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A113274 Record gaps between twin primes.

%I A113274 #79 Feb 16 2025 08:32:59
%S A113274 2,6,12,18,30,36,72,150,168,210,282,372,498,630,924,930,1008,1452,
%T A113274 1512,1530,1722,1902,2190,2256,2832,2868,3012,3102,3180,3480,3804,
%U A113274 4770,5292,6030,6282,6474,6552,6648,7050,7980,8040,8994,9312,9318,10200,10338,10668
%N A113274 Record gaps between twin primes.
%C A113274 a(n) mod 6 = 0 for each n>1.
%H A113274 Martin Raab, <a href="/A113274/b113274.txt">Table of n, a(n) for n = 1..82</a> (terms up to a(72) from Alexei Kourbatov, a(73)-a(75) from Tomás Oliveira e Silva)
%H A113274 Richard Fischer, <a href="http://www.fermatquotient.com/PrimLuecken/ZwillingsRekordLuecken">Maximale Lücken (Intervallen) von Primzahlenzwillingen</a>
%H A113274 G. H. Hardy and J. E. Littlewood, <a href="http://dx.doi.org/10.1007/BF02403921">Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
%H A113274 Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenktuples.htm#2tuples">Maximal gaps between prime k-tuples</a>
%H A113274 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; and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kourbatov/kourbatov3.html">J. Int. Seq. 16 (2013) #13.5.2</a>
%H A113274 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 A113274 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 [math.NT], 2014.
%H A113274 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 A113274 Mersenneforum, <a href="https://www.mersenneforum.org/showthread.php?t=24303">Gaps between prime pairs (Twin Primes)</a>.
%H A113274 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/twin_gaps.html">Gaps between twin primes</a>
%H A113274 Luis Rodriguez and Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_066.htm">Conjecture 66. Gaps between consecutive twin pairs</a>, The Prime Puzzles and Problems Connection.
%H A113274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%H A113274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimesConstant.html">Twin Prime Constant</a>
%F A113274 a(n) = A036063(n) + 2.
%F A113274 a(n) = A036062(n) - A113275(n).
%F A113274 From _Alexei Kourbatov_, Dec 29 2011: (Start)
%F A113274 (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 A113274 (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 Hardy-Littlewood k-tuple conjecture.
%F A113274 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 A113274 Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
%F A113274 (End)
%e A113274 The first twin primes are 3,5 and 5,7 so a(0)=5-3=2. The following pair is 11,13 so a(1)=11-5=6. The following pair is 17,19 so 6 remains the record and no terms are added.
%t A113274 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_, Oct 22 2005 *)
%t A113274 DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],2,1],#[[2]]-#[[1]] == 2&][[All,2]]],GreaterEqual] (* The program generates the first 27 terms of the sequence. *) (* _Harvey P. Dale_, Dec 31 2022 *)
%Y A113274 The smallest primes originating the sequence are given in A113275. Cf. A008407, A005250, A002386.
%K A113274 nonn
%O A113274 1,1
%A A113274 _Bernardo Boncompagni_, Oct 21 2005
%E A113274 More terms from _Robert G. Wilson v_, Oct 22 2005
%E A113274 Corrected terms based on A036063, cross-checked with independent computations by Carlos Rivera and Richard Fischer (linked).
%E A113274 Terms up to a(72) are given in Kourbatov (2013), terms up to a(75) in Oliveira e Silva website.