A329385 Dirichlet g.f.: 1 / (2 - Product_{k>=1} zeta(k*s)).

1, 1, 1, 3, 1, 3, 1, 8, 3, 3, 1, 11, 1, 3, 3, 22, 1, 11, 1, 11, 3, 3, 1, 36, 3, 3, 8, 11, 1, 13, 1, 59, 3, 3, 3, 45, 1, 3, 3, 36, 1, 13, 1, 11, 11, 3, 1, 116, 3, 11, 3, 11, 1, 36, 3, 36, 3, 3, 1, 57, 1, 3, 11, 160, 3, 13, 1, 11, 3, 13, 1, 164, 1, 3, 11, 11, 3, 13, 1, 116

Offset: 1

Author

Ilya Gutkovskiy, Jun 07 2020

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
• Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Crossrefs

Cf. A000688, A001358 (positions of 3's), A008578 (positions of 1's), A050354, A129667.

Programs

Mathematica

a[n_] := If[n == 1, n, Sum[If[d < n, FiniteAbelianGroupCount[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 80}]

Formula

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} A000688(k) * A(x^k).

a(1) = 1; a(n) = Sum_{d|n, d < n} A000688(n/d) * a(d).

Let f(s) = Product_{k>=1} zeta(k*s), then Sum_{k=1..n} a(k) ~ n^r / (-r*f'(r)), where r = A335494 = 1.8868691498777... is the root of the equation f(r) = 2 and f'(r) = -1.8255483309672084429580571100367977185868132697213762608374345719289... - Vaclav Kotesovec, Jun 11 2020