A294567 a(n) = Sum_{d|n} d^(1 + 2*n/d).

1, 9, 28, 97, 126, 588, 344, 2049, 2917, 6174, 1332, 53764, 2198, 52320, 258648, 430081, 4914, 2463429, 6860, 8352582, 15181712, 8560308, 12168, 242240964, 48843751, 134606598, 1167064120, 1651526120, 24390, 14202123408, 29792, 25905102849, 94162701936

Offset: 1

Author

Seiichi Manyama, Nov 02 2017

Keywords

• nonn

Comments

If p is prime, a(p) = 1 + p^3. - Robert Israel, Nov 03 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..3143

Crossrefs

Column k=2 of A294579.

Cf. A292164.

Programs

Maple

f:= n -> add(d^(1+2*n/d),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Nov 03 2017

Mathematica

sd[n_] := Module[{d = Divisors[n]}, Total[d^(1 + (2 n)/d)]]; Array[sd,40] (* Harvey P. Dale, Mar 17 2020 *)
a[n_] := DivisorSum[n, #^(1 + 2*n/#) &]; Array[a, 33] (* Amiram Eldar, Oct 04 2023 *)

PARI

a(n) = sumdiv(n, d, d^(1+2*n/d)); \\ Michel Marcus, Nov 02 2017

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-k^2*x^k))) \\ Seiichi Manyama, Jan 14 2023

Formula

L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018

G.f.: Sum_{k>0} k^3 * x^k / (1 - k^2 * x^k). - Seiichi Manyama, Jan 14 2023