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).

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

Offset: 1

Author

N. J. A. Sloane, based on correspondence with Ed Pegg Jr, Nov 23 2005

Keywords

• nonn

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

Examples

			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
		

References

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

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..39, extracted from pzktupel link.
• Jens Kruse Andersen, The Top-20 Prime Gaps
• Jens Kruse Andersen, Maximal gaps
• Jens Kruse Andersen, Record prime gaps, current version maintained by Norman Luhn.
• Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
• Eric Weisstein's World of Mathematics, Prime Gaps

Crossrefs

For the gaps, see A111871.

Cf. A002386, A111943, A214935, A277552.

Programs

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 *)

Formula

a(n) = A277552(n) - A111871(n). - Bobby Jacobs, Nov 13 2016

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