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A200503
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Record (maximal) gaps between prime sextuplets (p, p+4, p+6, p+10, p+12, p+16).
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9
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90, 15960, 24360, 1047480, 2605680, 2856000, 3605070, 4438560, 5268900, 17958150, 21955290, 23910600, 37284660, 40198200, 62438460, 64094520, 66134250, 70590030, 77649390, 83360970, 90070470, 93143820, 98228130, 117164040, 131312160, 151078830, 154904820
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OFFSET
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1,1
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COMMENTS
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Prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are densest permissible constellations of 6 primes. Average gaps between sextuplets (and, more generally, between prime k-tuples) can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(ln^6(p)). Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps are O(ln^7(p)).
A200504 lists initial primes in sextuplets 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..27.
Tony Forbes. List of all possible patterns of prime k-tuplets (up to k=50)
Alexei Kourbatov, Maximal gaps between prime k-tuples
Eric W. Weisstein, k-Tuple Conjecture
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FORMULA
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(1) Upper bound: gaps between prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are smaller than 0.058*(ln p)^7, 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)-1/3), where a = 0.058*(ln p)^6 is the average gap between sextuplets 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 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 ~0.058 is reciprocal to the Hardy-Littlewood 6-tuple constant 17.2986...
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EXAMPLE
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The gap of 15960 between sextuplets with initial primes 97 and 16057 is a maximal gap - larger than any preceding gap; therefore a(2)=15960.
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CROSSREFS
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Cf. A022008 (prime sextuplets), A113274, A113404, A200504, A201596, A201598, A201062, A201073, A201051, A201251, A202281, A202361, A008407, A002386.
Sequence in context: A036257 A203779 A201073 * A199229 A134648 A145413
Adjacent sequences: A200500 A200501 A200502 * A200504 A200505 A200506
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KEYWORD
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nonn,hard
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AUTHOR
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Alexei Kourbatov, Nov 18 2011
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STATUS
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approved
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