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A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

%I A344787 #13 Dec 16 2022 08:59:27
%S A344787 1,7,31,287,3131,47527,823551,16843583,387440266,10009772937,
%T A344787 285311670623,8918294639219,302875106592267,11112685050294387,
%U A344787 437893920912795941,18447025553014982271,827240261886336764195,39346558271492953948522,1978419655660313589123999
%N A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.
%C A344787 If p is prime, a(p) = p * Sum_{d|p} sigma_d(d) / d = p * (1 + (1^p + p^p)/p) = 1 + p + p^p.
%H A344787 Seiichi Manyama, <a href="/A344787/b344787.txt">Table of n, a(n) for n = 1..386</a>
%F A344787 G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k)^2. - _Seiichi Manyama_, Dec 16 2022
%e A344787 a(4) = 4 * Sum_{d|4} sigma_d(d) / d = 4 * ((1^1)/1 + (1^2 + 2^2)/2 + (1^4 + 2^4 + 4^4)/4) = 287.
%t A344787 Table[n*Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k])/k, {k, n}], {n, 20}]
%o A344787 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k)^2)) \\ _Seiichi Manyama_, Dec 16 2022
%Y A344787 Cf. A245466, A321141, A334874, A343781, A344434, A344480.
%Y A344787 Cf. A001001, A007429, A027847, A027848, A060640.
%K A344787 nonn
%O A344787 1,2
%A A344787 _Wesley Ivan Hurt_, May 28 2021