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A347405 a(n) = Sum_{d|n} 2^(tau(d) - 1).

%I A347405 #36 Oct 14 2021 08:48:18
%S A347405 1,3,3,7,3,13,3,15,7,13,3,49,3,13,13,31,3,49,3,49,13,13,3,185,7,13,15,
%T A347405 49,3,159,3,63,13,13,13,341,3,13,13,185,3,159,3,49,49,13,3,713,7,49,
%U A347405 13,49,3,185,13,185,13,13,3,2275,3,13,49,127,13,159,3,49,13,159,3,2525,3,13,49,49
%N A347405 a(n) = Sum_{d|n} 2^(tau(d) - 1).
%H A347405 Seiichi Manyama, <a href="/A347405/b347405.txt">Table of n, a(n) for n = 1..10000</a>
%F A347405 If p is prime, a(p^n) = 2^(n+1) - 1.
%F A347405 G.f.: Sum_{k>=1} 2^(tau(k) - 1) * x^k/(1 - x^k).
%t A347405 a[n_] := DivisorSum[n, 2^(DivisorSigma[0, #] - 1) &]; Array[a, 80] (* _Amiram Eldar_, Oct 08 2021 *)
%o A347405 (PARI) a(n) = sumdiv(n, d, 2^(numdiv(d)-1));
%o A347405 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 2^(numdiv(k)-1)*x^k/(1-x^k)))
%Y A347405 Cf. A000005 (tau), A000225, A347991, A347992.
%K A347405 nonn
%O A347405 1,2
%A A347405 _Seiichi Manyama_, Oct 08 2021