A211073 Primes p followed by a gap of at least 1/2 * log(p)^2.

2, 3, 5, 7, 13, 23, 31, 113, 1327, 19609, 25471, 31397, 34061, 43331, 44293, 155921, 188029, 212701, 265621, 338033, 360653, 370261, 396733, 404851, 492113, 544279, 576791, 604073, 838249, 860143, 1098847, 1139993, 1313467, 1349533, 1357201, 1388483, 1444309

Offset: 1

Author

Charles R Greathouse IV, Apr 26 2012

Keywords

• nonn

Comments

Primes followed by unusually long prime gaps.

The Cramér model suggests that there are about 2*sqrt(x/log^2 x) elements up to x. - Charles R Greathouse IV, Mar 18 2016

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002386, A082885, A111943, A182315.

Programs

Mathematica

Select[Prime[Range[10^4]], NextPrime[#] - # > (Log[#]^2)/2 &] (* Alonso del Arte, Jun 02 2013 *)

PARI

G=1; p=2; forprime(q=3, 1e7, if(q-p>=G && q-p>log(p)^2/2, G=ceil(log(p)^2/2); print1(p", ")); p=q)

Formula

Primes p such that all integers in (p, p + 0.5 * log(p)^2) are composite.