%I #101 Apr 06 2022 08:49:45

%S 2,3,7,113,1129,1327,19609,31397,155921,360653,370261,1357201,2010733,

%T 17051707,20831323,191912783,436273009,2300942549,3842610773,

%U 4302407359,10726904659,25056082087,304599508537,461690510011,1346294310749,1408695493609,1968188556461,2614941710599,13829048559701,19581334192423,218209405436543,1693182318746371

%N 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 As I understand it, the sequence refers to "Smallest prime p whose 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 in the sequence. - Jose Brox, Dec 31 2005

%C 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. - _N. J. A. Sloane_, Aug 27 2010

%C The sequence b(n) = (prime(n+1)/prime(n))^n is increasing for terms prime(n) of this sequence. - _Thomas Ordowski_, May 04 2012

%C The smallest prime(n) such that (prime(n+1)/prime(n))^n is increasing: 2, 3, 7, 23, 113, 1129, 1327, ... (A205827). - _Thomas Ordowski_, May 04 2012

%C (prime(n+1)/prime(n))^n > 1 + merit(n) for n > 2, where merit(n) = (prime(n+1)-prime(n))/log(prime(n)). - _Thomas Ordowski_, May 14 2012

%C Merit(1) + merit(2) + ... + merit(n) =: S(n) ~ n, where merit(n) is as above. - _Thomas Ordowski_, Aug 03 2012

%C For the index of a(n), see the comment at A214935. - _John W. Nicholson_, Nov 21 2013

%D Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23 2005

%H Jens Kruse Andersen, <a href="http://primerecords.dk/primegaps/gaps20.htm">The Top-20 Prime Gaps</a>

%H Jens Kruse Andersen, <a href="http://primerecords.dk/primegaps/maximal.htm">Maximal gaps</a>

%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>

%F a(n) = A277552(n) - A111871(n). - _Bobby Jacobs_, Nov 13 2016

%e The first few entries correspond to the following gaps. The table gives n, p, gap = q-p and the merit of the gap.

%e 1, 2, 1, 1.4427

%e 2, 3, 2, 1.82048

%e 3, 7, 4, 2.05559

%e 4, 113, 14, 2.96147

%e 5, 1129, 22, 3.12985

%e 6, 1327, 34, 4.72835

%e 7, 19609, 52, 5.26116

%e 8, 31397, 72, 6.95352

%e 9, 155921, 86, 7.19238

%e 10, 360653, 96, 7.50254

%e 11, 370261, 112, 8.73501

%e 12, 1357201, 132, 9.34782

%t With[{s = Map[(#2 - #1)/Log[#1] & @@ # &, Partition[Prime@ Range[10^6], 2, 1]]}, Map[Prime@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger_, Jul 19 2018 *)

%Y For the gaps, see A111871.

%Y Cf. A002386, A111943, A214935, A277552.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, based on correspondence with _Ed Pegg Jr_, Nov 23 2005

%E Corrected by Jose Brox, Dec 31 2005

%E Corrected and edited by _Daniel Forgues_, Oct 23 2009

%E Further edited by _Daniel Forgues_, Nov 01 2009, Nov 13 2009, Nov 24 2009