A308693 a(n) = Sum_{d|n} d^(3*(n/d - 1)).

1, 2, 2, 10, 2, 93, 2, 578, 731, 4223, 2, 56765, 2, 262489, 547068, 2359810, 2, 31173510, 2, 152949071, 387538140, 1073743157, 2, 20134371189, 244140627, 68719478935, 282430067924, 618515646977, 2, 12056339359929, 2, 39582552821762, 205891133866212

Offset: 1

Author

Seiichi Manyama, Jun 17 2019

Keywords

• nonn

Crossrefs

Column k=3 of A308694.

Programs

Mathematica

a[n_] := DivisorSum[n, #^(3*(n/# - 1)) &]; Array[a, 33] (* Amiram Eldar, May 09 2021 *)

PARI

{a(n) = sumdiv(n, d, d^(3*(n/d-1)))}

PARI

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k^3*x^k)^(1/k^4)))))

Formula

L.g.f.: -log(Product_{k>=1} (1 - k^3*x^k)^(1/k^4)) = Sum_{k>=1} a(k)*x^k/k.

a(p) = 2 for prime p.

G.f.: Sum_{k>=1} x^k/(1 - k^3*x^k). - Ilya Gutkovskiy, Jul 25 2019