A344044 a(n) = Sum_{d|n} sigma(d)^3.

1, 28, 65, 371, 217, 1820, 513, 3746, 2262, 6076, 1729, 24115, 2745, 14364, 14105, 33537, 5833, 63336, 8001, 80507, 33345, 48412, 13825, 243490, 30008, 76860, 66262, 190323, 27001, 394940, 32769, 283584, 112385, 163324, 111321, 839202, 54873, 224028, 178425, 812882, 74089

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k >= 1} sigma(k)^3 * x^k/(1 - x^k).

If p is prime, a(p) = 1 + (p+1)^3.

Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^10*zeta(3)/194400) * Product_{p prime} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 1.6422194986... . - Amiram Eldar, Nov 20 2022

Mathematica

a[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Array[a, 41] (* Amiram Eldar, May 08 2021 *)

Prog

(PARI) a(n) = sumdiv(n, d, sigma(d)^3);
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^3*x^k/(1-x^k)))

Crossrefs

Cf. A002117, A007429, A065018, A344043, A344047.

Sequence in context: A044130 A044511 A290543 * A183230 A246911 A323918

Adjacent sequences: A344041 A344042 A344043 * A344045 A344046 A344047

Keyword

nonn,mult

Author

Seiichi Manyama, May 08 2021

Status

approved