cp's OEIS Frontend

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).

%I A111870 #106 Feb 16 2025 08:32:58
%S A111870 2,3,7,113,1129,1327,19609,31397,155921,360653,370261,1357201,2010733,
%T A111870 17051707,20831323,191912783,436273009,2300942549,3842610773,
%U A111870 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 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 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. - _N. J. A. Sloane_, Aug 27 2010
%C A111870 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 A111870 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 A111870 (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 A111870 Merit(1) + merit(2) + ... + merit(n) =: S(n) ~ n, where merit(n) is as above. - _Thomas Ordowski_, Aug 03 2012
%C A111870 For the index of a(n), see the comment at A214935. - _John W. Nicholson_, Nov 21 2013
%D A111870 Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23 2005
%H A111870 Hugo Pfoertner, <a href="/A111870/b111870.txt">Table of n, a(n) for n = 1..39</a>, extracted from pzktupel link.
%H A111870 Jens Kruse Andersen, <a href="http://primerecords.dk/primegaps/gaps20.htm">The Top-20 Prime Gaps</a>
%H A111870 Jens Kruse Andersen, <a href="http://primerecords.dk/primegaps/maximal.htm">Maximal gaps</a>
%H A111870 Jens Kruse Andersen, <a href="https://www.pzktupel.de/JensKruseAndersen/risinggap.php">Record prime gaps</a>, current version maintained by Norman Luhn.
%H A111870 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%H A111870 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>
%F A111870 a(n) = A277552(n) - A111871(n). - _Bobby Jacobs_, Nov 13 2016
%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
%t A111870 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 A111870 For the gaps, see A111871.
%Y A111870 Cf. A002386, A111943, A214935, A277552.
%K A111870 nonn
%O A111870 1,1
%A A111870 _N. J. A. Sloane_, based on correspondence with _Ed Pegg Jr_, Nov 23 2005
%E A111870 Corrected by Jose Brox, Dec 31 2005
%E A111870 Corrected and edited by _Daniel Forgues_, Oct 23 2009
%E A111870 Further edited by _Daniel Forgues_, Nov 01 2009, Nov 13 2009, Nov 24 2009