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

Author

Wesley Ivan Hurt, May 19 2021

Keywords

• nonn

Comments

Inverse Möbius transform of sigma_n(n) (A023887). - Wesley Ivan Hurt, Mar 31 2025

Examples

			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.
		

Links

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

Crossrefs

Cf. A023887 (sigma_n(n)), A245466, A321141, A334874, A343781.

Programs

Mathematica

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

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

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