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A111870
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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).
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26
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23 2005
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LINKS
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Table of n, a(n) for n=1..32.
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
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FORMULA
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a(n) = A277552(n) - A111871(n). - Bobby Jacobs, Nov 13 2016
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, based on correspondence with Ed Pegg Jr, Nov 23 2005
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EXTENSIONS
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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
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STATUS
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approved
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