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A317528 Expansion of Sum_{k>=1} mu(k)^2*x^k/(1 + x^k), where mu() is the Moebius function (A008683).

%I A317528 #17 Nov 19 2022 04:46:33
%S A317528 1,0,2,-2,2,0,2,-2,2,0,2,-4,2,0,4,-2,2,0,2,-4,4,0,2,-4,2,0,2,-4,2,0,2,
%T A317528 -2,4,0,4,-4,2,0,4,-4,2,0,2,-4,4,0,2,-4,2,0,4,-4,2,0,4,-4,4,0,2,-8,2,
%U A317528 0,4,-2,4,0,2,-4,4,0,2,-4,2,0,4,-4,4,0,2,-4,2,0,2,-8,4,0,4,-4,2,0,4,-4,4,0,4
%N A317528 Expansion of Sum_{k>=1} mu(k)^2*x^k/(1 + x^k), where mu() is the Moebius function (A008683).
%H A317528 Antti Karttunen, <a href="/A317528/b317528.txt">Table of n, a(n) for n = 1..20000</a>
%F A317528 G.f.: Sum_{k>=1} x^A005117(k)/(1 + x^A005117(k)).
%F A317528 L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
%F A317528 a(n) = Sum_{d|n} (-1)^(n/d+1)*A008966(d).
%F A317528 If n is odd, a(n) = A034444(n).
%F A317528 Multiplicative with a(2) = 0, a(2^e) = -2 for e>1, and a(p^e) = 2 for p>2 and e>=1. - _Amiram Eldar_, Nov 19 2022
%p A317528 with(numtheory): seq(coeff(series(add(mobius(k)^2*x^k/(1+x^k),k=1..n), x,n+1),x,n),n=1..120); # _Muniru A Asiru_, Jul 30 2018
%t A317528 nmax = 95; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A317528 nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t A317528 Table[DivisorSum[n, (-1)^(n/# + 1) &, SquareFreeQ[#] &], {n, 95}]
%t A317528 f[p_, e_] := 2; f[2, e_] := If[e == 1, 0, -2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2022 *)
%o A317528 (PARI) A317528(n) = sumdiv(n,d,((-1)^(1+d))*issquarefree(n/d)); \\ _Antti Karttunen_, Dec 05 2021
%Y A317528 Cf. A005117, A008683, A008966, A034444, A048272, A100007, A300894, A305614.
%K A317528 sign,mult
%O A317528 1,3
%A A317528 _Ilya Gutkovskiy_, Jul 30 2018