

A113404


Record gaps between prime quadruplets.


12



6, 90, 630, 660, 1170, 2190, 3780, 6420, 8940, 9030, 13260, 16470, 24150, 28800, 29610, 39990, 56580, 56910, 71610, 83460, 94530, 114450, 157830, 159060, 171180, 177360, 190500, 197910, 268050, 315840, 395520, 435240, 440910, 513570, 536010, 539310, 557340, 635130
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OFFSET

1,1


COMMENTS

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the HardyLittlewood ktuple conjecture, average gaps between prime ktuples are O(log^k(p)), with k=4 for quadruplets. If a gap is larger than all preceding gaps, 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 prime quadruplets are O(log^5(p)).  Alexei Kourbatov, Jan 04 2012


REFERENCES

Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 170, 1923.


LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..71
Tony Forbes, Prime ktuplets
Alexei Kourbatov, Maximal gaps between prime ktuples (graphs, more terms)
A. Kourbatov, Maximal gaps between prime ktuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013.  From N. J. A. Sloane, Feb 09 2013
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959, 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
Eric W. Weisstein, HardyLittlewood Constants
Eric W. Weisstein, Prime Constellation


FORMULA

From Alexei Kourbatov, Jan 04 2012: (Start)
(1) Upper bound: gaps between prime quadruplets (p, p+2, p+6, p+8) are smaller than 0.241*(log p)^5, 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)0.55), where a = 0.241*(log p)^4 is the average gap between quadruplets 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 ~0.241 is reciprocal to the HardyLittlewood 4tuple constant 4.15118... (End)


EXAMPLE

The first prime quadruplets are (5,7,11,13) and (11,13,17,19), so a(1)=115=6. The next quadruplet is (101,103,107,109), so a(2)=10111=90. The following quadruplet is 191,193,197,199 so 90 remains the record and no terms are added.


CROSSREFS

A229907 lists initial primes in quadruplets preceding the maximal gaps. A113403 lists the corresponding primes at the end of the maximal gaps. Cf. A008407, A007530.
Sequence in context: A335275 A166782 A121249 * A177283 A121607 A100594
Adjacent sequences: A113401 A113402 A113403 * A113405 A113406 A113407


KEYWORD

nonn


AUTHOR

Bernardo Boncompagni, Oct 28 2005


EXTENSIONS

Terms 159060 to 635130 added by Alexei Kourbatov, Jan 04 2012


STATUS

approved



