OFFSET
1,1
COMMENTS
Prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are densest permissible constellations of 6 primes. Average gaps between sextuplets (and, more generally, between prime k-tuples) can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^6(p)). Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps are O(log^7(p)).
LINKS
Martin Raab, Table of n, a(n) for n = 1..83 (terms 1..56 from Alexei Kourbatov, terms 57..71 from Norman Luhn).
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.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei 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.
Norman Luhn, Record Gaps Between Prime Sextuplets.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
FORMULA
(1) Conjectured upper bound: gaps between prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are smaller than 0.058*(log p)^7, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)-1/3), where a = 0.058*(log p)^6 is the average gap between sextuplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
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 maximal gaps. 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 ~0.058 is reciprocal to the Hardy-Littlewood 6-tuple constant 17.2986...
EXAMPLE
The gap of 15960 between sextuplets with initial primes 97 and 16057 is a maximal gap - larger than any preceding gap; therefore a(2)=15960.
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Alexei Kourbatov, Nov 18 2011
STATUS
approved