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A111943 Prime p with prime gap q - p of n-th record Cramer-Shanks-Granville ratio, where q is smallest prime larger than p and C-S-G ratio is (q-p)/(log p)^2. 6
13, 23, 113, 1327, 31397, 370261, 2010733, 20831323, 25056082087, 2614941710599, 19581334192423, 218209405436543, 1693182318746371 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(13) was discovered by Bertil Nyman in 1999.

Shanks conjectures that the ratio will never reach 1. Granville conjectures the opposite: that the ratio will exceed or come arbitrarily close to 2/e^gamma = 1.1229....

Firoozbakht's conjecture implies that the ratio is below 1-1/log(p) for all primes p>=11; see Th.1 of arXiv:1506.03042. In Cramér's probabilistic model of primes, the ratio is below 1-1/log(p) for almost all maximal gaps between primes; see A235402. - Alexei Kourbatov, Jan 28 2016

REFERENCES

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, A8.

LINKS

Table of n, a(n) for n=1..13.

Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial J. 1 (1995), pp. 12-28.

Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015; J. Integer Sequences, 18 (2015), Article 15.11.2.

Thomas R. Nicely, First occurrence of a prime gap of 1000 or greater

Daniel Shanks, On maximal gaps between successive primes, Math. Comp. 18 (88) (1964), 646-651.

Eric Weisstein's World of Mathematics, Prime Gaps.

Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.

Eric Weisstein's World of Mathematics, Shanks Conjecture (and Wolf Conjecture).

EXAMPLE

   n  ratio              prime

  ==  ======  ================

   2  0.6103                23

   3  0.6264               113

   4  0.6575              1327

   5  0.6715             31397

   6  0.6812            370261

   7  0.7025           2010733

   8  0.7394          20831323

   9  0.7953       25056082087

  10  0.7975     2614941710599

  11  0.8177    19581334192423

  12  0.8311   218209405436543

  13  0.9206  1693182318746371

PROG

(PARI) r=CSG=0; p=13; forprime(q=17, 1e8, if(q-p>r, r=q-p; t=r/log(p)^2; if(t>CSG, CSG=t; print1(p", "))); p=q) \\ Charles R Greathouse IV, Apr 07 2013

CROSSREFS

Subsequence of A002386.

Cf. A111870, A166363.

Sequence in context: A147443 A131447 A110196 * A039448 A089768 A185684

Adjacent sequences:  A111940 A111941 A111942 * A111944 A111945 A111946

KEYWORD

nonn,hard

AUTHOR

N. J. A. Sloane, following emails from R. K. Guy and Ed Pegg Jr, Nov 27 2005

EXTENSIONS

Corrected and edited (p_n could be misinterpreted as the n-th prime) by Daniel Forgues, Nov 20 2009

Edited by Charles R Greathouse IV, May 14 2010

STATUS

approved

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Last modified May 29 03:15 EDT 2016. Contains 273487 sequences.