This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A111943 #70 Feb 16 2025 08:32:58 %S A111943 23,113,1327,31397,370261,2010733,20831323,25056082087,2614941710599, %T A111943 19581334192423,218209405436543,1693182318746371 %N 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. %C A111943 Primes less than 23 are anomalous and are excluded. %C A111943 a(12) was discovered by Bertil Nyman in 1999. %C A111943 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 A111943 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 A111943 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, A8. %H A111943 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 A111943 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 A111943 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101] %H A111943 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 A111943 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>. %H A111943 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cramer-GranvilleConjecture.html">Cramer-Granville Conjecture</a>. %H A111943 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ShanksConjecture.html">Shanks Conjecture</a> (and Wolf Conjecture). %e A111943 ----------------------------- %e A111943 n ratio a(n) %e A111943 ----------------------------- %e A111943 1 0.6103 23 %e A111943 2 0.6264 113 %e A111943 3 0.6575 1327 %e A111943 4 0.6715 31397 %e A111943 5 0.6812 370261 %e A111943 6 0.7025 2010733 %e A111943 7 0.7394 20831323 %e A111943 8 0.7953 25056082087 %e A111943 9 0.7975 2614941710599 %e A111943 10 0.8177 19581334192423 %e A111943 11 0.8311 218209405436543 %e A111943 12 0.9206 1693182318746371 %o A111943 (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 A111943 Subsequence of A002386. %Y A111943 Cf. A111870, A166363. %K A111943 nonn,hard %O A111943 1,1 %A A111943 _N. J. A. Sloane_, following emails from _R. K. Guy_ and _Ed Pegg Jr_, Nov 27 2005 %E A111943 Corrected and edited (p_n could be misinterpreted as the n-th prime) by _Daniel Forgues_, Nov 20 2009 %E A111943 Edited by _Charles R Greathouse IV_, May 14 2010