1,1

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).

A. Granville, Cramer's conjecture

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

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

Cf. A000040, A001223, A002386.

Sequence in context: A134197 A053001 A053607 this_sequence A101301 A103116 A075321

Adjacent sequences: A124126 A124127 A124128 this_sequence A124130 A124131 A124132

fini,nonn

Remi Eismann (reismann(AT)free.fr), Dec 10 2006