A308688 - OEIS

%I #22 May 09 2021 02:51:20

%S 1,3,4,13,6,66,8,201,253,648,12,5488,14,8550,22824,49681,18,316743,20,

%T 865578,1611152,2098506,24,27246276,1953151,33556656,129199240,

%U 202152908,30,1758141606,32,3223326753,10460514288,8589939540,1261056768,146050621105,38

%N a(n) = Sum_{d|n} d^(2*n/d - 1).

%H Seiichi Manyama, <a href="/A308688/b308688.txt">Table of n, a(n) for n = 1..1000</a>

%F L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)^(1/k^2)) = Sum_{k>=1} a(k)*x^k/k.

%F a(p) = p+1 for prime p.

%F G.f.: Sum_{k>=1} k*x^k/(1 - k^2*x^k). - _Ilya Gutkovskiy_, Jul 25 2019

%t a[n_] := DivisorSum[n, #^(2*n/# - 1) &]; Array[a, 37] (* _Amiram Eldar_, May 09 2021 *)

%o (PARI) {a(n) = sumdiv(n, d, d^(2*n/d-1))}

%o (PARI) N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k^2*x^k)^(1/k^2)))))

%Y Column k=2 of A308690.

%Y Cf. A087909, A308692.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jun 17 2019