

A201073


Record (maximal) gaps between prime quintuplets (p, p+2, p+6, p+8, p+12).


11



6, 90, 1380, 14580, 21510, 88830, 97020, 107100, 112140, 301890, 401820, 577710, 689850, 846210, 857010, 986160, 1655130, 2035740, 2266320, 2467290, 2614710, 3305310, 3530220, 3880050, 3885420, 5290440, 5713800, 6049890
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OFFSET

1,1


COMMENTS

Prime quintuplets (p, p+2, p+6, p+8, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime ktuples can be deduced from the HardyLittlewood ktuple conjecture and are O(log^k(p)), with k=5 for quintuplets. 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 are O(log^6(p)).
A201074 lists initial primes in quintuplets (p, p+2, p+6, p+8, p+12) preceding the maximal gaps. A233432 lists the corresponding primes at the end of the maximal gaps.


LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..64
Tony Forbes, Prime ktuplets
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 quintuplets (graphs/data up to 10^15)
A. Kourbatov, Maximal gaps between prime ktuples: a statistical approach, arXiv preprint arXiv:1301.2242, 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, kTuple Conjecture


FORMULA

(1) Upper bound: gaps between prime quintuplets are smaller than 0.0987*(log p)^6, 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)0.4), where a = 0.0987*(log p)^5 is the average gap between quintuplets 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 known 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 ~0.0987 is reciprocal to the HardyLittlewood 5tuple constant 10.1317...


EXAMPLE

The initial four gaps of 6, 90, 1380, 14580 (between quintuplets starting at p=5, 11, 101, 1481, 16061) form an increasing sequence of records. Therefore a(1)=6, a(2)=90, a(3)=1380, and a(4)=14580. The next gap (after 16061) is smaller, so a new term is not added.


CROSSREFS

Cf. A022006 (prime quintuplets p, p+2, p+6, p+8, p+12), A113274, A113404, A200503, A201596, A201598, A201051, A201251, A202281, A202361, A201062, A201074, A002386, A233432.
Sequence in context: A317487 A037959 A247150 * A006480 A138462 A002896
Adjacent sequences: A201070 A201071 A201072 * A201074 A201075 A201076


KEYWORD

nonn


AUTHOR

Alexei Kourbatov, Nov 26 2011


STATUS

approved



