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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). 23
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

Comment from Jose Brox, Dec 31 2005: As I understand it, the sequence refers to "Smallest prime p such that its 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 on the sequence.

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) 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. (edp(AT)wolfram.com), Posting to Seq Fan mailing list, Nov 23 2005

LINKS

Table of n, a(n) for n=1..32.

Jens Kruse Andersen, The Top-20 Prime Gaps

Jens Kruse Andersen, Maximal gaps

Thomas Nicely, Prime gaps

Eric Weisstein's World of Mathematics, Prime Gaps

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

CROSSREFS

For the gaps see A111871.

Cf. A002386, A111943, A214935.

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. (edp(AT)wolfram.com), 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 December 20 19:24 EST 2014. Contains 252289 sequences.