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%I #12 Sep 20 2019 21:12:07
%S 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,
%T -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,
%U 1,5,-4,3,1,1,-1,3,1,1,-5,1,1,5,-1,3,1,1,-5
%N Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).
%C Number of odd proper divisors of n minus number of even proper divisors of n.
%H Antti Karttunen, <a href="/A325937/b325937.txt">Table of n, a(n) for n = 1..65537</a>
%F G.f.: Sum_{k>=2} x^k / (1 + x^k).
%F a(n) = -Sum_{d|n, d<n} (-1)^d.
%F a(n) = A048272(n) + (-1)^n.
%t nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t Table[-DivisorSum[n, (-1)^# &, # < n &], {n, 1, 80}]
%o (PARI) A325937(n) = -sumdiv(n, d, if(d==n,0,((-1)^d))); \\ _Antti Karttunen_, Sep 20 2019
%Y Cf. A032741, A048272, A058344, A091954, A275495 (partial sums), A325939.
%K sign,look
%O 1,9
%A _Ilya Gutkovskiy_, Sep 09 2019