%I A111870
%S A111870 2,3,7,113,1129,1327,19609,31397,155921,360653,370261,1357201,
%T A111870 2010733,17051707,20831323,191912783,436273009,2300942549,
%U A111870 3842610773,4302407359,10726904659,25056082087,304599508537,461690510011,
               1346294310749,1408695493609,1968188556461,2614941710599,13829048559701,
               19581334192423,218209405436543,1693182318746371
%N A111870 Prime p with prime gap q-p of n_th record merit, where q is smallest 
               prime larger than p and the merit of a prime gap is (q-p)/log(p).
%C A111870 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.
%C A111870 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.
%D A111870 Ed Pegg, Jr. (edp(AT)wolfram.com), Posting to Seq Fan mailing list, Nov 
               23, 2005
%H A111870 Jens Kruse Andersen, <a href="http://users.cybercity.dk/~dsl522332/math/
               primegaps/gaps20.htm">Prime gaps</a>
%H A111870 Jens Kruse Andersen, <a href="http://users.cybercity.dk/~dsl522332/math/
               primegaps/maximal.htm">Maximal gaps</a>
%H A111870 Thomas Nicely, <a href="http://www.trnicely.net/">Prime gaps</a>
%H A111870 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeGaps.html">Prime Gaps</a>
%e A111870 The first few entries correspond to the following gaps. The table gives 
               n, p, gap = q-p and the merit of the gap.
%e A111870 1, 2, 1, 1.4427
%e A111870 2, 3, 2, 1.82048
%e A111870 3, 7, 4, 2.05559
%e A111870 4, 113, 14, 2.96147
%e A111870 5, 1129, 22, 3.12985
%e A111870 6, 1327, 34, 4.72835
%e A111870 7, 19609, 52, 5.26116
%e A111870 8, 31397, 72, 6.95352
%e A111870 9, 155921, 86, 7.19238
%e A111870 10, 360653, 96, 7.50254
%e A111870 11, 370261, 112, 8.73501
%e A111870 12, 1357201, 132, 9.34782
%Y A111870 For the gaps see A111871. Cf. A111943.
%Y A111870 Adjacent sequences: A111867 A111868 A111869 this_sequence A111871 A111872 
               A111873
%K A111870 nonn,new
%O A111870 1,1
%A A111870 N. J. A. Sloane (njas(AT)research.att.com), based on correspondence with 
               Ed Pegg, Jr. (edp(AT)wolfram.com), Nov 23 2005
%E A111870 Corrected by Jose Brox, Dec 31 2005
%E A111870 Corrected and edited by Daniel Forgues (squid(AT)zensearch.com), Oct 
               23 2009
%E A111870 Further edited by Daniel Forgues (squid(AT)zensearch.com), Nov 01 2009, 
               Nov 13 2009, Nov 24 2009