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A201596 Record (maximal) gaps between prime triples (p, p+4, p+6). 11
6, 24, 30, 90, 150, 156, 210, 240, 306, 366, 384, 444, 810, 834, 1086, 1200, 1326, 2316, 3876, 4230, 4350, 8244, 8880, 9450, 10686, 10950, 11784, 12816, 13554, 15504, 15576, 16254, 16506, 16596, 19446, 19944, 21516, 38340, 39990, 41556, 45786, 47190, 48246, 59856 (list; graph; refs; listen; history; text; internal format)
Prime triples (p, p+4, 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)).
A201597 lists initial primes p in triples (p, p+4, p+6) preceding the maximal gaps. A233435 lists the corresponding primes p at the end of the maximal gaps.
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture
Gaps between prime triples (p, p+4, 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.
The gap of 6 between triples starting at p=7 and p=13 is the very first gap, so a(1)=6. The gap of 24 between triples starting at p=13 and p=37 is a maximal gap - larger than any preceding gap; therefore a(2)=24. The gap of 30 between triples at p=37 and p=67 is again a maximal gap, so a(3)=30. The next gap is smaller, so it does not contribute to the sequence.
DeleteDuplicates[Differences[Select[Partition[Prime[Range[5*10^6]], 3, 1], Differences[#]=={4, 2}&][[;; , 1]]], GreaterEqual] (* Harvey P. Dale, Feb 26 2023 *)
Sequence in context: A348714 A209452 A275302 * A128459 A229292 A147826
Alexei Kourbatov, Dec 03 2011

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