

A111943


Prime p with prime gap q  p of nth record CramerShanksGranville ratio, where q is smallest prime larger than p and CSG ratio is (qp)/(log p)^2.


6



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

1,1


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 11/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 11/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, SpringerVerlag, Third Edition, 2004, A8.


LINKS

Table of n, a(n) for n=1..12.
Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial J. 1 (1995), pp. 1228.
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), 646651.
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, CramerGranville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture (and Wolf Conjecture).


EXAMPLE


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


PROG

(PARI) r=CSG=0; p=13; forprime(q=17, 1e8, if(qp>r, r=qp; 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: A233362 A288751 A070024 * A057877 A156568 A042026
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 nth prime) by Daniel Forgues, Nov 20 2009
Edited by Charles R Greathouse IV, May 14 2010


STATUS

approved



