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A111943
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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.
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7
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23, 113, 1327, 31397, 370261, 2010733, 20831323, 25056082087, 2614941710599, 19581334192423, 218209405436543, 1693182318746371
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, A8.
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LINKS
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Eric Weisstein's World of Mathematics, Prime Gaps.
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EXAMPLE
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-----------------------------
n ratio a(n)
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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EXTENSIONS
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Corrected and edited (p_n could be misinterpreted as the n-th prime) by Daniel Forgues, Nov 20 2009
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STATUS
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approved
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