%I
%S 2,6,12,18,30,36,72,150,168,210,282,372,498,630,924,930,1008,1452,
%T 1512,1530,1722,1902,2190,2256,2832,2868,3012,3102,3180,3480,3804,
%U 4770,5292,6030,6282,6474,6552,6648,7050,7980,8040,8994,9312,9318,10200,10338,10668,10710,11388,11982,12138,12288,12630,13050,14262,14436,14952,15396,15720,16362,16422,16590,16896,17082,18384,19746,19992,20532,21930,22548,23358,23382,25230,26268,28842
%N Record gaps between twin primes.
%C a(n) mod 6 = 0 for each n>1.
%D Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 170, 1923.
%H Richard Fischer, <a href="http://www.fermatquotient.com/PrimLuecken/ZwillingsRekordLuecken">Maximale Lücken (Intervallen) von Primzahlenzwillingen</a>
%H Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenktuples.htm#2tuples">Maximal gaps between prime ktuples</a>
%H A. Kourbatov, <a href="http://arxiv.org/abs/1301.2242">Maximal gaps between prime ktuples: a statistical approach</a>, arXiv preprint arXiv:1301.2242, 2013.
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1309.4053">Tables of record gaps between prime constellations</a>, arXiv preprint arXiv:1309.4053, 2013.
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1401.6959">The distribution of maximal prime gaps in Cramer's probabilistic model of primes</a>, arXiv preprint arXiv:1401.6959, 2014
%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/twin_gaps.html">Gaps between twin primes</a>
%H Luis Rodriguez and Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_066.htm">Gaps between consecutive twin pairs</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/kTupleConjecture.html">kTuple Conjecture</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/TwinPrimesConstant.html">Twin Prime Constant</a>
%F a(n) = A036063(n) + 2.
%F a(n) = A036062(n)  A113275(n).
%F From _Alexei Kourbatov_, Dec 29 2011: (Start)
%F (1) Upper bound: gaps between twin primes are smaller than 0.76*(log p)^3, where p is the prime at the end of the gap.
%F (2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)1.2), where a = 0.76*(log p)^2 is the average gap between twin primes near p, as predicted by the HardyLittlewood ktuple conjecture.
%F Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
%F Both formulas (1) and (2) are derived from the HardyLittlewood ktuple conjecture via probabilitybased heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the ktuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
%F (End)
%e The first twin primes are 3,5 and 5,7 so a(0)=53=2. The following pair is 11,13 so a(1)=115=6. The following pair is 17,19 so 6 remains the record and no terms are added.
%t NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k]  !PrimeQ[k + 2], k+=6]; k]; p = 5; r = 2; t = {2}; Do[ q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, q  p]; Print[{p, q  p}]; r = q  p]; p = q, {n, 10^9}]; t (* _Robert G. Wilson v_ *)
%Y The smallest primes originating the sequence are given in A113275. Cf. A008407, A005250, A002386.
%K nonn
%O 1,1
%A _Bernardo Boncompagni_, Oct 21 2005
%E More terms from _Robert G. Wilson v_, Oct 22 2005
%E Corrected terms based on A036063, crosschecked with independent computations by Carlos Rivera and Richard Fischer (linked).
%E Terms up to a(72) are given in Kourbatov (2013), terms up to a(75) in Oliveira e Silva website.
