A013697 Second term in continued fraction for zeta(n).
1, 4, 12, 27, 57, 119, 245, 497, 1005, 2023, 4063, 8149, 16327, 32692, 65435, 130938, 261965, 524050, 1048259, 2096730, 4193742, 8387859, 16776218, 33553102, 67107091, 134215364, 268432305, 536866711, 1073736223, 2147476180, 4294957340, 8589921317, 17179851485
Offset: 2
Links
- Alois P. Heinz, Table of n, a(n) for n = 2..1000 (terms n = 2..100 from Vincenzo Librandi)
- Tal Barnea, On the Riemann Zeta Function and the fractional part of rational powers, arXiv:1808.06653 [math.NT], 2018.
- Tal Barnea, The Riemann Zeta Function and the Fractional Part of Rational Powers, J. Int. Seq., Vol. 22 (2019), Article 19.3.6.
Programs
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Mathematica
a[n_] := ContinuedFraction[ Zeta[n], 2] // Last; Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Feb 26 2013 *) -
Maxima
A013697(n):=floor(1/(zeta(n)-1))$ makelist(A013697(n),n,2,30); /* Martin Ettl, Nov 03 2012 */
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Python
from sympy import zeta print([1//(zeta(n) - 1) for n in range(2, 32)]) # Karl V. Keller, Jr., Jul 21 2020
Formula
From Franklin T. Adams-Watters, Mar 23 2010: (Start)
a(n) = floor(1/(zeta(n)-1)).
a(n) = 2^n - (4/3)^n + O(1). It appears that a(n) = 2^n - floor((4/3)^n) - k, where k is usually 2 but is sometimes 1. Up to n=1000, the only values of n where k = 1 are 4, 5, 13, 14, and 17. (End)
Extensions
More terms from Vladeta Jovovic, Apr 22 2001