cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Ilya Gutkovskiy, May 23 2018

Keywords

Examples

			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, ...
		

Crossrefs

Programs

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

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.