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%I #56 Nov 24 2022 04:22:15
%S 6,24,60,84,114,180,210,264,390,564,630,1050,1200,1530,2016,2844,3426,
%T 3756,3864,3936,4074,4110,6090,8250,9240,9270,10344,10506,10734,10920,
%U 12930,15204,20190,20286,21216,25746,34920,38820,39390,41754,43020,44310,52500,71346
%N Record (maximal) gaps between prime triples (p, p+2, p+6).
%C Prime triples (p, p+2, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=3 for triples. If a gap is larger than any preceding gap, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps between triples are O(log^4(p)).
%C A201599 lists initial primes p in triples (p, p+2, p+6) preceding the maximal gaps. A233434 lists the corresponding primes p at the end of the maximal gaps.
%H Alexei Kourbatov, <a href="/A201598/b201598.txt">Table of n, a(n) for n = 1..72</a>
%H G. H. Hardy and J. E. Littlewood, <a href="https://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, 1923.
%H Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenktuples.htm">Maximal gaps between prime k-tuples</a>
%H A. 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 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 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 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 Norman Luhn, <a href="https://pzktupel.de/RecordGaps/GAP03.php">Record Gaps Between Prime Triplets</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%F Gaps between prime triples (p, p+2, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.
%e The gap of 6 between triples starting at p=5 and p=11 is the very first gap, so a(1)=6. The gap of 6 between triples starting at p=11 and p=17 is not a record, so it does not contribute to the sequence. The gap of 24 between triples starting at p=17 and p=41 is a maximal gap - larger than any preceding gap; therefore a(2)=24.
%Y Cf. A022004 (prime triples p, p+2, p+6), A113274, A113404, A200503, A201596, A201062, A201073, A201051, A201251, A202281, A202361, A201599, A233434.
%K nonn
%O 1,1
%A _Alexei Kourbatov_, Dec 03 2011
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