cp's OEIS Frontend

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

%I A094500 #30 Oct 01 2023 06:00:59
%S A094500 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,
%T A094500 20,21,22,22,23,24,24,25,26,26,27,28,29,29,30,31,31,32,33,33,34,35,36,
%U A094500 36,37,38,38,39,40,40,41,42,42,43,44,45,45,46,47,47,48,49,49,50,51,51
%N A094500 Least number k such that (n+1)^k / n^k >= 2.
%C A094500 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
%C A094500 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
%C A094500 The limit n -> inf. a(n)/n = log 2. - _Robert G. Wilson v_, May 13 2014
%H A094500 Jon Eivind Vatne, <a href="https://doi.org/10.1016/j.jnt.2016.08.015">The sequence of middle divisors is unbounded</a>, Journal of Number Theory, Volume 172, March 2017, Pages 413-415. See n(i) p. 414.
%F A094500 a(n) = n*log(2) + O(1). - _Charles R Greathouse IV_, Sep 02 2015
%e A094500 a(3) = 3 because (4/3)^2 < 2 and (4/3)^3 > 2.
%t A094500 f[n_] := Block[{k = 1}, While[((n + 1)/n)^k < 2, k++]; k]; Array[f, 75]
%t A094500 (* to view the limit *) Array[ f/# &, 1000] (* _Robert G. Wilson v_, May 13 2014 *)
%o A094500 (PARI) a(n)=ceil(log(2)/log(1+1/n)) \\ _Charles R Greathouse IV_, Sep 02 2015
%Y A094500 Cf. A002379, A094969-A094999.
%K A094500 easy,nonn
%O A094500 1,2
%A A094500 _Robert G. Wilson v_, May 26 2004
%E A094500 Edited by _Jon E. Schoenfield_, Apr 26 2014