# Programs for A050000-A099999

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.
- A068