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

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, -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, 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

Offset: 1

Author

Ilya Gutkovskiy, Jul 30 2018

Keywords

• sign
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A005117, A008683, A008966, A034444, A048272, A100007, A300894, A305614.

Programs

Maple

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

Mathematica

nmax = 95; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
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]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, SquareFreeQ[#] &], {n, 95}]
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 *)

PARI

A317528(n) = sumdiv(n,d,((-1)^(1+d))*issquarefree(n/d)); \\ Antti Karttunen, Dec 05 2021

Formula

G.f.: Sum_{k>=1} x^A005117(k)/(1 + x^A005117(k)).

L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.

a(n) = Sum_{d|n} (-1)^(n/d+1)*A008966(d).

If n is odd, a(n) = A034444(n).

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