program): Multiplication table of the factorial numbers read by antidiagonals.
  • A098365 (program): Sums of two squares and divisible by 17.
  • A098378 (program): Number of characters needed to write number n in the traditional Ethiopic (Geez) number system.
  • A098383 (program): Define a function f on the positive integers by: if n is 1 or composite, stop; but if n = prime(k) then f(n) = k; a(n) = sum of terms in trajectory of n under repeated application of f.
  • A098385 (program): Ordered factorizations over hook-type prime signatures with exactly three distinct primes (third column of A098348).
  • A098386 (program): a(n) = prime(n)-Log2(n), where Log2 = A000523.
  • A098387 (program): Prime(n)+Log2(n), where Log2=A000523.
  • A098388 (program): a(n) = floor(log_2(prime(n))).
  • A098389 (program): Prime(n) - floor(log_2(prime(n))).
  • A098390 (program): Prime(n)+Log2(prime(n)), where Log2=A000523.
  • A098391 (program): a(n) = Log2(Log2(prime(n))), where Log2 = A000523.
  • A098392 (program): Prime(n)-Log2(Log2(prime(n))), where Log2=A000523.
  • A098393 (program): Prime(n)+Log2(Log2(prime(n))), where Log2=A000523.
  • A098394 (program): Next larger integer including the smallest digit of n.
  • A098399 (program): a(n) = 3^n*binomial(2n+1, n).
  • A098400 (program): a(n) = 4^n*binomial(2n+1, n).
  • A098401 (program): a(n) = (0^n + 3^n*binomial(2n,n))/2.
  • A098402 (program): a(n) = (0^n + 4^n * binomial(2*n,n))/2.
  • A098405 (program): Expansion of ((1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x))).
  • A098406 (program): a(n) = (10^n + 17)/9.
  • A098407 (program): Number of different hierarchical orderings that can be formed from n unlabeled elements with no repetition of subhierarchies.
  • A098409 (program): Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).
  • A098410 (program): Expansion of 1/(sqrt(1-4*x)*sqrt(1-8*x)).
  • A098411 (program): Expansion of 1/(sqrt(1-4x)sqrt(1-12x)).
  • A098413 (program): Greatest members p of prime triples (p-6, p-2, p).
  • A098416 (program): (A007529(n) + A098415(n)) / 4.
  • A098424 (program): Number of prime triples (p,q,r) <= n with p<q<r=p+6.
  • A098428 (program): Number of sexy prime pairs (p, p+6) with p <= n.
  • A098429 (program): Number of cousin prime pairs (p, p+4) with p <= n.
  • A098430 (program): a(n) = 4^n*(2*n)!/(n!)^2.
  • A098439 (program): Expansion of 1/sqrt(1-2x-47x^2).
  • A098440 (program): Expansion of 1/sqrt(1-2x-59x^2).
  • A098441 (program): Expansion of 1/sqrt(1 - 2*x - 63*x^2).
  • A098442 (program): Expansion of 1/sqrt(1-2x-95x^2).
  • A098443 (program): Expansion of 1/sqrt(1-8x-4x^2).
  • A098444 (program): Expansion of 1/sqrt(1-6x-11x^2).
  • A098445 (program): Coefficients of powers of q^5 in (q)_infty^5 = (q_5)^infty (mod 5).
  • A098452 (program): One of three ordered sets of positive integers that solves the minimal magic die puzzle.
  • A098453 (program): Expansion of 1/sqrt(1 - 4*x - 12*x^2).
  • A098455 (program): Expansion of 1/sqrt(1-4x-36x^2).
  • A098456 (program): Expansion of 1/sqrt(1-4x-64x^2).
  • A098457 (program): Farey Bisection Expansion of sqrt(7).
  • A098460 (program): Expansion of e.g.f. 1/sqrt(1-2x-2x^2).
  • A098461 (program): Expansion of E.g.f.: 1/sqrt(1-2*x-3*x^2).
  • A098462 (program): a(n) = n^n + (n+1)^n.
  • A098464 (program): Numbers k such that lcm(1,2,3,…,k) equals the denominator of the k-th harmonic number H(k).
  • A098465 (program): Expansion of (sqrt(1+3*x)-sqrt(1-5*x))/(4*x*sqrt(1-x)).
  • A098469 (program): A sequence related to the even-indexed Catalan numbers.
  • A098470 (program): Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 5th column from the center.
  • A098473 (program): Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.
  • A098474 (program): Triangle read by rows, T(n,k) = C(n,k)*C(2*k,k)/(k+1), n >= 0, 0 <= k <= n.
  • A098477 (program): Expansion of 1/sqrt(1-2*x-7*x^2+8*x^3).
  • A098478 (program): Expansion of 1/sqrt(1-2x-11x^2+12x^3).
  • A098479 (program): Expansion of 1/sqrt((1-x)^2 - 4*x^3).
  • A098480 (program): Expansion of 1/sqrt((1-x)^2-8x^3).
  • A098481 (program): Expansion of 1/sqrt((1-x)^2 - 12*x^3).
  • A098482 (program): Expansion of 1/sqrt((1-x)^2-4*x^4).
  • A098483 (program): Expansion of 1/sqrt((1-x)^2-8x^4).
  • A098484 (program): Expansion of 1/sqrt((1-x)^2-12x^4).
  • A098486 (program): Odd numbers with replacement of all prime factors 3 by 2.
  • A098491 (program): Number of partitions of n with parts occurring at most thrice and an even number of parts. Row sums of A098489.
  • A098492 (program): Number of partitions of n with parts occurring at most thrice and an odd number of parts. Row sums of A098490.
  • A098493 (program): Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.
  • A098495 (program): Array T(r,c) read by antidiagonals: values of triangle A098493 interpreted as polynomials, r >= 0.
  • A098496 (program): Antidiagonal sums of triangle A098495.
  • A098497 (program): Main diagonal of triangle A098495.
  • A098500 (program): Number of squares on infinite quarter chessboard at <=n knight moves from the corner.
  • A098502 (program): 16*n - 4.
  • A098508 (program): Second column of the inverse of a Catalan scaled binomial matrix.
  • A098509 (program): Denominators of the inverse of a Catalan scaled binomial matrix.
  • A098510 (program): Row sums of a matrix associated to the inverse of a Catalan scaled binomial matrix.
  • A098512 (program): Second column and subdiagonal of number triangle A098509.
  • A098518 (program): E.g.f. exp(x)*BesselI(1,2*sqrt(2)*x)/sqrt(2).
  • A098519 (program): E.g.f. exp(x)*BesselI(1,2*sqrt(3)*x)/sqrt(3).
  • A098520 (program): E.g.f. exp(x)*BesselI(1,4*x)/2.
  • A098521 (program): E.g.f. exp(x)*BesselI(2,2*sqrt(2)*x)/2.
  • A098522 (program): E.g.f. exp(x)*BesselI(2,2*sqrt(3)*x)/3.
  • A098523 (program): Expansion of (1+x^2)/(1-x-x^5) = (1+x^2)/((1-x+x^2)*(1-x^2-x^3)).
  • A098524 (program): Expansion of (1+2x^2)/(1-x-4x^5).
  • A098525 (program): Expansion of (1+3x^2)/(1-x-9x^5).
  • A098526 (program): Expansion of (1+4x^2)/(1-x-16x^5).
  • A098527 (program): Expansion (1+x^3)/(1-x-x^7).
  • A098528 (program): Expansion of (1+2*x^3)/(1-x-4*x^7).
  • A098531 (program): Sum of fifth powers of first n Fibonacci numbers.
  • A098532 (program): Sum of sixth powers of first n Fibonacci numbers.
  • A098533 (program): Sum of seventh powers of first n Fibonacci numbers.
  • A098534 (program): Mod 3 analog of Stern’s diatomic series.
  • A098547 (program): a(n) = n^3 + n^2 + 1.
  • A098554 (program): G.f.: x*(1-x^2)/((1+x^2)*(1+x+x^2)).
  • A098557 (program): E.g.f. (1/2)*(1+x)*log((1+x)/(1-x)).
  • A098558 (program): Expansion of e.g.f. (1+x)/(1-x).
  • A098559 (program): E.g.f. (1+3x)/(1-3x).
  • A098560 (program): E.g.f. (1+4*x)/(1-4*x).
  • A098568 (program): Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.
  • A098569 (program): Row sums of the triangle of triangular binomial coefficients given by A098568.
  • A098574 (program): a(n) = Sum_{k=0..floor(n/7)} C(n-5*k,2*k).
  • A098575 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-2*k,2*k)*2^k.
  • A098576 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-2*k,2*k) * 3^k.
  • A098577 (program): a(n) = Sum_{k=0..floor(n/5)} C(n-3*k,2*k) * 2^k.
  • A098578 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-3*k,k+1).
  • A098579 (program): Expansion of sqrt(1-8*x).
  • A098580 (program): Expansion of (sqrt(1-8*x)-4*x)/sqrt(1-8*x).
  • A098581 (program): Expansion of (1+2*x+4*x^2)/(1-x-8*x^4).
  • A098582 (program): Expansion of (1+2*x+4*x^2+8*x^3)/(1-x-16*x^5).
  • A098583 (program): Expansion of (1+2*x+4*x^2+8*x^3+16*x^4)/(1-x-32*x^6).
  • A098586 (program): a(n) = (1/2) * (5*P(n+1) + P(n) - 1), where P(k) are the Pell numbers A000129.
  • A098588 (program): a(n) = 2^n for n = 0..4; for n > 4, a(n) = 2*a(n-1) + a(n-5).
  • A098589 (program): a(n) = 3*a(n-1) + 2*a(n-3), with a(0)=1, a(1)=3.
  • A098590 (program): a(n) = 4^n for n = 0..3; for n > 3, a(n) = 4*a(n-1) + a(n-4).
  • A098592 (program): Number of primes between n*30 and (n+1)*30.
  • A098593 (program): A triangle of Krawtchouk coefficients.
  • A098597 (program): Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. Odd part of the n-th Catalan number.
  • A098599 (program): Riordan array ((1+2x)/(1+x),(1+x)).
  • A098600 (program): a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.
  • A098601 (program): Expansion of (1+2x)/((1+x)(1-x^2-x^3)).
  • A098602 (program): a(n) = A001652(n) * A046090(n).
  • A098603 (program): a(n) = n*(n+10).
  • A098605 (program): Positive integers n such that 2n - 17 is prime.
  • A098608 (program): 100^n.
  • A098609 (program): a(n) = 100^n-1.
  • A098610 (program): a(n) = 10^n + (-1)^n.
  • A098611 (program): a(n) = 10^n-(-1)^n.
  • A098613 (program): Expansion of psi(x^2) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
  • A098614 (program): Product of Fibonacci and Catalan numbers: a(n) = A000045(n+1)*A000108(n).
  • A098615 (program): G.f. A(x) satisfies: A(x*G(x)) = G(x), where G(x) is the g.f. for A098614(n) = Fibonacci(n+1)*Catalan(n).
  • A098616 (program): Product of Pell and Catalan numbers: a(n) = A000129(n+1)*A000108(n).
  • A098617 (program): G.f. A(x) satisfies: A(x*G(x)) = G(x), where G(x) is the g.f. for A098616(n) = Pell(n+1)*Catalan(n).
  • A098618 (program): Products of A007482 and Catalan numbers: a(n) = A007482(n)*A000108(n).
  • A098619 (program): G.f. A(x) satisfies: A(x*G098618(x)) = G098618(x), where G098618 is the g.f. for A098618(n) = A007482(n)*Catalan(n).
  • A098630 (program): Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled loops and arcs
  • A098631 (program): Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled arcs.
  • A098640 (program): a(n) = 2^p + 1 where p is the n-th prime.
  • A098646 (program): Trace sequence of 3 X 3 Krawtchouk matrix.
  • A098647 (program): Trace sequence associated to the 4 X 4 Krawtchouk matrix and its transpose.
  • A098648 (program): Expansion of (1-3*x)/(1 - 6*x + 4*x^2).
  • A098649 (program): Primes of the form 2(p+q) + 1, where p and q are consecutive primes.
  • A098655 (program): Trace sequence of 3 X 3 symmetric Krawtchouk matrix.
  • A098656 (program): Expansion of x(1-4x)/((1-2x)(1-8x^2)).
  • A098657 (program): Expansion of (1-x-4x^2)/((1-2x)(1-8x^2)).
  • A098658 (program): a(n) = 3^n*(2*n)!/(n!)^2.
  • A098659 (program): Expansion of 1/sqrt((1-7*x)^2-24*x^2).
  • A098660 (program): E.g.f. BesselI(0,2*sqrt(2)*x) + BesselI(1,2*sqrt(2)*x)/sqrt(2).
  • A098662 (program): E.g.f. BesselI(0,2*sqrt(3)*x) + BesselI(1,2*sqrt(3)*x)/sqrt(3).
  • A098663 (program): a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+1,k+1) * 3^k.
  • A098664 (program): E.g.f. BesselI(0,4x)+BesselI(1,4x)/2.
  • A098665 (program): a(n) = Sum_{k = 0..n} binomial(n,k) * binomial(n+1,k+1) * 4^k.
  • A098667 (program): Max{m: A098666(n,k)=1 for 1<=k<=m<=n}.
  • A098668 (program): Right edge of triangle A098666.
  • A098669 (program): Numbers of rows ending with ones in triangle A098666.
  • A098689 (program): Decimal expansion of Sum_{n>=0} Fibonacci(n)/n!.
  • A098693 (program): G.f.: q*Product_{k>0} (1-q^(12k))(1+q^(12k-1))(1+q^(12k-11))/(1-q^k).
  • A098695 (program): a(n) = 2^(n(n-1)/2) * Product_{k=1..n} k!.
  • A098696 (program): Main diagonal of array in A073146.
  • A098697 (program): Euler-Seidel matrix T(k,n) with start sequence A000248, read by antidiagonals.
  • A098698 (program): Main diagonal of array in A098697.
  • A098703 (program): a(n) = (3^n + phi^(n-1) + (-phi)^(1-n)) / 5, where phi denotes the golden ratio A001622.
  • A098704 (program): Decimal form of the binary numbers 10, 100010, 1000100010, 10001000100010, 100010001000100010,…
  • A098706 (program): a(n) = 2*A076218(n).
  • A098710 (program): a(n) = Product_{k|n} k!.
  • A098713 (program): a(n) = (2n+1)*2^(2n+1) - 1.
  • A098714 (program): Only one Pythagorean triangle of this perimeter exists.
  • A098721 (program): a(n) = C(n, 2)^(n-3) = (n(n-1)/2)^(n-3).
  • A098722 (program): a(n) = C(n, 3)^(n-4).
  • A098723 (program): a(n) = C(n, 4)^(n-5).
  • A098724 (program): a(n) = C(n, 5)^(n-6).
  • A098725 (program): a(4n) = 0, a(2n+1) = 1, a(4n+2) = a(n+1).
  • A098726 (program): Take three consecutive primes starting with the n-th prime. Calculate d(i,j) = abs(prime(i) - prime(j)), for all {i,j}, i.e., all possible differences. a(n) is the number of distinct differences (which can be either 3 or 2).
  • A098731 (program): Numbers k such that 2*R_k is a happy number (A007770), where R_k = 11…1 is the repunit (A002275) of length k.
  • A098735 (program): Numerator of sum of all matrix elements M(i,j) = i^2 + j^2 (i,j = 1..n) divided by n!.
  • A098736 (program): a(n) = product of n and all its digits.
  • A098737 (program): Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.
  • A098739 (program): This sequence is constructed using only the four single-digit primes 2,3,5,7 in that order.
  • A098740 (program): Start with the sequence of natural numbers S(0)={1,2,3,…} and define, for i>0, S(i)=D(i)S(i-1), where D(i)A denotes the operation of deleting the a(1+[i/2])th term of A={a(1),a(2),a(3),…}. E.g. D(3){1,2,4,6,9,10,…} means to delete the (a(1+[3/2])th = 2nd term of {1,2,4,9,10,…}, giving {1,4,9,10,…}. The given sequence is the limit of S(i) as i->inf.
  • A098741 (program): a(n) = (p^2*(p+1)*(p+2))/6 where p is n-th prime.
  • A098742 (program): Number of indecomposable set partitions of [1..n] without singletons.
  • A098746 (program): Number of permutations of [1..n] which avoid 4231 and 42513.
  • A098747 (program): Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU’s at low level.
  • A098748 (program): a(n) = floor((n^4-n^3-1)/(n^2-n-1)).
  • A098749 (program): Let f[n_]=((n^4-n^3-1)/ (n^3-n-1))^2; then a(n) = Floor[f[n]].
  • A098750 (program): a(n+1) = a(n) + 10’s complement of first digit of a(n); a(0) = 0.
  • A098751 (program): a(n+1) = a(n) + 10’s complement of each of the digits of a(n); a(0) = 0.
  • A098764 (program): a(n) = 3p - q where p and q are consecutive primes.
  • A098772 (program): a(n) = Sum_{k=0..n} binomial(2*n,2*k)^2.
  • A098773 (program): p*2^p + 1 where p is prime.
  • A098780 (program): First n numbers in binary representation concatenated in reverse order.
  • A098790 (program): a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.
  • A098792 (program): a(n) = A056188(n)/n.
  • A098796 (program): a(n) = (Catalan(P_n-1)+1)/P_n where P_n is the n-th prime and Catalan(k) is the Catalan number binomial(2k, k)/(k+1).
  • A098798 (program): a(n) = Sum_{1<k<n and k not dividing n} floor(n/k).
  • A098802 (program): Greatest prime factors in Pascal’s triangle (read by rows).
  • A098803 (program): a(n) = n^7 * 7^n.
  • A098805 (program): Array read by antidiagonals: Numerical sequences of Fibonacci-like polynomials produced by m-ary Huffman trees of maximum height for absolutely ordered sequences.
  • A098808 (program): a(n) = 2^(n + 11) - 11.
  • A098809 (program): a(n) = 2^(n+23) - 23.
  • A098810 (program): Array read by antidiagonals: Costs E[m,N] of m-ary Huffman trees of maximum height with N internal nodes (non-leaves) for minimizing absolutely ordered sequences of size n=2N+1; m > 1, N > 0.
  • A098815 (program): 2^p - 7 where p is prime.
  • A098820 (program): Periodicity of entries in the first row of a Laver Table of size 2^n.
  • A098821 (program): a(n) = (n-2) * 2^(n-1) + 5.
  • A098823 (program): a(n) = 16*(8*prime(n) + 7).
  • A098824 (program): Array read by antidiagonals: Minimizing absolutely ordered sequences of m-ary Huffman trees of maximum height; m > 1.
  • A098825 (program): Triangle read by rows: T(n,k) = number of partial derangements, that is, the number of permutations of n distinct, ordered items in which exactly k of the items are in their natural ordered positions, for n >= 0, k = n, n-1, …, 1, 0.
  • A098827 (program): Squarefree oblong (pronic) numbers having an odd number of prime factors.
  • A098828 (program): Primes of the form 3x^2 - y^2, where x and y are two consecutive numbers.
  • A098832 (program): Square array read by antidiagonals: even-numbered rows of the table are of the form n*(n+m) and odd-numbered rows are of the form n*(n+m)/2.
  • A098833 (program): Numbers n such that the sum of primes dividing n (with repetition) is a Fibonacci number.
  • A098842 (program): Number of n-digit Fibonacci numbers.
  • A098844 (program): a(1)=1, a(n) = n*a(floor(n/2)).
  • A098847 (program): a(n) = n*(n + 12).
  • A098848 (program): a(n) = n*(n + 14).
  • A098849 (program): a(n) = n*(n + 16).
  • A098850 (program): a(n) = n*(n + 18).
  • A098868 (program): Numbers n where A098018(n)=0.
  • A098870 (program): Sum of the cubes of the digits of the previous term, starting with 2.
  • A098871 (program): Sums of distinct powers of 4 plus 1.
  • A098875 (program): Decimal expansion of Sum_{n>0} n/exp(n).
  • A098878 (program): a(n) = (2^n - 1)^3 - 2.
  • A098879 (program): a(n) = (2^n - 1)^5 - 2.
  • A098880 (program): a(n) = 11^n * n^11.
  • A098884 (program): Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6.
  • A098890 (program): a(n) = A001652(n)*A046090(n)+1 = A098602(n)+1.
  • A098893 (program): Sum of number of prime-factors of all prefixes in decimal representation of n.
  • A098894 (program): Values of k such that {s(1),…,s(k)} is a palindrome, where {s(1),s(2),…} is the fixed point of the substitutions 0->1 and 1->110.
  • A098897 (program): Palindromic deficient numbers.
  • A098902 (program): Even numbers whose number of distinct prime factors is also even.
  • A098903 (program): Odd numbers whose number of distinct prime factors is also odd.
  • A098904 (program): Even numbers whose number of distinct prime factors is odd.
  • A098905 (program): Odd numbers whose number of distinct prime factors is positive and even.
  • A098909 (program): Triangle T(n,k) of numbers of connected (unicyclic) graphs with unique cycle of length k (3<=k<=n), on n labeled nodes.
  • A098916 (program): Permanent of the n X n (0,1)-matrices with ij-th entry equal to zero iff (i=1,j=1),(i=1,j=n),(i=n,j=1) and (i=n,j=n).
  • A098923 (program): 33-gonal numbers: n(31n-29)/2.
  • A098924 (program): 45-gonal numbers: n*(43*n-41)/2.
  • A098925 (program): Distribution of the number of ways for a child to climb a staircase having r steps (one step or two steps at a time).
  • A098931 (program): a(0) = 1, a(n) = 1 + 2*3 + 4*5 + 6*7 + … + (2n)*(2n+1) for n > 0.
  • A098933 (program): Primes of the form p+14, where p is a prime.
  • A098939 (program): a(n) = (2^n - 1)^11 - 2.
  • A098940 (program): a(n) = (2^n - 1)^7 - 2.
  • A098955 (program): Numbers with property that the last digit is the length of the number (written in base 10).
  • A098957 (program): Decimal value of the reverse binary expansion of the prime numbers.
  • A098961 (program): Sums of two squares and divisible by 13.
  • A098971 (program): a(0)=1; for n > 0, a(n)=a(floor(n/2))+2*a(floor(n/4)).
  • A098974 (program): Primes p such that q-p = 24, where q is the next prime after p.
  • A098975 (program): Nonzero elements of table A071675; also counts selected ordered multisets of the values {1,2,3}.
  • A098976 (program): Upper prime of a difference of 22 between consecutive primes.
  • A098984 (program): Numerators in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).
  • A098985 (program): Denominators in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).
  • A098986 (program): Numerators (divided by 2) in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).
  • A098987 (program): Numerators in series expansion of log(Product_{m>=0} (1+q^m)).
  • A098988 (program): Denominators in series expansion of log(Product_{m>=1} (1+q^m)).
  • A098992 (program): Number of permutations of [n] with exactly 2 descents which avoid the pattern 1324.
  • A098996 (program): p(p+1)(2p+1) where p is prime.
  • A098997 (program): (1/30)*(p(p+1)(2p+1)(3p^2+3p-1)) where p is prime.
  • A098998 (program): p(11p-7) where p is prime.
  • A098999 (program): Sum of cubes of the first n primes.
  • A099003 (program): Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0).
  • A099007 (program): Primes of the form 6n^2 - 2n - 1.
  • A099012 (program): a(n) = 3^(n-1)*Fibonacci(n).
  • A099013 (program): a(n) = Sum_{k=0..n} 3^(k-1)*Fibonacci(k).
  • A099014 (program): a(n) = Fibonacci(n)*(Fibonacci(n-1)^2 + Fibonacci(n+1)^2).
  • A099015 (program): a(n) = Fib(n+1)*(2*Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).
  • A099016 (program): a(n) = 3*L(2*n)/5 - (-1)^n/5, where L = A000032.
  • A099019 (program): Odd composite numbers n such that n-2 and n+2 are also composite.
  • A099021 (program): Main diagonal of array in A099020.
  • A099022 (program): a(n) = Sum_{k=0..n} C(n,k)*(2*n-k)!.
  • A099023 (program): Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).
  • A099024 (program): A033466(5n+2). Values of A033466(n) that differ from A058031(n+1)+1.
  • A099025 (program): Expansion of 1 / ((1+x) * (1-5*x+x^2)).
  • A099026 (program): Array x AND NOT y, read by rising antidiagonals.
  • A099027 (program): a(n) = Sum_{k=0..n} n-k AND NOT k.
  • A099033 (program): a(n) = Sum_{k=1..n} (-1)^A000120(3*k).
  • A099034 (program): a(n) = Sum_{k=1..n} (-1)^A000120(5*k).
  • A099035 (program): a(n) = (n+1)*2^(n-1) - 1.
  • A099036 (program): a(n) = 2^n - Fibonacci(n).
  • A099037 (program): Triangle of diagonals of symmetric Krawtchouk matrices.
  • A099039 (program): Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.
  • A099040 (program): Riordan array (1, 2+2x).
  • A099041 (program): Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (10;1).
  • A099044 (program): a(n) = (2*0^n + 3^n*binomial(2*n,n))/3.
  • A099045 (program): a(n) = (3*0^n + 4^n*binomial(2*n,n))/4.
  • A099046 (program): a(n) = (4*0^n + 5^n*binomial(2*n,n))/5.
  • A099047 (program): Numbers n such that n-1 and n+1 are both composite.
  • A099048 (program): Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0).
  • A099049 (program): Numbers n such that n-1 and n+1 are both prime or both composite.
  • A099050 (program): a(1)=1; for n>=2, a(n)=sum(1<=i<=j<=n-1, gcd(a(i),a(j))).
  • A099051 (program): p*2^p - 1 where p is prime.
  • A099054 (program): Arshon’s sequence: start from 1 and replace the letters in odd positions using 1 -> 123, 2 -> 231, 3 -> 312 and the letters in even positions using 1 -> 321, 2-> 132, 3 -> 213.
  • A099055 (program): A bisection of A054519.
  • A099056 (program): A bisection of A054519.
  • A099059 (program): The odd bisection of A000594.
  • A099060 (program): The even bisection of A000594.
  • A099061 (program): A bisection of A000960.
  • A099062 (program): A bisection of A000960.
  • A099063 (program): Binomial transform of A000960.
  • A099064 (program): Inverse binomial transform of A000960.
  • A099072 (program): First differences of A000960, divided by 2.
  • A099074 (program): Partial sums of A000960.
  • A099076 (program): a(n) = A000960(n) mod 3.
  • A099087 (program): Expansion of 1/(1 - 2*x + 2*x^2).
  • A099089 (program): Riordan array (1, 2+x).
  • A099091 (program): Riordan array (1,2+3x).
  • A099092 (program): Riordan array (1,2+4x).
  • A099093 (program): Riordan array (1, 3+3x).
  • A099094 (program): a(n) = 3a(n-2) + 3a(n-3), a(0)=1, a(1)=0, a(2)=3.
  • A099095 (program): Riordan array (1,3+2x).
  • A099096 (program): Riordan array (1,2-x).
  • A099097 (program): Riordan array (1, 3+x).
  • A099098 (program): Quadrisection of a Padovan sequence.
  • A099099 (program): Quadrisection of a generalized Padovan sequence.
  • A099100 (program): a(n) = Fibonacci(5*n+1).
  • A099101 (program): Quintisection of 1/(1-x^3-x^4).
  • A099110 (program): Expansion of 1 / ((1+x)*(1-2x)*(1-3x)*(1-4x)).
  • A099111 (program): Expansion of 1 / ((1+x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
  • A099121 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2}.
  • A099122 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3}.
  • A099123 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4}.
  • A099124 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5}.
  • A099125 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6}.
  • A099126 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7}.
  • A099127 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7,8}.
  • A099128 (program): Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6,7,8,9}.
  • A099131 (program): Quintisection and binomial transform of 1/(1-x^4-x^5).
  • A099132 (program): Quintisection of 1/(1-x^5-x^6).
  • A099133 (program): 4^(n-1)*Fibonacci(n).
  • A099134 (program): Expansion of x/(1-2x-19x^2).
  • A099138 (program): 6^(n-1)*J(n).
  • A099139 (program): 18^n-(-6)^n)/24.
  • A099140 (program): a(n) = 4^n * T(n,3/2) where T is the Chebyshev polynomial of the first kind.
  • A099141 (program): a(n) = 5^n * T(n,7/5) where T is the Chebyshev polynomial of the first kind.
  • A099142 (program): a(n) = 6^n * T(n, 4/3) where T is the Chebyshev polynomial of the first kind.
  • A099150 (program): Positive integers k such that f(k)+f(k)=concatenation of k and k, where f(k)=k(k+3)/2 (A000096).
  • A099151 (program): Positive integers a such that f(3a)+f(a)=concatenation of 3a and a, where f(k)=k(k+3)/2 (A000096).
  • A099156 (program): a(n) = 2^(n-1)*U(n-1, 2).
  • A099157 (program): a(n) = 4^(n-1)*U(n-1, 3/2) where U is the Chebyshev polynomial of the second kind.
  • A099158 (program): a(n) = 5^(n-1) * U(n-1, 7/5) where U is the Chebyshev polynomial of the second kind.
  • A099159 (program): (L(n-2)+2*3^n)/5.
  • A099163 (program): Expansion of (1-2*x^2)/((1-2*x)*(1+x-x^2)).
  • A099164 (program): (L(n+2)+2*3^n)/5.
  • A099166 (program): G.f.: (1+x^2)/((1-2x)(1-x-x^2); a(n)=3a(n-1)-a(n-2)-2a(n-3); a(n)=5*2^n-L(n+3); a(n)=sum{k=0..n, (L(k)-0^k)2^(n-k)}.
  • A099167 (program): G.f.: (1+x^2)/((1-3x)(1-x-x^2)).
  • A099168 (program): a(n) = 3^n * 5^binomial(n,2).
  • A099169 (program): a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.
  • A099170 (program): Generalized Motzkin paths with no hills and 3-horizontal steps.
  • A099171 (program): Generalized Motzkin paths with no hills and 4-horizontal steps (even coefficients).
  • A099173 (program): Array T(k,n) read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2).
  • A099174 (program): Triangle read by rows: coefficients of modified Hermite polynomials.
  • A099176 (program): a(n)=2a(n-1)+4a(n-2)-4a(n-3)-4a(n-4).
  • A099177 (program): a(n)=2a(n-1)+4a(n-2)-4a(n-3)-4a(n-4).
  • A099180 (program): Primes p which are greater than p reversed.
  • A099184 (program): Heavy primes: primes p such that p-1 has more than 2 divisors with multiplicity.
  • A099188 (program): a(n) = 2*ceiling(n/sqrt(2)).
  • A099193 (program): Figurate numbers based on the 7-dimensional regular convex polytope called the 7-dimensional cross-polytope, or 7-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 4}. It is the dual of the 7-dimensional hypercube.
  • A099195 (program): Figurate numbers based on the 8-dimensional regular convex polytope called the 8-dimensional cross-polytope, or 8-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 3, 4}. It is the dual of the 8-dimensional hypercube.
  • A099196 (program): Figurate numbers based on the 9-dimensional regular convex polytope called the 9-dimensional cross-polytope, or 9-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 3, 3, 4}. It is the dual of the 9-dimensional hypercube.
  • A099197 (program): Figurate numbers based on the 10-dimensional regular convex polytope called the 10-dimensional cross-polytope, or 10-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 3, 3, 3, 4}. It is the dual of the 10-dimensional hypercube.
  • A099198 (program): A bisection of A002807.
  • A099201 (program): A bisection of A002807.
  • A099202 (program): 2^floor(n^2/2).
  • A099204 (program): A variation on Flavius’s sieve (A000960): Start with the natural numbers; at the k-th sieving step, remove every p-th term of the sequence remaining after the (k-1)-st sieving step, where p is the k-th prime; iterate.
  • A099207 (program): A variation on Flavius’s sieve (A000960): Start with the primes; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate.
  • A099210 (program): a(n)=(P(n)*P(n+2))^5 with P(i)=i-th prime.
  • A099211 (program): a(n)=-2a(n-1)+4a(n-3).
  • A099212 (program): a(n)=-2a(n-1)+4a(n-3).
  • A099213 (program): a(n) = a(n-1)+a(n-2)+3a(n-3), with a(0)=a(1)=a(2)=1.
  • A099214 (program): a(n)=4a(n-1)-4a(n-2)+4a(n-3).
  • A099215 (program): a(n)=4a(n-1)-4a(n-2)+3a(n-3).
  • A099216 (program): a(n)=4a(n-1)-4a(n-2)+2a(n-3).
  • A099217 (program): Decimal expansion of Li_3(1/2).
  • A099232 (program): a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3).
  • A099233 (program): Square array read by antidiagonals associated to sections of 1/(1-x-x^k).
  • A099234 (program): A trisection of 1/(1-x-x^4).
  • A099235 (program): A quadrisection of 1/(1-x-x^5).
  • A099236 (program): Sums of antidiagonals of A099233.
  • A099237 (program): a(n) = Sum_{k=0..n} binomial(n*(n-k), k).
  • A099238 (program): Square array read by antidiagonals with rows generated by 1/(1-x-x^(k+1)).
  • A099239 (program): Square array read by antidiagonals associated with sections of 1/(1-x-x^k).
  • A099240 (program): Main diagonal of A099239.
  • A099241 (program): Sums of antidiagonals of A099239.
  • A099242 (program): (6n+5)-th terms of expansion of 1/(1 - x - x^6).
  • A099243 (program): A variation on Flavius’s sieve (A000960): Start with the primes; at the k-th sieving step, remove every p-th term of the sequence remaining after the (k-1)-st sieving step, where p is the k-th prime; iterate.
  • A099244 (program): Greatest common divisor of length of n in binary representation and its number of ones.
  • A099245 (program): Numerator of relative frequency of number of ones in binary representation of n.
  • A099246 (program): Denominator of relative frequency of number of ones in binary representation of n.
  • A099247 (program): Numbers such that, in binary representation, the length and the number of ones are coprime.
  • A099248 (program): Numbers such that length in binary representation and number of ones have a common divisor greater than 1.
  • A099249 (program): Number of numbers not greater than n such that length in binary representation and number of ones are coprime.
  • A099250 (program): Bisection of Motzkin numbers A001006.
  • A099251 (program): Bisection of Motzkin sums (A005043).
  • A099252 (program): Bisection of A005043.
  • A099253 (program): (7*n+6)-th terms of expansion of 1/(1-x-x^7).
  • A099254 (program): Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot).
  • A099255 (program): G.f. (7+6*x-6*x^2-3*x^3)/((x^2+x-1)*(x^2-x-1)).
  • A099256 (program): G.f.: (3-x)(1+3x+x^2)/((1-x-x^2)(1+x-x^2)).
  • A099257 (program): a(1)=1, a(n+1) = if a(n)=n then 2*n+1 else smallest number not occurring earlier.
  • A099258 (program): Inverse of A099257.
  • A099259 (program): A100287(n+1) - A000960(n).
  • A099260 (program): Number of decimal digits in (10^n)-th prime number.
  • A099261 (program): Length in bits of (10^n)-th prime number.
  • A099262 (program): a(n) = (1/5040)*7^n + (1/240)*5^n + (1/72)*4^n + (1/16)*3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).
  • A099263 (program): a(n) = (1/40320)*8^n + (1/1440)*6^n + (1/360)*5^n + (1/64)*4^n + (11/180)*3^n + (53/288)*2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).
  • A099264 (program): A000960(n) - A100287(n).
  • A099265 (program): Partial sums of A056272.
  • A099266 (program): Partial sums of A056273.
  • A099267 (program): Numbers generated by the golden sieve.
  • A099269 (program): A sequence derived from a matrix using “0,1,2,3,4,5,6”.
  • A099270 (program): Unsigned member r=-12 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099271 (program): Unsigned member r=-13 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099272 (program): Unsigned member r=-14 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099273 (program): Unsigned member r=-15 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099275 (program): Unsigned member r=-17 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099276 (program): Unsigned member r=-18 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099277 (program): Unsigned member r=-19 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099278 (program): Unsigned member r=-20 of the family of Chebyshev sequences S_r(n) defined in A092184.
  • A099279 (program): Squares of A001076 (generalized Fibonacci).
  • A099281 (program): Decimal expansion of the sine integral at 1.
  • A099283 (program): Decimal expansion of the hyperbolic sine integral at 1.
  • A099301 (program): Arithmetic derivative of d(n), the number of divisors of n.
  • A099306 (program): n’’’, the third arithmetic derivative of n.
  • A099308 (program): Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.
  • A099309 (program): Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308.
  • A099310 (program): Arithmetic derivative of Euler’s totient function phi(n).
  • A099322 (program): An inverse Catalan transform of J(3n)/J(3).
  • A099323 (program): Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)).
  • A099324 (program): Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).
  • A099325 (program): Expansion of (sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(3/2)).
  • A099326 (program): Expansion of ((1-2x)*sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(5/2)).
  • A099327 (program): Expansion of ((1-x)sqrt(1+2x)+(1+x)sqrt(1-2x))/(2(1-2x)^(5/2)).
  • A099358 (program): a(n) = sum of digits of k^4 as k runs from 1 to n.
  • A099359 (program): a(n) = (2^n + 1)^3 - 2.
  • A099360 (program): a(n) = (2^n + 1)^4 - 2.
  • A099363 (program): An inverse Chebyshev transform of 1-x.
  • A099364 (program): An inverse Chebyshev transform of (1-x)^2.
  • A099365 (program): Squares of A052918(n-1) (generalized Fibonacci).
  • A099366 (program): Squares of A005668(n) (generalized Fibonacci).
  • A099367 (program): Squares of A054413(n-1), n>=1,(generalized Fibonacci).
  • A099368 (program): Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at x=51/2.
  • A099369 (program): Squares of A041025(n-1), n>=1, (generalized Fibonacci).
  • A099370 (program): Chebyshev polynomial of the first kind, T(n,x), evaluated at x=33.
  • A099371 (program): Expansion of g.f.: x/(1 - 9*x - x^2).
  • A099372 (program): Squares of A099371(n) (generalized Fibonacci).
  • A099373 (program): Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at 83/2.
  • A099374 (program): Squares of A041041(n-1), n>=1 (generalized Fibonacci).
  • A099375 (program): Sequence matrix for odd numbers.
  • A099376 (program): An inverse Chebyshev transform of x^3.
  • A099377 (program): Numerators of the harmonic means of the divisors of the positive integers.
  • A099378 (program): Denominators of the harmonic means of the divisors of the positive integers.
  • A099389 (program): Subsequence of primes in sequence b(n) = 3*prime(n) - prime(n+1) - 3 (A100021).
  • A099392 (program): a(n) = floor((n^2 - 2*n + 3)/2).
  • A099393 (program): a(n) = 4^n + 2^n - 1.
  • A099394 (program): Triangle T(k,n) by rows: n! * A075499(k,n).
  • A099395 (program): One if odd part of n is 3, zero otherwise.
  • A099396 (program): a(n) = floor(log_2(2/3*n)) for n >= 2, a(1) = 0.
  • A099397 (program): Chebyshev polynomial of the first kind, T(n,x), evaluated at x=51.
  • A099398 (program): Numerators of rationals (in lowest terms) used in a certain high temperature expansion.
  • A099399 (program): Denominators of rationals (in lowest terms) used in a certain high temperature expansion.
  • A099407 (program): Number of points of self-intersection of the path of a billiard ball traveling at a 45-degree angle on a prime(n) X prime(n+1) billiard table. Also equal to 1/2 the number of the lattice points lying within a prime(n) X prime(n+1) rectangle.
  • A099423 (program): Lean quaternary temporal logic [LQTL] (emergent) cumulative column frequencies from the zeroth to 22nd iteration of LQTL logic in A094266. Note that with an initial False in the zeroth iteration Murphy’s Law holds in LQTL in all but six iterations (the 6th, 7th, 14th, 15th, 21st and 22nd).
  • A099425 (program): Expansion of (1+x^2)/(1-2*x-x^2).
  • A099427 (program): a(1) = 1; for n > 1, a(n) = 1 + greatest common divisor of n and a(n-1).
  • A099429 (program): A Jacobsthal-Lucas convolution.
  • A099430 (program): 2^n+(-1)^n-1.
  • A099431 (program): Expansion of x(1-2x+3x^2)/(1-x-2x)^2;.
  • A099432 (program): Convolution of A030195(n) (generalized (3,3)-Fibonacci) with itself.
  • A099443 (program): A Chebyshev transform of Fib(n+1).
  • A099444 (program): A Chebyshev transform of Fib(2n+2).
  • A099445 (program): An Alexander sequence for the Miller Institute knot.
  • A099446 (program): A Chebyshev transform of A057083.
  • A099447 (program): An Alexander sequence for the knot 6_3.
  • A099448 (program): A Chebyshev transform of A030191 associated to the knot 7_6.
  • A099449 (program): An Alexander sequence for the knot 7_6.
  • A099450 (program): Expansion of 1/(1 - 5x + 7x^2).
  • A099451 (program): A Chebyshev transform of A099450 associated to the knot 7_7.
  • A099452 (program): An Alexander sequence for the knot 7_7.
  • A099453 (program): Expansion of 1/(1 - 7*x + 11*x^2).
  • A099454 (program): A Chebyshev transform of A099453 associated to the knot 8_12.
  • A099455 (program): An Alexander sequence for the knot 8_12.
  • A099456 (program): Expansion of 1/(1 - 4*x + 5*x^2).
  • A099457 (program): A Chebyshev transform of A099456 associated to the knot 9_44.
  • A099458 (program): An Alexander sequence for the knot 9_44.
  • A099459 (program): Expansion of 1/(1 - 7*x + 9*x^2).
  • A099460 (program): A Chebyshev transform of A099459 associated to the knot 9_48.
  • A099461 (program): An Alexander sequence for the knot 9_48.
  • A099462 (program): Expansion of x/(1 - 4*x^2 - 4*x^3).
  • A099463 (program): Bisection of tribonacci numbers.
  • A099464 (program): Trisection of tribonacci numbers.
  • A099467 (program): a(1) = a(2) = 1; for n > 2, a(n) is the smallest number > a(n-1) which is not the sum of 2 consecutive elements of the sequence.
  • A099469 (program): Least common multiple of n and its digit sum.
  • A099470 (program): A sequence generated from the Quadrifoil.
  • A099471 (program): A sequence generated from the Quadrifoil (flat knot).
  • A099474 (program): Numbers n such that 3*prime(n) - prime(n+1) - 3 is prime.
  • A099475 (program): Number of divisors d of n such that d+2 is also a divisor of n.
  • A099477 (program): Numbers having no divisors d such that also d+2 is a divisor.
  • A099479 (program): Count, repeating 4n three times for n > 0.
  • A099480 (program): Count from 1, repeating 2n five times.
  • A099483 (program): A Fibonacci convolution.
  • A099484 (program): A Fibonacci convolution.
  • A099485 (program): A Fibonacci convolution.
  • A099486 (program): Expansion of x/((1 + x^2)*(1 - 4*x + x^2)).
  • A099487 (program): Expansion of (1-3x+x^2)/((1+x^2)(1-4x+x^2)).
  • A099488 (program): Expansion of (1-x)^2/((1+x^2)(1-4x+x^2)).
  • A099489 (program): Expansion of (1-x+x^2)/((1+x^2)(1-4x+x^2)).
  • A099491 (program): A Chebyshev transform of Padovan numbers.
  • A099492 (program): A Chebyshev transform of the Padovan-Jacobsthal numbers.
  • A099493 (program): Expansion of (1+x^2)^2/(1+x^2-2x^3+x^4+x^6).
  • A099494 (program): A Chebyshev transform of Fib(n)+(-1)^n.
  • A099495 (program): A Chebyshev transform of Fib(n)^2.
  • A099496 (program): a(n) = (-1)^n * Fibonacci(2*n+1).
  • A099503 (program): Expansion of 1/(1-4*x+x^3).
  • A099504 (program): Expansion of 1/(1-5x+x^3).
  • A099505 (program): A transform of the Fibonacci numbers.
  • A099508 (program): A transform of the Jacobsthal numbers.
  • A099510 (program): Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + 2*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.
  • A099511 (program): Row sums of triangle A099510, so that a(n) = Sum_{k=0..n} coefficient of z^k in (1 + 2*z + z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.
  • A099513 (program): Row sums of triangle A099512, so that a(n) = Sum_{k=0..n} coefficient of z^k in (1 + 3*z + z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.
  • A099515 (program): Row sums of triangle A099514, so that a(n) = Sum_{k=0..n} coefficient of z^k in (1 + z + 2*z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.
  • A099516 (program): A transform of the Pell numbers.
  • A099517 (program): A transform of (1-x)/(1-2x).
  • A099524 (program): Expansion of 1/(1-5*x-x^3).
  • A099525 (program): Expansion of 1/(1-2x-3x^3).
  • A099526 (program): Expansion of 1/(1-2x-3x^4).
  • A099528 (program): Row sums of triangle A099527, so that a(n) = Sum_{k=0..n} coefficient of z^k in (2 + 3*z + z^2)^(n-[k/2]), where [k/2] is the integer floor of k/2.
  • A099529 (program): Expansion of (1+x)^2/((1+x)^2+x^3).
  • A099530 (program): Expansion of 1/(1-x+x^4).
  • A099531 (program): Expansion of (1+x)^3/((1+x)^3+x^4).
  • A099534 (program): a(n)=Sum of the first n decimal places of e.
  • A099535 (program): Sum of the first n decimal places of log(2).
  • A099536 (program): Sum of the first n digits of Zeta(3) (Apery’s constant), including the initial 1.
  • A099537 (program): Sum of the first n decimal places of zeta(3) (Apery’s constant). This does not include the initial “1.” of zeta(3).
  • A099538 (program): Sum of the first n digits of sqrt(2), including the initial “1”.
  • A099539 (program): Sum of the first n decimal places of sqrt(2).
  • A099544 (program): Odd part of n modulo 3.
  • A099545 (program): Odd part of n, modulo 4.
  • A099546 (program): Odd part of n modulo 5.
  • A099547 (program): Odd part of n modulo 6.
  • A099548 (program): Odd part of n modulo 7.
  • A099549 (program): Odd part of n modulo 8.
  • A099550 (program): Odd part of n modulo 9.
  • A099551 (program): Odd part of n modulo 10. Final digit of A000265(n).
  • A099555 (program): Triangle, read by rows, where T(n,k) = (n-floor(k/2))^k for k = 0..2*n.
  • A099557 (program): Slanted Pascal’s triangle, read by rows, such that T(n,k) = binomial(n-[k/2],k) for [n*2/3]>=k>=0, where [x]=floor(x).
  • A099558 (program): Antidiagonal sums of the triangle A099557.
  • A099559 (program): a(n) = Sum_{k=0..floor(n/5)} C(n-4k,k+1).
  • A099560 (program): a(n) = Sum_{k=0..floor(n/3)} C(n-2k,k-1).
  • A099561 (program): Sum C(n-3k,k-1), k=0..floor(n/4).
  • A099562 (program): Sum C(n-4k,k-1), k=0..floor(n/5).
  • A099563 (program): a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),…, where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.
  • A099564 (program): a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),…, where f(n,d)=Floor(n/F(d+1)), with F denoting the Fibonacci numbers (A000045).
  • A099565 (program): Location of records in A099564.
  • A099566 (program): A099565(n)/n.
  • A099567 (program): Riordan array (1/(1-x-x^3), 1/(1-x)).
  • A099568 (program): Expansion of (1-x)/((1-2*x)*(1-x-x^3)).
  • A099569 (program): Riordan array ((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).
  • A099570 (program): Expansion of ((1+x)^2 - x^3)/(1+x)^2.
  • A099571 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, k).
  • A099572 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, k).
  • A099573 (program): Reverse of number triangle A054450.
  • A099574 (program): Diagonal sums of triangle A099573.
  • A099575 (program): Number triangle T(n,k) = binomial(n + floor(k/2) + 1, n + 1), 0 <= k <= n.
  • A099576 (program): Row sums of triangle A099575.
  • A099577 (program): Diagonal sums of triangle A099575.
  • A099578 (program): a(n) = binomial(floor((3n+2)/2), floor(n/2)).
  • A099579 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 3^(k-1).
  • A099580 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 4^(k-1).
  • A099581 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*3^(n-k-1).
  • A099582 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1).
  • A099583 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*2^(n-k-1)*(3/2)^(k-1).
  • A099584 (program): Exponent of 3 in factorization of prime(n) - 1.
  • A099585 (program): Remove all 3s from prime(n) - 1.
  • A099586 (program): Constant term in (1+x)^n mod (1+x^4).
  • A099587 (program): a(n) = coefficient of x in (1+x)^n mod (1+x^4).
  • A099588 (program): Coefficient of x^2 in (1+x)^n mod 1+x^4.
  • A099589 (program): Expansion of x^3 / (1 - 4x + 6x^2 - 4x^3 + 2x^4).
  • A099590 (program): 2^(n-1) times coefficient of x in (1+x)^n mod U(n,x), U the Chebyshev polynomials.
  • A099597 (program): Array T(k,n) read by antidiagonals: expansion of exp(x+y)/(1-xy).
  • A099598 (program): Antidiagonal sums of array A099597.
  • A099601 (program): Quotient of de Bruijn sums S(4,n)/S(2,n).
  • A099602 (program): Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.
  • A099603 (program): Row sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).
  • A099604 (program): Antidiagonal sums of triangle A099602, in which row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).
  • A099606 (program): Row sums of triangle A099605, in which row n equals the inverse Binomial transform of column n of the triangle A034870 of even-indexed rows of Pascal’s triangle.
  • A099608 (program): Table of crystal ball sequences for A_n lattices read by antidiagonals.
  • A099609 (program): Naive list of twin primes (A077800 prefixed by 2, 3).
  • A099612 (program): Numerators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.
  • A099615 (program): Triangle read by rows, 2<=k<=n: T(n,k) = denominator of (1+1/n)^k-(1+k/n) and of (1-1/n)^k-(1-k/n).
  • A099617 (program): Denominators of the coefficients in the Taylor expansion of sec(x) + tan(x) around x=0.
  • A099618 (program): a(n) = 1 if the n-th prime == 1 mod 6, otherwise a(n) = 0.
  • A099621 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k+1) * 3^(n-k-1)*(4/3)^k.
  • A099622 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1)*(5/4)^k.
  • A099623 (program): Sum C(n-k,k+2)2^(n-k-2)(3/2)^k, k=0..floor(n/2).
  • A099624 (program): Sum C(n-k,k+2)3^(n-k-2)(4/3)^k, k=0..floor(n/2).
  • A099625 (program): Sum C(n-k,k+2)2^(n-k-2)(1/2)^k, k=0..floor(n/2).
  • A099626 (program): A transform of the Pell numbers.
  • A099627 (program): Triangle read by rows: T(n,k)=2^n+2^k-1 with n>=k>=0.
  • A099628 (program): Numbers m where m-th Catalan number A000108(m) = binomial(2m,m)/(m+1) is divisible by 2 but not by 4, i.e., where A048881(m) = 1.
  • A099634 (program): a(n) = gcd(P+p, P*p) where P is the largest and p the smallest prime factor of n.
  • A099635 (program): a(n) = gcd(sum of all prime factors of n, n).
  • A099636 (program): a(n) = gcd(sum of distinct prime factors of n, product of distinct prime factors of n).
  • A099638 (program): a[n]=A098210[n]/15.
  • A099644 (program): a[n]=Mod[q(n),PrimePi[q(n)]]=Mod[A001359(n), A000720[A001359(n))] where q(n) is the n-th lesser-twin-prime. A004648 restricted to lesser twins.
  • A099654 (program): a(n) is the number of n-subsets [n=1,2,…,10] of the 10 decimal digits from which no prime numbers can be constructed. See also A099653.
  • A099669 (program): Partial sums of repdigits of A002276.
  • A099670 (program): Partial sums of repdigits of A002277.
  • A099671 (program): Partial sums of repdigits of A002278.
  • A099672 (program): Partial sums of repdigits of A002279.
  • A099673 (program): Partial sums of repdigits of A002280.
  • A099674 (program): Partial sums of repdigits of A002281.
  • A099675 (program): Partial sums of repdigits of A002282.
  • A099676 (program): Partial sums of repdigits of A002283.
  • A099677 (program): Primes arising in A032682.
  • A099679 (program): Least m such that repunit R_m is a multiple of A045572(n) (i.e., odd numbers not divisible by 5).
  • A099721 (program): a(n) = n^2*(2*n+1).
  • A099726 (program): Sum of remainders of the n-th prime mod k, for k = 1,2,3,…,n.
  • A099730 (program): Array read by antidiagonals. Rows contain odd numbers reaching same odd successor in Collatz function iteration.
  • A099738 (program): a(n) = 2*Sum_{k=1..n} (n+1-k) (Sum_{j|k} 1/floor(n/j)).
  • A099739 (program): a(n) = Sum_{k=1..n} k!*mu(k), where mu() is the Moebius function.
  • A099743 (program): Number of permutations with exactly 1 valley which avoid the pattern 1324.
  • A099751 (program): Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.
  • A099753 (program): a(n) = (2*n+1)^(n+2).
  • A099754 (program): a(n) = (3^n +1)/2 + 2^n.
  • A099760 (program): a(n+1) = 2*n*a(n) + 2 with a(0)=1.
  • A099761 (program): a(n) = ( n*(n+2) )^2.
  • A099762 (program): a(n) = n^2 * (n+1)^3.
  • A099764 (program): a(n) = n^2 * (n+1)^2 * (n+2)^2 = 36*A001249(n-1).
  • A099765 (program): a(n) = (1/Pi)*(2^n/n)*(n-1)!*Integral_{t>=0} (sin(t)/t)^n dt.
  • A099767 (program): Number of n-digit palindromes in base n.
  • A099770 (program): Expansion of 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^6)).
  • A099773 (program): Number of partitions of n into odd prime parts.
  • A099774 (program): Number of divisors of 2*n-1.
  • A099776 (program): Length of the hypotenuse of an integer right triangle with the hypotenuse being one more than the longer side. The shorter sides are just consecutive odd numbers 3, 5, 7, …
  • A099777 (program): Number of divisors of 2n.
  • A099779 (program): a(n) = ceiling( 1/2 + (Sum_{i=0..n-1}C(n,i)*C(n,i+1))/2^(n+1) ).
  • A099780 (program): a(n) = Sum_{k=0..floor(n/3)} C(n-k, 2*k) * 2^k*3^(n-3*k).
  • A099781 (program): a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 4^(n-3*k).
  • A099782 (program): a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * 4^(n-3*k).
  • A099783 (program): a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k)*3^(n-2*k).
  • A099784 (program): a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * (-2)^(n-3*k).
  • A099785 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^(n-3*k).
  • A099786 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k)*3^(n-4*k).
  • A099788 (program): a(n) = Product_{i=1..2n} prime(i).
  • A099795 (program): Least common multiple of 1, 2, 3, …, prime(n)-1.
  • A099800 (program): Bisection of A002110.
  • A099801 (program): PrimePi(2n+1), the number of primes less than or equal to 2n+1.
  • A099802 (program): Bisection of A000720.
  • A099812 (program): Number of distinct primes dividing 2n (i.e., omega(2n)).
  • A099813 (program): Bisection of A007318.
  • A099814 (program): Bisection of A002275.
  • A099816 (program): Bisection of A000796 (decimal expansion of Pi).
  • A099817 (program): Bisection of A000796 (decimal expansion of Pi).
  • A099820 (program): Even nonnegative integers in base 2 (bisection of A007088).
  • A099821 (program): Odd positive integers in base 2 (bisection of A007088).
  • A099822 (program): Bisection of A005117.
  • A099823 (program): G.f. is the continued fraction: A(x) = 1/[1 - x/[1 - (x-x^2)/[1 - (x^2-x^4)/[1 - (x^3-x^6)/[1-… - (x^n-x^(2n))/[1 - … ]]]]]]].
  • A099827 (program): Generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5 multiplied by (n!)^5.
  • A099828 (program): Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.
  • A099831 (program): Perimeters of Pythagorean triangles that can be constructed in exactly 2 different ways.
  • A099835 (program): Bisection of A005117.
  • A099836 (program): Bisection of A000961.
  • A099837 (program): Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.
  • A099838 (program): Expansion of (1-x)^2(1+x)/(1+x+x^2).
  • A099839 (program): a(0) = 1, a(1) = -11 and a(n) = -10*a(n-1) - 5*a(n-2), n >= 2.
  • A099840 (program): Expansion of (1-6*x)/(1-20*x^2).
  • A099841 (program): Expansion of (1-16*x)/(1-20*x+80*x^2).
  • A099842 (program): Expansion of (1-x)/(1 + 6*x - 3*x^2).
  • A099843 (program): A transform of the Fibonacci numbers.
  • A099844 (program): An Alexander sequence for the knot 8_2.
  • A099845 (program): A Chebyshev transform of A090400 related to the knot 8_2.
  • A099846 (program): An Alexander sequence for the knot 8_5.
  • A099847 (program): Bisection of A000961.
  • A099848 (program): All natural numbers occur in their order as many times as they have ordered prime factorizations.
  • A099849 (program): Partial sums of A008480.
  • A099850 (program): Sum of the first n terms of A004648 (the remainder after dividing n-th prime by n).
  • A099854 (program): A Chebyshev transform of A048739 related to the knot 8_5.
  • A099855 (program): a(n) = n*2^n - 2^(n/2)*sin(Pi*n/4).
  • A099856 (program): Expansion of (1+3x)/(1-3x).
  • A099857 (program): Expansion of (1+3x+x^2)/(1-3x+x^2).
  • A099858 (program): A Chebyshev transform of (1+3x)/(1-3x).
  • A099859 (program): A Chebyshev transform of A006053 related to the knot 7_1.
  • A099860 (program): A Chebyshev transform related to the knot 7_1.
  • A099861 (program): Bisection of A002808.
  • A099862 (program): Bisection of A002808.
  • A099865 (program): Numerator of sum of all elements M(i,j,k) = i*j/k, (i,j,k = 1..n). a(n) = Numerator[Sum[Sum[Sum[i*j/k,{i,1,n}],{j,1,n}],{k,1,n}]].
  • A099866 (program): Denominator of sum of all elements M(i,j,k) = i*j/k, (i,j,k = 1..n). a(n) = Denominator[Sum[Sum[Sum[i*j/k,{i,1,n}],{j,1,n}],{k,1,n}]].
  • A099867 (program): a(n) = 5*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=9.
  • A099868 (program): a(n) = 5*a(n-1) - a(n-2), a(0) = 3, a(1) = 25.
  • A099869 (program): Bisection of A014137.
  • A099880 (program): Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements).
  • A099884 (program): XOR difference triangle of the powers of 2, read by rows; Square array A(row,col): A(0,col) = 2^col, A(row,col) = A048724(A(row-1, col)) for row > 0, read by descending antidiagonals.
  • A099885 (program): Central terms of the rows of the XOR difference triangle of the powers of 2 (A099884) so that a(n) = A099884(n,[n/2]).
  • A099886 (program): XOR binomial transform of A099885.
  • A099890 (program): XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.
  • A099892 (program): XOR BINOMIAL transform of A003188 (Gray code numbers); also the main diagonal of the XOR difference triangle A099891.
  • A099893 (program): XOR BINOMIAL transform of A006068 (inverse Gray code).
  • A099894 (program): XOR BINOMIAL transform of A038712.
  • A099895 (program): XOR BINOMIAL transform of A000069 (Odious numbers).
  • A099896 (program): A permutation of the natural numbers where a(n) = n XOR [n/2] XOR [n/4].
  • A099898 (program): Shifts left and divides by 4 under the XOR BINOMIAL transform (A099899).
  • A099899 (program): Multiplies by 4 and shifts right under the XOR BINOMIAL transform (A099898).
  • A099901 (program): Shifts left and divides by 2 under the XOR BINOMIAL transform (A099902).
  • A099902 (program): Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).
  • A099903 (program): Sum of all matrix elements of n X n matrix M(i,j) = i^3+j^3, (i,j = 1..n). a(n) = n^3*(n+1)^2/2.
  • A099904 (program): Numerator of sum of all matrix elements of N X N matrix M(i,j) = i^3+j^3, (i,j = 1..n) divided by n!.
  • A099905 (program): a(n) = binomial(2n-1, n-1) mod n.
  • A099906 (program): a(n) = binomial(2n-1,n-1) mod n^2.
  • A099907 (program): a(n) = C(2n-1,n-1) mod n^3.
  • A099908 (program): C(2n-1,n-1) mod n^4.
  • A099909 (program): (prime(n)*(prime(n+1)-1) + (prime(n)-1)*prime(n+1)) / 2.
  • A099910 (program): Number of distinct prime-factors of ((prime(n)*(prime(n+1)-1)+(prime(n)-1)*prime(n+1))/2).
  • A099911 (program): Primes of the form (p*(q-1) + (p-1)*q)/2, where p and q are consecutive odd primes.
  • A099912 (program): Number of closed walks on the Herschel graph.
  • A099913 (program): Related to the Herschel graph.
  • A099914 (program): Expansion of (1+3x)/((1-x)(1-10x)).
  • A099915 (program): Expansion of (1+4x)/((1-x)(1-10x)).
  • A099916 (program): Expansion of (1+x^2)^2/(1-x^3+x^6).
  • A099917 (program): Expansion of (1+x^2)^2/(1+x^3+x^6).
  • A099918 (program): A Chebyshev transform related to the 7th cyclotomic polynomial.
  • A099919 (program): a(n) = F(3) + F(6) + F(9) + … + F(3n), F(n) = Fibonacci numbers A000045.
  • A099920 (program): a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.
  • A099921 (program): a(n) = 5*Fibonacci(n)^2.
  • A099922 (program): a(n) = F(4n) - 2n, where F(n) = Fibonacci numbers A000045.
  • A099923 (program): Fourth powers of Lucas numbers A000032.
  • A099924 (program): Self-convolution of Lucas numbers.
  • A099925 (program): a(n) = Lucas(n) + (-1)^n.
  • A099930 (program): a(n) = Pell(n)*Pell(n-1)*Pell(n-2) / 10.
  • A099931 (program): a(n) = Pell(n)*Pell(n-1)*Pell(n-2)*Pell(n-3) / 120.
  • A099933 (program): Main diagonal of array A007754.
  • A099934 (program): Antidiagonal sums of array A007754.
  • A099935 (program): Decimal expansion of Sum_{k>=0} (-1)^(k+1)*A000045(k)/k!.
  • A099938 (program): Consider the sequence of circles C0, C1, C2, C3 …, where C0 is a half-circle of radius 1. C1 is the largest circle that fits into C0 and has radius 1/2. C(n+1) is the largest circle that fits inside C0 but outside C(n), etc. Sequence gives the curvatures (reciprocals of the radii) of the circles.
  • A099942 (program): Start with 1, then alternately double or add 2.
  • A099943 (program): Number of 5 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0).
  • A099944 (program): Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0).
  • A099945 (program): Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0).
  • A099946 (program): a(n) = lcm{1, 2, …, n}/(n*(n-1)), n >= 2.
  • A099953 (program): a(n) = A076795(n) - 1.
  • A099957 (program): a(n) = Sum_{i=0..n-1} phi(2i+1).
  • A099958 (program): (1/2)*number of distinct angular positions under which an observer positioned at the center of an edge of a square lattice can see the (2n)X(2n-1) points symmetrically surrounding his position.
  • A099971 (program): Write (sqrt(5)-1)/2 as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.
  • A099972 (program): Write 1/sqrt(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.
  • A099975 (program): Bisection of A014137.
  • A099976 (program): Bisection of A000984.
  • A099977 (program): Bisection of Bell numbers, A000110.
  • A099978 (program): Bisection of A001157: a(n) = sigma_2(2n-1).
  • A099979 (program): Bisection of A001157: sigma_2(2n).
  • A099980 (program): Bisection of A001358.
  • A099981 (program): Bisection of A001358.
  • A099984 (program): Bisection of A007947.
  • A099985 (program): a(n) = rad(2n), where rad = A007947.
  • A099986 (program): Bisection of A001113 (digits of e).
  • A099987 (program): Bisection of A001113 (digits of e).
  • A099990 (program): a(n) = Moebius(2n+1).
  • A099991 (program): a(n) = Moebius(2n).
  • A099994 (program): Bisection of A002113.
  • A099995 (program): Bisection of A002113.
  • A099996 (program): a(n) = lcm{1, 2, …, 2*n}.

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