1,1

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)).

A200504 lists initial primes in sextuplets preceding the maximal gaps. A233426 lists the corresponding primes at the end of the maximal gaps.

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, Patterns of prime k-tuplets & the Hardy-Littlewood constants.

Norman Luhn, Record Gaps Between Prime Sextuplets.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

(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...

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.

DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]], 6, 1], Differences[#]=={4, 2, 4, 2, 4}&][[;; , 1]]], GreaterEqual] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, May 08 2025 *)

Cf. A022008 (prime sextuplets), A113274, A113404, A200504, A201596, A201598, A201062, A201073, A201051, A201251, A202281, A202361, A008407, A002386, A233426.

Sequence in context: A203779 A359842 A295594 * A199229 A278411 A278631

Adjacent sequences: A200500 A200501 A200502 * A200504 A200505 A200506

nonn,hard

Alexei Kourbatov, Nov 18 2011

approved