A325939 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k).

0, 1, -1, 2, -1, 1, -1, 3, -2, 1, -1, 3, -1, 1, -3, 4, -1, 1, -1, 3, -3, 1, -1, 5, -2, 1, -3, 3, -1, 1, -1, 5, -3, 1, -3, 4, -1, 1, -3, 5, -1, 1, -1, 3, -5, 1, -1, 7, -2, 1, -3, 3, -1, 1, -3, 5, -3, 1, -1, 5, -1, 1, -5, 6, -3, 1, -1, 3, -3, 1, -1, 7, -1, 1, -5, 3, -3, 1, -1, 7

Offset: 1

Author

Ilya Gutkovskiy, Sep 09 2019

Keywords

• sign
• look

Comments

Number of even divisors of n minus number of odd strong divisors of n (i.e. odd divisors > 1).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A032741, A048272, A075997 (partial sums), A325937.

Programs

Mathematica

nmax = 80; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^# &, # > 1 &], {n, 1, 80}]

PARI

A325939(n) = sumdiv(n, d, if(1==d,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

Formula

G.f.: Sum_{k>=2} (-1)^k * x^k / (1 - x^k).

a(n) = Sum_{d|n, d>1} (-1)^d.

a(n) = 1 - A048272(n).