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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(ln^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(ln^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, 1-70, 1923.

LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..71

Tony Forbes, Prime k-tuplets

Alexei Kourbatov, Maximal gaps between prime k-tuples (graphs, more terms)

A. Kourbatov, Maximal gaps between prime k-tuples: 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

Eric W. Weisstein, k-Tuple Conjecture

Eric W. Weisstein, Hardy-Littlewood Constants

Eric W. Weisstein, Prime Constellation

FORMULA

Contribution 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*(ln 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*(ln(p/a)-0.55), where a = 0.241*(ln p)^4 is the average gap between quadruplets 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 known 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 ~0.241 is reciprocal to the Hardy-Littlewood 4-tuple constant 4.15118... (End)

EXAMPLE

The first prime quadruplets are (5,7,11,13) and (11,13,17,19), so a(1)=11-5=6. The next quadruplet is (101,103,107,109), so a(2)=101-11=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: A177573 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

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Last modified September 21 14:10 EDT 2014. Contains 247026 sequences.