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A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

%I A344081 #14 May 11 2021 10:32:35
%S A344081 1,5,9,86,33,4109,129,65622,19692,1048613,2049,2176786526,8193,
%T A344081 268435589,1073741865,152587956247,131073,101559956692208,524289,
%U A344081 3656158441111670,4398046511241,17592186046469,8388609,4722366482871822065758
%N A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.
%H A344081 Seiichi Manyama, <a href="/A344081/b344081.txt">Table of n, a(n) for n = 1..719</a>
%F A344081 G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - x^k).
%F A344081 If p is prime, a(p) = 1 + 2^p.
%t A344081 a[n_] := DivisorSum[n, DivisorSigma[0, #]^# &]; Array[a, 24] (* _Amiram Eldar_, May 09 2021 *)
%o A344081 (PARI) a(n) = sumdiv(n, d, numdiv(d)^d);
%o A344081 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-x^k)))
%Y A344081 Cf. A007425, A062367, A097988, A279789, A344047, A344080.
%K A344081 nonn
%O A344081 1,2
%A A344081 _Seiichi Manyama_, May 09 2021