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 n, p, gap = q-p and the merit of the gap.

1, 2, 1, 1.4427

2, 3, 2, 1.82048

3, 7, 4, 2.05559

4, 113, 14, 2.96147

5, 1129, 22, 3.12985

6, 1327, 34, 4.72835

7, 19609, 52, 5.26116

8, 31397, 72, 6.95352

9, 155921, 86, 7.19238

10, 360653, 96, 7.50254

11, 370261, 112, 8.73501

12, 1357201, 132, 9.34782

For the gaps see A111871. Cf. A111943.

Adjacent sequences: A111867 A111868 A111869 this_sequence A111871 A111872 A111873

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

Further edited by Daniel Forgues (squid(AT)zensearch.com), Nov 01 2009, Nov 13 2009, Nov 24 2009