A344434 - OEIS

%I #19 Jul 25 2022 10:42:16

%S 1,6,29,279,3127,47484,823545,16843288,387440202,10009769782,

%T 285311670613,8918294591103,302875106592255,11112685049470800,

%U 437893920912789563,18447025552998138393,827240261886336764179,39346558271492566413252,1978419655660313589123981

%N 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.

%H Seiichi Manyama, <a href="/A344434/b344434.txt">Table of n, a(n) for n = 1..386</a>

%F 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.

%F G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k). - _Seiichi Manyama_, Jul 25 2022

%e 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.

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

%o (PARI) a(n) = sumdiv(n, d, sigma(d, d)); \\ _Michel Marcus_, May 19 2021

%o (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

%Y Cf. A245466, A321141, A334874, A343781.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, May 19 2021