A124129 - OEIS

%I #19 Dec 10 2018 02:32:52

%S 3,7,13,23,31,113

%N Primes p for which there are no primes between p and p+sqrt(p).

%C Conjecture: there are no other terms.

%C The finiteness of this sequence would follow from Cramer's conjecture that lim sup (p(n+1)-p(n))/log(p(n))^2 = 1. - _Dean Hickerson_, Dec 12 2006

%C The finiteness of this sequence would imply that, for every sufficiently large positive integer n, there is a prime between n^2 and (n+1)^2. Except for the "sufficiently large", that's Legendre's conjecture, which is still unproved. - _Dean Hickerson_, Dec 12 2006

%C There are no other terms less than 218034721194214273 (assuming that the extended table of terms in A002386 is correct). - _Dean Hickerson_, Dec 12 2006

%C The evidence suggests that for any k, the number of primes with p < gap(p)^k is finite (this sequence being the special case k = 2), where gap(p) is the difference between p and the next prime. - _David W. Wilson_, Dec 13 2006

%C Primes for which sqrt(A000040(n)) < A001223(n).

%C Also primes p(n) for which the remainder of the division of p(n)^2 by p(n+1) is different from the remainder of the division of p(n+1)^2 by p(n).

%H A. Granville, <a href="http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf">Harald Cramér and the distribution of prime numbers</a>, Scandinavian Actuarial Journal, Volume 1995, 1995 - Issue 1.

%H Matt Visser, <a href="https://arxiv.org/abs/1812.02762">Strong version of Andrica's conjecture</a>, arXiv:1812.02762 [math.NT], 2018.

%e a(1) = 3 because sqrt(3) < 2. a(6) = 113 because sqrt(113) < 14.

%t Select[ Prime@ Range@100, PrimePi[ # + Sqrt@# ] - PrimePi@# == 0 &] (* _Robert G. Wilson v_, Dec 18 2006 *)

%Y Cf. A000040, A001223, A002386.

%K fini,nonn

%O 1,1

%A _Rémi Eismann_, Dec 10 2006