A124129 Primes p for which there are no primes between p and p+sqrt(p).

3, 7, 13, 23, 31, 113

Offset: 1

Author

Rémi Eismann, Dec 10 2006

Keywords

• fini
• nonn

Comments

Conjecture: there are no other terms.

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

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

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

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

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

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).

Examples

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

Links

• A. Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal, Volume 1995, 1995 - Issue 1.
• Matt Visser, Strong version of Andrica's conjecture, arXiv:1812.02762 [math.NT], 2018.

Crossrefs

Cf. A000040, A001223, A002386.

Programs

Mathematica

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