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

2,2

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.

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)

Mathematica

a[n_] := ContinuedFraction[ Zeta[n], 2] // Last; Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Feb 26 2013 *)

Prog

(Maxima) A013697(n):=floor(1/(zeta(n)-1))$
makelist(A013697(n), n, 2, 30); /* Martin Ettl, Nov 03 2012 */
(Python)
from sympy import zeta
print([1//(zeta(n) - 1) for n in range(2, 32)]) # Karl V. Keller, Jr., Jul 21 2020

Crossrefs

Bisections: A190297, A190584.

Sequence in context: A007009 A188814 A104384 * A306055 A212522 A207408

Adjacent sequences: A013694 A013695 A013696 * A013698 A013699 A013700

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from Vladeta Jovovic, Apr 22 2001

Status

approved