A278425 - OEIS

%I #31 Nov 30 2016 05:34:57

%S 1,1,1,2,1,4,2,4,1,2,3,4,9,1,6,3,7,5,6,10,4,2,5,5,8,7,2,5,11,4,3,10,9,

%T 6,15,6,8,4,3,8,5,7,5,12,2,7,3,11,6,6,10,9,10,6,2,3,5,23,9,6,4,10,4,8,

%U 6,8,20,5,9,19,4,12,7,18,7,7,2,6,17,6,14,6,16,16,6,9,13,19,15,14,18,4,18,5,14,14,13,4,9,8

%N Largest k such that there are no primes between kn and k(n+1); -1 if no such k exists.

%C This sequence deals with the question of whether there is always a prime between nk and n(k+1). For n<=3 the answer has been proven to be yes (see links and examples). For n>3 the problem remains open, however we can conjecture the values of a(n) by checking the first few hundred k.

%C Conjecture: For every n, there exists a finite m such that for every k>m there is at least one prime between kn and k(n+1). In other words, a(n) is never -1.

%C Conjecture follows from the Prime Number Theorem: for fixed n, the number of primes between kn and k(n+1) is asymptotic to k/log(k) as k -> infinity, and in particular is nonzero for all sufficiently large k. - _Robert Israel_, Nov 28 2016

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bertrand&#39;s_postulate">Bertrand's Postulate</a>

%H M. El Bachraoui, <a href="http://qc.fengyuan.com/random/elbachraouiIJCMS13-16-2006.pdf">Primes in the interval [2n,3n]</a>, International Journal of Contemporary Mathematical Sciences, volume 1, number 13, pages 617-621, 2006.

%H Andy Loo, <a href="http://www.m-hikari.com/ijcms-2011/37-40-2011/looIJCMS37-40-2011.pdf">On the primes in the interval [3n,4n]</a>, International Journal of Contemporary Mathematical Sciences, volume 6, number 38, pages 1871-1882, 2011.

%e Bertrand's postulate shows that for k>1 there is always a prime between k and 2k. Hence a(1) = 1.

%e In 2006, M. El Bachraoui showed that for k>1 there is always a prime between 2k and 3k. Hence a(2) = 1.

%e In 2011, Andy Loo showed that for k>1 there is always a prime between 3k and 4k. Hence a(3) = 1.

%Y Cf. A060715.

%K nonn

%O 1,4

%A _Dmitry Kamenetsky_, Nov 28 2016