This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A278425 #31 Nov 30 2016 05:34:57 %S A278425 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 A278425 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 A278425 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 A278425 Largest k such that there are no primes between kn and k(n+1); -1 if no such k exists. %C A278425 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 A278425 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 A278425 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 A278425 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bertrand's_postulate">Bertrand's Postulate</a> %H A278425 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 A278425 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 A278425 Bertrand's postulate shows that for k>1 there is always a prime between k and 2k. Hence a(1) = 1. %e A278425 In 2006, M. El Bachraoui showed that for k>1 there is always a prime between 2k and 3k. Hence a(2) = 1. %e A278425 In 2011, Andy Loo showed that for k>1 there is always a prime between 3k and 4k. Hence a(3) = 1. %Y A278425 Cf. A060715. %K A278425 nonn %O A278425 1,4 %A A278425 _Dmitry Kamenetsky_, Nov 28 2016