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a(n) = Sum_{d|n} d^(tau(d) - 1).
1

%I #12 Oct 15 2021 08:38:24

%S 1,3,4,19,6,222,8,531,85,1008,12,249070,14,2754,3384,66067,18,1889871,

%T 20,3201024,9272,10662,24,4586721006,631,17592,19768,17213138,30,

%U 21870004602,32,33620499,35952,39324,42888,2821112046175,38,54894,59336,163843201536

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

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

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

%t a[n_] := DivisorSum[n, #^(DivisorSigma[0, #] - 1) &]; Array[a, 40] (* _Amiram Eldar_, Oct 14 2021 *)

%o (PARI) a(n) = sumdiv(n, d, d^(numdiv(d)-1));

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

%Y Cf. A000005 (tau), A174937, A347405, A347992, A348350.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Oct 14 2021