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%I #69 Oct 28 2021 12:37:08
%S 23,113,1327,31397,370261,2010733,20831323,25056082087,2614941710599,
%T 19581334192423,218209405436543,1693182318746371
%N 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.
%C Primes less than 23 are anomalous and are excluded.
%C a(12) was discovered by Bertil Nyman in 1999.
%C 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....
%C 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
%D R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, A8.
%H Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/cramer.pdf">Harald Cramér and the distribution of prime numbers</a>, Scandinavian Actuarial J. 1 (1995), pp. 12-28.
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1506.03042">Upper bounds for prime gaps related to Firoozbakht's conjecture</a>, arXiv:1506.03042 [math.NT], 2015; J. Integer Sequences, 18 (2015), Article 15.11.2.
%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%H Daniel Shanks, <a href="http://dx.doi.org/10.2307/2002951">On maximal gaps between successive primes</a>, Math. Comp. 18 (88) (1964), 646-651.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cramer-GranvilleConjecture.html">Cramer-Granville Conjecture</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ShanksConjecture.html">Shanks Conjecture</a> (and Wolf Conjecture).
%e -----------------------------
%e n ratio a(n)
%e -----------------------------
%e 1 0.6103 23
%e 2 0.6264 113
%e 3 0.6575 1327
%e 4 0.6715 31397
%e 5 0.6812 370261
%e 6 0.7025 2010733
%e 7 0.7394 20831323
%e 8 0.7953 25056082087
%e 9 0.7975 2614941710599
%e 10 0.8177 19581334192423
%e 11 0.8311 218209405436543
%e 12 0.9206 1693182318746371
%o (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
%Y Subsequence of A002386.
%Y Cf. A111870, A166363.
%K nonn,hard
%O 1,1
%A _N. J. A. Sloane_, following emails from _R. K. Guy_ and _Ed Pegg Jr_, Nov 27 2005
%E Corrected and edited (p_n could be misinterpreted as the n-th prime) by _Daniel Forgues_, Nov 20 2009
%E Edited by _Charles R Greathouse IV_, May 14 2010
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