A344434 a(n) = Sum_{d|n} sigma_d(d), where sigma_k(n) is the sum of the k-th powers of the divisors of n.

1, 6, 29, 279, 3127, 47484, 823545, 16843288, 387440202, 10009769782, 285311670613, 8918294591103, 302875106592255, 11112685049470800, 437893920912789563, 18447025552998138393, 827240261886336764179, 39346558271492566413252, 1978419655660313589123981

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..386

Formula

If p is prime, a(p) = Sum_{d|p} sigma_d(d) = sigma_1(1) + sigma_p(p) = 1^1 + (1^p + p^p) = p^p + 2.

G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 25 2022

Example

a(6) = Sum_{d|6} sigma_d(d) = (1^1) + (1^2 + 2^2) + (1^3 + 3^3) + (1^6 + 2^6 + 3^6 + 6^6) = 47484.

Mathematica

Table[Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 20}]

Prog

(PARI) a(n) = sumdiv(n, d, sigma(d, d)); \\ Michel Marcus, May 19 2021
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k))) \\ Seiichi Manyama, Jul 25 2022

Crossrefs

Cf. A245466, A321141, A334874, A343781.

Sequence in context: A205811 A143563 A266205 * A321141 A359053 A122802

Adjacent sequences: A344431 A344432 A344433 * A344435 A344436 A344437

Keyword

nonn

Author

Wesley Ivan Hurt, May 19 2021

Status

approved