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A202281 Record (maximal) gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32). 9
33081664140, 50040961320, 211797665730, 278538937950, 314694286830, 446820068310, 589320949140, 1135263664920, 1154348695500, 1280949740070, 1340804150070, 1458168320490, 1539906870810, 1858581264540, 2590180927950, 3182865274050, 4949076176310, 5719502339670 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are one of the two types of densest permissible constellations of 10 primes (A027569 and A027570).

Average gaps between prime k-tuples are O(log^k(p)), with k=10 for decuplets, by the Hardy-Littlewood k-tuple conjecture. 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^11(p)).

A202282 lists initial primes in decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding 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

Table of n, a(n) for n=1..18.

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.

Eric W. Weisstein, k-Tuple Conjecture

FORMULA

(1) Upper bound: gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are smaller than 0.00059*(log p)^11, where p is the prime at the end of the gap.

(2) Estimate for the actual size of maximal gaps near p: max gap = a(log(p/a)-0.2), where a = 0.00059(log p)^10 is the average gap between 10-tuples near p.

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.

EXAMPLE

The gap of 33081664140 after the first decuplet starting at p=11 is the term a(1). The next three gaps of 50040961320, 211797665730, 278538937950 form an increasing sequence, each setting a new record; therefore each of these gaps is in the sequence, as a(2), a(3), a(4). The next gap is not a record, so it is not in this sequence.

CROSSREFS

Cf. A027569 (prime decuplets p+0,2,6,8,12,18,20,26,30,32), A202282, A202361, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251

Sequence in context: A234392 A212942 A233794 * A034655 A099600 A115498

Adjacent sequences:  A202278 A202279 A202280 * A202282 A202283 A202284

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Dec 15 2011

STATUS

approved

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Last modified November 23 16:11 EST 2014. Contains 249851 sequences.