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%I #17 Jun 06 2023 06:26:45
%S 1,-1,-1,-1,-1,1,-1,1,-1,1,-1,1,-1,1,1,0,-1,1,-1,1,1,1,-1,-1,-1,1,1,1,
%T -1,-1,-1,0,1,1,1,1,-1,1,1,-1,-1,-1,-1,1,1,1,-1,0,-1,1,1,1,-1,-1,1,-1,
%U 1,1,-1,-1,-1,1,1,0,1,-1,-1,1,1,-1,-1,-1,-1,1,1
%N Dirichlet g.f.: 1 / (zeta(s) * zeta(2*s)).
%C Dirichlet convolution of A008966 and A007427.
%C Dirichlet convolution of A008683 and A271102.
%H Vaclav Kotesovec, <a href="/A307445/b307445.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletGeneratingFunction.html">Dirichlet Generating Function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_series">Dirichlet series</a>.
%F Multiplicative with a(p) = a(p^2) = -1, a(p^3) = 1, and a(p^e) = 0 for e >= 4. - _Amiram Eldar_, Dec 25 2022
%t nmax = 100; A271102 = Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, MoebiusMu[n/#]*A271102[[#]] &], {n, 1, nmax}]
%t nmax = 100; A007427 = Table[DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#] &], {n, 1, nmax}]; Table[DivisorSum[n, Abs[MoebiusMu[n/#]]*A007427[[#]] &], {n, 1, nmax}]
%t f[p_, e_] := 0; f[p_, 1] = f[p_, 2] = -1; f[p_, 3] = 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 25 2022 *)
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X^2)*(1-X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%Y Cf. A007427, A008683, A008966, A271102.
%K sign,mult
%O 1
%A _Vaclav Kotesovec_, Apr 08 2019
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