A328487 Dirichlet g.f.: zeta(s)^2 * zeta(s-1)^2 * (1 - 2^(1 - s))^2.

1, 2, 8, 3, 12, 16, 16, 4, 42, 24, 24, 24, 28, 32, 96, 5, 36, 84, 40, 36, 128, 48, 48, 32, 98, 56, 184, 48, 60, 192, 64, 6, 192, 72, 192, 126, 76, 80, 224, 48, 84, 256, 88, 72, 504, 96, 96, 40, 178, 196, 288, 84, 108, 368, 288, 64, 320, 120, 120, 288, 124, 128, 672, 7, 336

Offset: 1

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

• nonn
• easy
• mult

Comments

Dirichlet convolution of A000593 with itself.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A001620, A034761, A328486.

Programs

Mathematica

nmax = 65; A000593 = Table[DivisorSum[n, Mod[#, 2] # &], {n, 1, nmax}]; Table[DivisorSum[n, A000593[[#]] A000593[[n/#]] &], {n, 1, nmax}]
f[p_, e_] := ((e+1)*p^(e+3) - (e+3)*(p^(e+2) - p + 1) + 2)/(p-1)^3; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

Formula

a(n) = Sum_{d|n} A000593(d) * A000593(n/d).

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (Pi^2 * (log(n)/2 + log(2) + gamma - 1/4) + 6*zeta'(2)) / 144, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019

Multiplicative with a(2^e) = e+1, and a(p^e) = ((e+1)*p^(e+3) - (e+3)*(p^(e+2) - p + 1) + 2)/(p-1)^3 for an odd prime p. - Amiram Eldar, Sep 15 2023