A027848 a(n) = Sum_{ d|n } sigma(n/d)*d^4.

1, 19, 85, 311, 631, 1615, 2409, 4991, 6898, 11989, 14653, 26435, 28575, 45771, 53635, 79887, 83539, 131062, 130341, 196241, 204765, 278407, 279865, 424235, 394406, 542925, 558778, 749199, 707311, 1019065, 923553, 1278255, 1245505, 1587241, 1520079, 2145278, 1874199, 2476479, 2428875, 3149321, 2825803, 3890535, 3418845, 4557083, 4352638, 5317435

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001001, A027847, A288391.

Programs

Mathematica

f[p_, e_] := (1 + p + p^2 - p^(e+1) - p^(e+2) - p^(e+3) - p^(e+4) + p^(4*e+7))/(1 - p^3 - p^4 + p^7); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 03 2023 *)

PARI

N=66; x='x+O('x^N); /* that many terms */
c=sum(j=1,N,j*x^j);
t=log(1/prod(j=1,N, eta(x^(j))^(j^3)));
Vec(serconvol(t,c)) /* show terms */
/* Joerg Arndt, May 03 2008 */

Formula

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-4). [corrected by Michael Shamos, May 03 2025]

Multiplicative with a(p^e) = (p^(4e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2+p+1)/(p^7 - (p^3+p^2+p+1)*p + p^2+p+1). - Mitch Harris, Jun 27 2005

L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma_3(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

Sum_{k=1..n} a(k) ~ zeta(5) * Pi^4 * n^5 / 450. - Vaclav Kotesovec, Feb 16 2020, [corrected May 04 2025, according to the corrected DGF]