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A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

%I A113275 #52 Dec 25 2024 09:17:28
%S A113275 3,5,17,41,71,311,347,659,2381,5879,13397,18539,24419,62297,187907,
%T A113275 687521,688451,850349,2868959,4869911,9923987,14656517,17382479,
%U A113275 30752231,32822369,96894041,136283429,234966929,248641037,255949949
%N A113275 Lesser of twin primes for which the gap before the following twin primes is a record.
%H A113275 Martin Raab, <a href="/A113275/b113275.txt">Table of n, a(n) for n = 1..82</a> (first 75 terms from Max Alekseyev)
%H A113275 Harvey Dubner, <a href="/A007534/a007534.pdf">Twin Prime Conjectures</a>, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
%H A113275 Alexei Kourbatov, <a href="http://arxiv.org/abs/1301.2242">Maximal gaps between prime k-tuples: a statistical approach</a>, arXiv preprint arXiv:1301.2242 [math.NT], 2013.
%H A113275 Alexei Kourbatov, <a href="http://arxiv.org/abs/1309.4053">Tables of record gaps between prime constellations</a>, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
%H A113275 Alexei Kourbatov and Marek Wolf, <a href="http://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%H A113275 Mersenneforum, <a href="https://www.mersenneforum.org/showthread.php?t=24303">Gaps between prime pairs (Twin Primes)</a>.
%H A113275 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/twin_gaps.html">Gaps between twin primes</a>
%F A113275 a(n) = A036061(n) - 2.
%F A113275 a(n) = A036062(n) - A113274(n).
%e A113275 The smallest twin prime pair is 3, 5, then 5, 7 so a(1) = 3; the following pair is 11, 13 so a(2) = 5 because 11 - 5 = 6 > 5 - 3 = 2; the following pair is 17, 19: since 17 - 11 = 6 = 11 - 5 nothing happens; the following pair is 29, 31 so a(3)= 17 because 29 - 17 = 12 > 11 - 5 = 6.
%t A113275 NextLowerTwinPrim[n_] := Block[{k = n + 2}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k++ ]; k]; p = 3; r = 0; t = {3}; Do[q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, p]; r = q - p]; p = q, {n, 10^9}] (* _Robert G. Wilson v_, Oct 22 2005 *)
%Y A113275 Record gaps are given in A113274. Cf. A002386.
%K A113275 nonn
%O A113275 1,1
%A A113275 _Bernardo Boncompagni_, Oct 21 2005
%E A113275 a(22)-a(30) from _Robert G. Wilson v_, Oct 22 2005
%E A113275 Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) in Oliveira e Silva's website, added by _Max Alekseyev_, Nov 06 2015