A379742 a(n) is the number of divisors of A000178(n).

1, 1, 2, 6, 18, 72, 234, 1088, 3600, 10416, 28080, 124080, 387828, 1921024, 6926400, 20941344, 54934880, 251328000, 810152280, 4254092800, 15266200950, 46208448000, 129674387920, 640501862400, 2197261252368, 6404827161600, 17436935577600, 43314231340800, 101062601640000, 436914124416000, 1355859833328000, 7074064925491200

Offset: 0

Author

Tsuyoshi Hanatate, Dec 31 2024

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000005, A000178, A027423.

Programs

Mathematica

DivisorSigma[0, FoldList[Times, Range[0, 30]!]] (* Paolo Xausa, Jan 06 2025 *)

PARI

a(n)=numdiv(prod(k=1,n,k!));
vector(30,n,a(n-1)) \\ Joerg Arndt, Jan 02 2025

PARI

a(n) = {my(prd = 1); forprime(p = 2, n, prd *= (1 + (n*(n+1)/2 - 1 - sum(i = 2, n, sumdigits(i, p)))/(p-1))); prd;} \\ Amiram Eldar, Jan 15 2025

Formula

a(n) = A000005(A000178(n)).

a(n) = Product_{primes p <= n} (1 + (n*(n+1)/2 - 1 - Sum_{i=2..n} s_p(i))/(p-1)), where s_p(i) is the sum of digits of i in base p. - Amiram Eldar, Jan 15 2025