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.

23, 113, 1327, 31397, 370261, 2010733, 20831323, 25056082087, 2614941710599, 19581334192423, 218209405436543, 1693182318746371

Offset: 1

Author

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

Keywords

• nonn
• hard

Comments

Primes less than 23 are anomalous and are excluded.

a(12) 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

Examples

			-----------------------------
n   ratio                a(n)
-----------------------------
1   0.6103                23
2   0.6264               113
3   0.6575              1327
4   0.6715             31397
5   0.6812            370261
6   0.7025           2010733
7   0.7394          20831323
8   0.7953       25056082087
9   0.7975     2614941710599
10  0.8177    19581334192423
11  0.8311   218209405436543
12  0.9206  1693182318746371
		

References

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

Links

• 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 prime gaps [For local copy see A000101]
• 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).

Crossrefs

Subsequence of A002386.

Cf. A111870, A166363.

Programs

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

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