A325937 Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).

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

Offset: 1

Author

Ilya Gutkovskiy, Sep 09 2019

Keywords

• sign
• look

Comments

Number of odd proper divisors of n minus number of even proper divisors of n.

Links

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

Crossrefs

Cf. A032741, A048272, A058344, A091954, A275495 (partial sums), A325939.

Programs

Mathematica

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

PARI

A325937(n) = -sumdiv(n, d, if(d==n,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

Formula

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

a(n) = -Sum_{d|n, d

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

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