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A307427 Dirichlet g.f.: zeta(3*s) / (zeta(s) * zeta(2*s)).

%I A307427 #18 Feb 16 2025 08:33:55
%S A307427 1,-1,-1,-1,-1,1,-1,2,-1,1,-1,1,-1,1,1,-1,-1,1,-1,1,1,1,-1,-2,-1,1,2,
%T A307427 1,-1,-1,-1,-1,1,1,1,1,-1,1,1,-2,-1,-1,-1,1,1,1,-1,1,-1,1,1,1,-1,-2,1,
%U A307427 -2,1,1,-1,-1,-1,1,1,2,1,-1,-1,1,1,-1,-1,-2,-1,1
%N A307427 Dirichlet g.f.: zeta(3*s) / (zeta(s) * zeta(2*s)).
%C A307427 Dirichlet convolution of A210826 and A271102.
%C A307427 Dirichlet convolution of A307424 and A008683.
%H A307427 Vaclav Kotesovec, <a href="/A307427/b307427.txt">Table of n, a(n) for n = 1..10000</a>
%H A307427 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletGeneratingFunction.html">Dirichlet Generating Function</a>.
%H A307427 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_series">Dirichlet series</a>.
%F A307427 Multiplicative with a(p^e) = 2 if e == 0 (mod 3), and -1 otherwise. - _Amiram Eldar_, Dec 25 2022
%t A307427 nmax = 100; A271102 = Table[DivisorSum[n, Abs[MoebiusMu[#]] * MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, Mod[DivisorSigma[0, n/#], 3, -1] * A271102[[#]] &], {n, 1, nmax}]
%t A307427 nmax = 100; A307424 = Table[DivisorSum[n, Abs[MoebiusMu[#]] * Mod[DivisorSigma[0, n/#], 3, -1]&], {n, 1, nmax}]; Table[DivisorSum[n, MoebiusMu[#] * A307424[[n/#]] &], {n, 1, nmax}]
%t A307427 f[p_, e_] := If[Divisible[e, 3], 2, -1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 25 2022 *)
%o A307427 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X)*(1-X^2)/(1-X^3))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%Y A307427 Cf. A008683, A010057, A210826, A271102, A307424.
%K A307427 sign,mult
%O A307427 1,8
%A A307427 _Vaclav Kotesovec_, Apr 08 2019