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a(n) = Sum_{d|n} sigma(d)^d.
4

%I #12 May 08 2021 07:41:09

%S 1,10,65,2411,7777,2986058,2097153,2562893036,10604499438,

%T 3570467234410,743008370689,232218265092200875,793714773254145,

%U 21035720123170684938,504857282956046114465,727423121747187826721517,2185911559738696531969,43567528752021332763809905512,5242880000000000000000001

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

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

%F If p is prime, a(p) = 1 + (p+1)^p.

%t a[n_] := DivisorSum[n, DivisorSigma[1, #]^# &]; Array[a, 19] (* _Amiram Eldar_, May 08 2021 *)

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k)*x)^k/(1-x^k)))

%Y Cf. A065018, A342473, A344044.

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 08 2021