A094500 Least number k such that (n+1)^k / n^k >= 2.

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47, 48, 49, 49, 50, 51, 51

Offset

1,2

Comments

This sequence also describes the minimum number of (n+1)-player games, where each player has an equal chance of winning, that must be played for a given player to have at least a 50% chance of winning at least once. E.g., a(3) = 3 because in a 4-player random game, a given player will have a greater than 50% chance of winning at least once if 3 games are played. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 28 2006

Also, a(n) denotes a median m of the geometric random variable on the positive integers with mean value n+1. The median is obtained by solving 1-(n/n+1)^m >= 1/2 for least integer m. - Dennis P. Walsh, Aug 13 2012

The limit n -> inf. a(n)/n = log 2. - Robert G. Wilson v, May 13 2014

Links

Table of n, a(n) for n=1..73.

Jon Eivind Vatne, The sequence of middle divisors is unbounded, Journal of Number Theory, Volume 172, March 2017, Pages 413-415. See n(i) p. 414.

Formula

a(n) = n*log(2) + O(1). - Charles R Greathouse IV, Sep 02 2015

Example

a(3) = 3 because (4/3)^2 < 2 and (4/3)^3 > 2.

Mathematica

f[n_] := Block[{k = 1}, While[((n + 1)/n)^k < 2, k++]; k]; Array[f, 75]
(* to view the limit *) Array[ f/# &, 1000] (* Robert G. Wilson v, May 13 2014 *)

Prog

(PARI) a(n)=ceil(log(2)/log(1+1/n)) \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Cf. A002379, A094969-A094999.

Sequence in context: A039708 A189730 A249569 * A049473 A154951 A095769

Adjacent sequences: A094497 A094498 A094499 * A094501 A094502 A094503

Keyword

easy,nonn

Author

Robert G. Wilson v, May 26 2004

Extensions

Edited by Jon E. Schoenfield, Apr 26 2014

Status

approved