A294645 - OEIS

%I #39 Feb 16 2025 08:33:51

%S 1,9,82,1057,15626,282252,5764802,134480385,3486843451,100048830174,

%T 3138428376722,107006334784468,3937376385699290,155572843119354936,

%U 6568408508343827972,295150156996346511361,14063084452067724991010,708236696816416252145973

%N a(n) = Sum_{d|n} d^(n+1).

%H Seiichi Manyama, <a href="/A294645/b294645.txt">Table of n, a(n) for n = 1..385</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>

%F G.f.: Sum_{k>0} k^(k+1)*x^k/(1-(k*x)^k).

%F L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)) = Sum_{k>=1} a(k)*x^k/k. - _Seiichi Manyama_, Jun 02 2019

%F a(n) ~ n^(n+1). - _Vaclav Kotesovec_, Oct 07 2020

%t Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* _Vaclav Kotesovec_, Oct 07 2020 *)

%o (PARI) {a(n) = sigma(n, n+1)}

%o (PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, k^(k+1)*x^k/(1-(k*x)^k)))

%o (PARI) N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ _Seiichi Manyama_, Jun 02 2019

%Y Column k=1 of A308504.

%Y Cf. A023882, A023887, A082245, A158095, A292312.

%K nonn,changed

%O 1,2

%A _Seiichi Manyama_, Nov 05 2017