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A201596 Record (maximal) gaps between prime triplets (p, p+4, p+6). 11
6, 24, 30, 90, 150, 156, 210, 240, 306, 366, 384, 444, 810, 834, 1086, 1200, 1326, 2316, 3876, 4230, 4350, 8244, 8880, 9450, 10686, 10950, 11784, 12816, 13554, 15504, 15576, 16254, 16506, 16596, 19446, 19944, 21516, 38340, 39990, 41556, 45786, 47190, 48246, 59856 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Prime triplets (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=3 for triplets. 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 between triplets are O(log^4(p)).

A201597 lists initial primes p in triplets (p, p+4, p+6) preceding the maximal gaps. A233435 lists the corresponding primes p at the end of the maximal gaps.

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..79

T. Forbes, Prime k-tuplets

Alexei Kourbatov, Maximal gaps between prime k-tuples

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

FORMULA

Gaps between prime triplets (p, p+4, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

EXAMPLE

The gap of 6 between triplets starting at p=7 and p=13 is the very first gap, so a(1)=6. The gap of 24 between triplets starting at p=13 and p=37 is a maximal gap - larger than any preceding gap; therefore a(2)=24. The gap of 30 between triplets at p=37 and p=67 is again a maximal gap, so a(3)=30. The next gap is smaller, so it does not contribute to the sequence.

CROSSREFS

Cf. A022005 (prime triplets p, p+4, p+6), A113274, A113404, A200503, A201598, A201062, A201073, A201051, A201251, A202281, A202361, A201597, A233435.

Sequence in context: A147778 A209452 A275302 * A128459 A229292 A147826

Adjacent sequences:  A201593 A201594 A201595 * A201597 A201598 A201599

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Dec 03 2011

STATUS

approved

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Last modified December 10 11:36 EST 2016. Contains 279001 sequences.