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a(n) = Sum_{d|n} sigma(d)^(n/d).
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%I #12 May 08 2021 23:07:02

%S 1,4,5,17,7,56,9,146,78,298,13,1501,15,2276,1265,9219,19,25716,21,

%T 77519,16929,177328,25,739582,7808,1594562,264382,5611241,31,15699452,

%U 33,48863172,4196081,129140542,312753,447589422,39,1162261928,67111665,3771805472,43,10764897556,45

%N a(n) = Sum_{d|n} sigma(d)^(n/d).

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

%F If p is prime, a(p) = 2 + p.

%t a[n_] := DivisorSum[n, DivisorSigma[1 , #]^(n/#) &]; Array[a, 43] (* _Amiram Eldar_, May 08 2021 *)

%o (PARI) a(n) = sumdiv(n, d, sigma(d)^(n/d));

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

%Y Cf. A007429, A055225, A279789, A309369, A342471, A344060.

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 08 2021