A305007 Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 18, 19, 2, 21, 11, 23, 6, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 36, 37, 19, 39, 20, 41, 21, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 27, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35, 71, 72, 73, 37, 75

Offset

1,2

Links

Table of n, a(n) for n=1..75.

Formula

Denominators 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))).

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

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

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

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

a(p^k) = p^k where p is a 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 = 75; Rest[Denominator[CoefficientList[Series[Sum[x^k/(k (1 + x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Log[Product[(1 - x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Log[EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))], {x, 0, nmax}], x]]]
Denominator[Table[Sum[(-1)^(n/d + 1) 1/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[DivisorSum[n, -(-1)^# # &]/n, {n, 75}]]

Prog

(PARI) a(n) = denominator(sumdiv(n, d, (-1)^(d+1)*d/n)); \\ Michel Marcus, May 24 2018
(Magma) [Denominator(&+[(-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, A305006 (numerators).

Sequence in context: A294650 A323129 A053585 * A353667 A098988 A274346

Adjacent sequences: A305004 A305005 A305006 * A305008 A305009 A305010

Keyword

nonn,frac

Author

Ilya Gutkovskiy, May 23 2018

Status

approved