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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....

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, Harold Cramér and the distribution of prime numbers, Scandanavian Actuarial J. 1 (1995), pp. 12-28.

Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture, arXiv preprint, 2015.

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

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 August 31 17:10 EDT 2015. Contains 261250 sequences.