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A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).

%I A326417 #16 Dec 02 2020 03:15:12
%S A326417 1,3,4,6,4,12,4,10,10,12,4,24,4,12,16,15,4,30,4,24,16,12,4,40,10,12,
%T A326417 20,24,4,48,4,21,16,12,16,60,4,12,16,40,4,48,4,24,40,12,4,60,10,30,16,
%U A326417 24,4,60,16,40,16,12,4,96,4,12,40,28,16,48,4,24,16,48,4,100,4,12,40
%N A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).
%C A326417 Inverse Moebius transform applied twice to A001227.
%H A326417 Amiram Eldar, <a href="/A326417/b326417.txt">Table of n, a(n) for n = 1..10000</a>
%F A326417 G.f.: Sum_{k>=1} tau_3(k) * x^k / (1 - x^(2*k)), where tau_3 = A007425.
%F A326417 a(n) = tau_4(n) if n odd, tau_4(n) - tau_4(n/2) if n even, where tau_4 = A007426.
%F A326417 a(n) = Sum_{d|n, n/d odd} tau_3(d).
%F A326417 a(n) = Sum_{d|n} A000005(n/d) * A001227(d).
%F A326417 Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A280486.
%F A326417 Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (e+1)*(e+2)*(e+3)/6 for odd primes p. - _Amiram Eldar_, Dec 02 2020
%t A326417 Table[Sum[DivisorSigma[0, n/d] Total[Mod[Divisors[d], 2]], {d, Divisors[n]}], {n, 1, 75}]
%t A326417 nmax = 75; A007425 = Table[DivisorSum[n, DivisorSigma[0, #] &], {n, 1, nmax}]; Table[DivisorSum[n, A007425[[#]] &, OddQ[n/#] &], {n, 1, nmax}]
%t A326417 f[2, e_] := (e + 1)*(e + 2)/2; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 02 2020 *)
%Y A326417 Cf. A000005, A001227, A007425, A007426, A133700, A280486, A318845.
%K A326417 nonn,mult
%O A326417 1,2
%A A326417 _Ilya Gutkovskiy_, Oct 18 2019