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A202361
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Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).
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9
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12102794130, 141702673770, 424052301750, 699699330330, 714303547230, 739544215410, 1623198312120, 2691533434590, 4207848555330, 4936074819480, 5887574660310, 6562654104930, 7205070907650, 8129061524010, 8362548652500, 9741706748970, 9967327212570
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OFFSET
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1,1
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COMMENTS
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Prime decuplets (p+0,2,6,12,14,20,24,26,30,32) are one of the two types of densest permissible constellations of 10 primes (A027569 and A027570).
Average gaps between prime k-tuples are O(ln^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(ln^11(p)).
A202362 lists initial primes in decuplets (p+0,2,6,12,14,20,24,26,30,32) preceding the maximal gaps.
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REFERENCES
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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.
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
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LINKS
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Table of n, a(n) for n=1..17.
T. Forbes, Prime k-tuplets
Eric W. Weisstein, k-Tuple Conjecture
Alexei Kourbatov, Maximal gaps between prime k-tuples
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FORMULA
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(1) Upper bound: gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32) are smaller than 0.00059*(ln 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(ln(p/a)-0.2), where a = 0.00059*(ln 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.
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EXAMPLE
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The gap of 12102794130 between the very first decuplets starting at p=9853497737 and p=21956291867 is the initial term a(1)=12102794130. The next gap after the decuplet starting at p=21956291867 is smaller, so it is not in this sequence. The next gap of 141702673770 between the decuplets at p=22741837817 and p=164444511587 is a new record; therefore the next term is a(2)=141702673770.
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CROSSREFS
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Cf. A027570 (prime decuplets p+0,2,6,12,14,20,24,26,30,32), A202362, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251, A202281
Sequence in context: A172587 A172581 A213534 * A015430 A135496 A113642
Adjacent sequences: A202358 A202359 A202360 * A202362 A202363 A202364
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KEYWORD
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nonn
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AUTHOR
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Alexei Kourbatov, Dec 18 2011
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STATUS
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approved
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