A308692 a(n) = Sum_{d|n} d^(2*(n/d - 1)).

1, 2, 2, 6, 2, 27, 2, 82, 83, 283, 2, 2047, 2, 4147, 7188, 20546, 2, 125964, 2, 343407, 533844, 1048699, 2, 10076747, 390627, 16777387, 43053284, 84003927, 2, 667311413, 2, 1342439682, 3486799044, 4294967587, 249905428, 52916914768, 2, 68719477099, 282429565044

Offset: 1

Author

Seiichi Manyama, Jun 17 2019

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..3143

Crossrefs

Column k=2 of A308694.

Programs

Maple

N:=100: # for a(1)..a(N)
g:= add(x^k/(1-k^2*x^k),k=1..N):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 05 2020

Mathematica

a[n_] := DivisorSum[n, #^(2*(n/# - 1)) &]; Array[a, 39] (* Amiram Eldar, May 09 2021 *)

PARI

{a(n) = sumdiv(n, d, d^(2*(n/d-1)))}

PARI

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k^2*x^k)^(1/k^3)))))

Formula

L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)^(1/k^3)) = Sum_{k>=1} a(k)*x^k/k.

a(p) = 2 for prime p.

G.f.: Sum_{k>=1} x^k/(1 - k^2*x^k). - Ilya Gutkovskiy, Jul 25 2019