A277552 Primes q with prime gap q - p of n-th record merit.

3, 5, 11, 127, 1151, 1361, 19661, 31469, 156007, 360749, 370373, 1357333, 2010881, 17051887, 20831533, 191913031, 436273291, 2300942869, 3842611109, 4302407713, 10726905041, 25056082543, 304599509051, 461690510543, 1346294311331, 1408695494197, 1968188557063, 2614941711251, 13829048560417, 19581334193189

Offset: 1

Author

Bobby Jacobs, Nov 07 2016

Keywords

• nonn

Comments

The merit of the gap between consecutive primes p and q is (q-p)/log(p).

This sequence is infinite (Westzynthius 1931). - Charles R Greathouse IV, Nov 10 2016

Crossrefs

Cf. A111870, A111871.

Programs

Mathematica

p = 2; q = 3; lmt = 0; lst = {}; While[p < 10^12, If[q > p + lmt*Log[p], AppendTo[lst, q]; Print[q]; lmt = (q - p)/Log[p]]; p = q; q = NextPrime@ p]; lst (* or *)
(* set lst = the terms in A111870 *) NextPrime[ lst] (* Robert G. Wilson v, Nov 07 2016 *)

PARI

r=rm=0; p=2; forprime(q=3,, t=q-p; if(t>r, r=t; t/=log(p); if(t>rm, rm=t; print1(q", "))); p=q) \\ Charles R Greathouse IV, Nov 11 2016

Formula

a(n) = A111870(n) + A111871(n). - Charles R Greathouse IV, Nov 11 2016

Extensions

More terms from Robert G. Wilson v, Nov 07 2016