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A307428 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(3*s)).

%I A307428 #18 Feb 16 2025 08:33:55
%S A307428 1,-1,-1,1,-1,1,-1,-2,1,1,-1,-1,-1,1,1,2,-1,-1,-1,-1,1,1,-1,2,1,1,-2,
%T A307428 -1,-1,-1,-1,-2,1,1,1,1,-1,1,1,2,-1,-1,-1,-1,-1,1,-1,-2,1,-1,1,-1,-1,
%U A307428 2,1,2,1,1,-1,1,-1,1,-1,2,1,-1,-1,-1,1,-1,-1,-2,-1
%N A307428 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(3*s)).
%C A307428 Dirichlet convolution of A307423 and A008683.
%H A307428 Vaclav Kotesovec, <a href="/A307428/b307428.txt">Table of n, a(n) for n = 1..10000</a>
%H A307428 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletGeneratingFunction.html">Dirichlet Generating Function</a>.
%H A307428 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_series">Dirichlet series</a>.
%F A307428 Multiplicative with a(p) = -1, a(p^2) = 1, and a(p^e) = 2*(-1)^e for e >= 3. - _Amiram Eldar_, Dec 25 2022
%t A307428 nmax = 100; A307423 = Table[DivisorSum[n, Boole[Max[FactorInteger[#][[All, 2]]] < 3] * LiouvilleLambda[n/#]&], {n, 1, nmax}]; Table[DivisorSum[n, MoebiusMu[#] * A307423[[n/#]] &], {n, 1, nmax}]
%t A307428 f[p_, e_] := 2*(-1)^e; f[p_, 1] := -1; f[p_, 2] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 25 2022 *)
%o A307428 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X^3)/(1+X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%Y A307428 Cf. A008683, A010052, A056624, A307423.
%K A307428 sign,mult
%O A307428 1,8
%A A307428 _Vaclav Kotesovec_, Apr 08 2019