A317531 - OEIS

%I #11 Oct 05 2018 11:21:32

%S 0,1,1,0,1,0,1,-1,2,0,1,-1,1,0,2,-2,1,-1,1,-1,2,0,1,-2,2,0,3,-1,1,-1,

%T 1,-3,2,0,2,-2,1,0,2,-2,1,-1,1,-1,3,0,1,-3,2,-1,2,-1,1,-2,2,-2,2,0,1,

%U -2,1,0,3,-4,2,-1,1,-1,2,-1,1,-3,1,0,3,-1,2,-1,1,-3,4,0,1,-2,2,0,2,-2,1,-2,2,-1,2,0,2

%N Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)).

%H Antti Karttunen, <a href="/A317531/b317531.txt">Table of n, a(n) for n = 1..65537</a>

%F G.f.: Sum_{k>=1} x^A246655(k)/(1 + x^A246655(k)).

%F L.g.f.: log(Product_{p prime, k>=1} (1 + x^(p^k))^(1/p^k)) = Sum_{n>=1} a(n)*x^n/n.

%F a(n) = Sum_{d|n} (-1)^(n/d+1)*A069513(d).

%F If n is odd, a(n) = A001222(n).

%t nmax = 95; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

%t nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + Boole[PrimePowerQ[k]] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

%t Table[DivisorSum[n, (-1)^(n/# + 1) &, PrimePowerQ[#] &], {n, 95}]

%o (PARI) A317531(n) = sumdiv(n,d,((-1)^(n/d+1))*(1==omega(d))); \\ _Antti Karttunen_, Sep 30 2018

%Y Cf. A001222, A048272, A069513, A246655, A288571, A305614, A317528.

%K sign

%O 1,9

%A _Ilya Gutkovskiy_, Jul 30 2018