A305006 Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).

1, -1, 4, -5, 6, -2, 8, -13, 13, -3, 12, -5, 14, -4, 8, -29, 18, -13, 20, -3, 32, -6, 24, -13, 31, -7, 40, -10, 30, -4, 32, -61, 16, -9, 48, -65, 38, -10, 56, -39, 42, -16, 44, -15, 26, -12, 48, -29, 57, -31, 24, -35, 54, -20, 72, -13, 80, -15, 60, -2, 62, -16, 104, -125, 84

Offset

1,3

Links

Table of n, a(n) for n=1..65.

Formula

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))).

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

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

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

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

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

Example

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, ...

Mathematica

nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[x^k/(k (1 + x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Log[Product[(1 - x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Log[EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))], {x, 0, nmax}], x]]]
Numerator[Table[Sum[(-1)^(n/d + 1) 1/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[DivisorSum[n, -(-1)^# # &]/n, {n, 65}]]

Prog

(PARI) a(n) = numerator(sumdiv(n, d, (-1)^(d+1)*d/n)); \\ Michel Marcus, May 24 2018
(Magma) [Numerator(&+[(-1)^(d+1)*d/n: d in Divisors(n)]): n in [1..100]]; // Vincenzo Librandi, May 24 2018

Crossrefs

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

Sequence in context: A106592 A106593 A276036 * A338688 A010665 A346832

Adjacent sequences: A305003 A305004 A305005 * A305007 A305008 A305009

Keyword

sign,frac

Author

Ilya Gutkovskiy, May 23 2018

Status

approved