A383795 Dirichlet g.f.: zeta(2*s-2) * zeta(s)^2.

1, 2, 2, 7, 2, 4, 2, 12, 12, 4, 2, 14, 2, 4, 4, 33, 2, 24, 2, 14, 4, 4, 2, 24, 28, 4, 22, 14, 2, 8, 2, 54, 4, 4, 4, 84, 2, 4, 4, 24, 2, 8, 2, 14, 24, 4, 2, 66, 52, 56, 4, 14, 2, 44, 4, 24, 4, 4, 2, 28, 2, 4, 24, 139, 4, 8, 2, 14, 4, 8, 2, 144, 2, 4, 56, 14, 4, 8, 2, 66, 113

Offset: 1

Author

Vaclav Kotesovec, May 10 2025

Keywords

• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A035316, A057521.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], Sum[(e+1-2*k) * p^(2*k), {k, 0, (e-1)/2}], Sum[(e+1-2*k) * p^(2*k), {k, 0, e/2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 24 2025 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/((1-p^2*X^2)*(1-X)^2))[n], ", "))

Formula

Sum_{k=1..n} a(k) ~ zeta(3/2)^2 * n^(3/2)/3 - n*(log(n) + 2*log(2*Pi) + 2*gamma - 1)/2, where gamma is the Euler-Mascheroni constant A001620.

Multiplicative with a(p^e) = Sum_{k=0..(e-1)/2} (e+1-2*k) * p^(2*k) if e is odd, and Sum_{k=0..e/2} (e+1-2*k) * p^(2*k) if e is even. - Amiram Eldar, May 24 2025