1,1

Comment from Jose Brox, Dec 31, 2005: As I understand it, the sequence refers to "Smallest prime p such that its following gap has bigger merit than the other primes smaller than p." If that is the case, then it has an error. The sequence starts: 2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 4652353, 2010733... but you can see that 4652353 > 2010733, so in any case it should be listed after, not before it. But above that, its merit is 10.03 < 10.20, the merit of 2010733, so it is not in a mistaken position: it shouldn't appear on the sequence.

The logarithmic (base 10) graph seems to be linearly asymptotic to n with slope ~ 1/log(10) which would imply that: log(prime with n_th record merit) ~ n as n goes to infinity.

Ed Pegg, Jr. (edp(AT)wolfram.com), Posting to Seq Fan mailing list, Nov 23, 2005

Jens Kruse Andersen, Prime gaps

Jens Kruse Andersen, Maximal gaps

Thomas Nicely, Prime gaps

Eric Weisstein's World of Mathematics, Prime Gaps

The first few entries correspond to the following gaps. The table gives p_n, gap = p_{n+1}-p_n and the merit of the gap.

2, 1, 1.4427

3, 2, 1.82048

7, 4, 2.05559

113, 14, 2.96147

1129, 22, 3.12985

1327, 34, 4.72835

19609, 52, 5.26116

31397, 72, 6.95352

155921, 86, 7.19238

360653, 96, 7.50254

370261, 112, 8.73501

1357201, 132, 9.34782

For the gaps see A111871. Cf. A111943.

nonn,new

N. J. A. Sloane (njas(AT)research.att.com), based on correspondence with Ed Pegg, Jr. (edp(AT)wolfram.com), Nov 23 2005

Corrected by Jose Brox, Dec 31 2005

Corrected and edited by Daniel Forgues (squid(AT)zensearch.com), Oct 23 2009

Edited by Daniel Forgues (squid(AT)zensearch.com), Nov 01 2009

Definition corrected (p_n could be misinterpreted as the n_th prime) by Daniel Forgues (squid(AT)zensearch.com), Nov 13 2009