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 A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)). 4
 1, -1, 1, 0, 1, -1, 1, -2, 2, -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -1, 1, -2, 2, -1, 2, 0, 1, -1, 1, -3, 1, -1, 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 2, -1, 1, -1, 2, -2, 1, 0, 1, -2, 1, -2, 1, -1, 1, 0, 1, -1, 2, -2, 1, -1, 1, 0, 1, -1, 1, -4, 1, -1, 2, 0, 1, -1, 1, -1, 3, -1, 1, 0, 1, -1, 1, -2, 1, -2, 1, 0, 1, -1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 LINKS Antti Karttunen, Table of n, a(n) for n = 1..11025 Antti Karttunen, Data supplement: n, a(n) computed for n =  1..100000 FORMULA G.f.: Sum_{k>=1} x^A000290(k)/(1 + x^A000290(k)). L.g.f.: log(Product_{k>=1} (1 + x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. a(n) = Sum_{d|n} (-1)^(n/d+1)*A010052(d). If n is odd, a(n) = A046951(n). Multiplicative with a(2^e) = -floor(e/2+1) for odd e, -floor((e-1)/2) for even e, and a(p^e) = floor(e/2+1) for an odd prime p. - Amiram Eldar, Oct 25 2020 MAPLE seq(coeff(series(add(x^(k^2)/(1+x^(k^2)), k=1..n), x, n+1), x, n), n=1..100); # Muniru A Asiru, Jul 30 2018 MATHEMATICA nmax = 95; Rest[CoefficientList[Series[Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]] nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]] Table[DivisorSum[n, (-1)^(n/# + 1) &, IntegerQ[#^(1/2)] &], {n, 95}] f[p_, e_] := If[p == 2, If[OddQ[e], -Floor[e/2 + 1], -Floor[(e - 1)/2]], Floor[e/2 + 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *) PROG (PARI) A317529(n) = sumdiv(n, d, ((-1)^(1+(n/d)))*issquare(d)); \\ Antti Karttunen, Nov 07 2018 CROSSREFS Cf. A000290, A010052, A046951, A048272, A300853. Sequence in context: A326918 A328698 A359936 * A285194 A039978 A099918 Adjacent sequences: A317526 A317527 A317528 * A317530 A317531 A317532 KEYWORD sign,mult,easy AUTHOR Ilya Gutkovskiy, Jul 30 2018 STATUS approved

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Last modified December 1 14:09 EST 2023. Contains 367476 sequences. (Running on oeis4.)