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Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).
1

%I #12 Sep 08 2022 08:46:21

%S 1,-1,4,-5,6,-2,8,-13,13,-3,12,-5,14,-4,8,-29,18,-13,20,-3,32,-6,24,

%T -13,31,-7,40,-10,30,-4,32,-61,16,-9,48,-65,38,-10,56,-39,42,-16,44,

%U -15,26,-12,48,-29,57,-31,24,-35,54,-20,72,-13,80,-15,60,-2,62,-16,104,-125,84

%N Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).

%F Numerators of coefficients in expansion of log(Sum_{k>=0} x^(k*(k+1)/2)) = log(Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1))).

%F Numerators of coefficients in expansion of log(theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.

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

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

%F a(n) = numerator of A002129(n)/n.

%F a(p) = p + 1 where p is an odd prime.

%e 1, -1/2, 4/3, -5/4, 6/5, -2/3, 8/7, -13/8, 13/9, -3/5, 12/11, -5/3, 14/13, -4/7, 8/5, -29/16, 18/17, -13/18, 20/19, ...

%t nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[x^k/(k (1 + x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]

%t nmax = 65; Rest[Numerator[CoefficientList[Series[Log[Product[(1 - x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x]]]

%t nmax = 65; Rest[Numerator[CoefficientList[Series[Log[EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))], {x, 0, nmax}], x]]]

%t Numerator[Table[Sum[(-1)^(n/d + 1) 1/d, {d, Divisors[n]}], {n, 65}]]

%t Numerator[Table[DivisorSum[n, -(-1)^# # &]/n, {n, 65}]]

%o (PARI) a(n) = numerator(sumdiv(n, d, (-1)^(d+1)*d/n)); \\ _Michel Marcus_, May 24 2018

%o (Magma) [Numerator(&+[(-1)^(d+1)*d/n: d in Divisors(n)]): n in [1..100]]; // _Vincenzo Librandi_, May 24 2018

%Y Cf. A002129, A010054, A017665, A017666, A098987, A098988, A305007 (denominators).

%K sign,frac

%O 1,3

%A _Ilya Gutkovskiy_, May 23 2018