OFFSET
1,4
COMMENTS
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.
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.
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
LINKS
Wikipedia, Bertrand's Postulate
M. El Bachraoui, Primes in the interval [2n,3n], International Journal of Contemporary Mathematical Sciences, volume 1, number 13, pages 617-621, 2006.
Andy Loo, On the primes in the interval [3n,4n], International Journal of Contemporary Mathematical Sciences, volume 6, number 38, pages 1871-1882, 2011.
EXAMPLE
Bertrand's postulate shows that for k>1 there is always a prime between k and 2k. Hence a(1) = 1.
In 2006, M. El Bachraoui showed that for k>1 there is always a prime between 2k and 3k. Hence a(2) = 1.
In 2011, Andy Loo showed that for k>1 there is always a prime between 3k and 4k. Hence a(3) = 1.
CROSSREFS
KEYWORD
nonn
AUTHOR
Dmitry Kamenetsky, Nov 28 2016
STATUS
approved