A027847 a(n) = Sum_{d|n} sigma(n/d)*d^3.

1, 11, 31, 95, 131, 341, 351, 775, 850, 1441, 1343, 2945, 2211, 3861, 4061, 6231, 4931, 9350, 6879, 12445, 10881, 14773, 12191, 24025, 16406, 24321, 22990, 33345, 24419, 44671, 29823, 49911, 41633, 54241, 45981, 80750, 50691, 75669, 68541, 101525, 68963, 119691, 79551, 127585, 111350

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

Links

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

Crossrefs

Cf. A001001 (Dirichlet convolution of sigma and n^2), A275585.

Programs

Mathematica

a[n_] := DivisorSum[n, DivisorSigma[1, n/#]*#^3&]; Array[a, 45] (* Jean-François Alcover, Dec 07 2015 *)
f[p_, e_] := (p^(3 e + 5) - (p^2 + p + 1)*p^(e + 1) + p + 1)/((p^3 - 1)*(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Aug 27 2023 *)

PARI

N=66; x='x+O('x^N);
c=sum(j=1,N,j*x^j);
t=log(1/prod(j=1,N, eta(x^(j))^(j^2)));
Vec(serconvol(t,c)) \\ Joerg Arndt, May 03 2008

PARI

a(n) = sumdiv(n, d, sigma(n/d)*d^3); \\ Michel Marcus, Feb 24 2015

Formula

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-3). - Corrected by Álvar Ibeas, Jan 31 2015

Multiplicative with a(p^e) = (p^(3e+5) - (p^2+p+1)*p^(e+1) + p + 1) / ((p^3-1)*(p^2-1)). - Mitch Harris, Jun 27 2005

L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma_2(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

Sum_{k=1..n} a(k) ~ Pi^4 * zeta(3) * n^4 / 360. - Vaclav Kotesovec, Jan 31 2019