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 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). 26
 2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 2010733, 17051707, 20831323, 191912783, 436273009, 2300942549, 3842610773, 4302407359, 10726904659, 25056082087, 304599508537, 461690510011, 1346294310749, 1408695493609, 1968188556461, 2614941710599, 13829048559701, 19581334192423, 218209405436543, 1693182318746371 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 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 The sequence b(n) = (prime(n+1)/prime(n))^n is increasing for terms prime(n) of this sequence. - Thomas Ordowski, May 04 2012 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 (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 Merit(1) + merit(2) + ... + merit(n) =: S(n) ~ n, where merit(n) is as above. - Thomas Ordowski, Aug 03 2012 For the index of a(n), see the comment at A214935. - John W. Nicholson, Nov 21 2013 REFERENCES Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23 2005 LINKS Jens Kruse Andersen, The Top-20 Prime Gaps Jens Kruse Andersen, Maximal gaps Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101] Eric Weisstein's World of Mathematics, Prime Gaps FORMULA a(n) = A277552(n) - A111871(n). - Bobby Jacobs, Nov 13 2016 EXAMPLE 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 MATHEMATICA 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 *) CROSSREFS For the gaps, see A111871. Cf. A002386, A111943, A214935, A277552. Sequence in context: A163152 A088120 A230778 * A182514 A062935 A083436 Adjacent sequences:  A111867 A111868 A111869 * A111871 A111872 A111873 KEYWORD nonn AUTHOR N. J. A. Sloane, based on correspondence with Ed Pegg Jr, Nov 23 2005 EXTENSIONS Corrected by Jose Brox, Dec 31 2005 Corrected and edited by Daniel Forgues, Oct 23 2009 Further edited by Daniel Forgues, Nov 01 2009, Nov 13 2009, Nov 24 2009 STATUS approved

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Last modified May 17 23:07 EDT 2022. Contains 353779 sequences. (Running on oeis4.)