

A200503


Record (maximal) gaps between prime sextuplets (p, p+4, p+6, p+10, p+12, p+16).


11



90, 15960, 24360, 1047480, 2605680, 2856000, 3605070, 4438560, 5268900, 17958150, 21955290, 23910600, 37284660, 40198200, 62438460, 64094520, 66134250, 70590030, 77649390, 83360970, 90070470, 93143820, 98228130, 117164040, 131312160, 151078830, 154904820
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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 ktuples) can be deduced from the HardyLittlewood ktuple 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.


LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..56
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. 170, 1923.
Alexei Kourbatov, Maximal gaps between prime ktuples
Alexei Kourbatov, Maximal gaps between prime ktuples: 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 ktuplets & the HardyLittlewood constants.
Norman Luhn, Record Gaps Between Prime Sextuplets, up to 10^17
Eric Weisstein's World of Mathematics, kTuple 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 HardyLittlewood ktuple 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 HardyLittlewood ktuple conjecture via probabilitybased heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the ktuple conjecture itself has no formal proof either). In both formulas, the constant ~0.058 is reciprocal to the HardyLittlewood 6tuple 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

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


KEYWORD

nonn,hard


AUTHOR

Alexei Kourbatov, Nov 18 2011


STATUS

approved



