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A201073
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Record (maximal) gaps between prime quintuplets (p, p+2, p+6, p+8, p+12)
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9
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6, 90, 1380, 14580, 21510, 88830, 97020, 107100, 112140, 301890, 401820, 577710, 689850, 846210, 857010, 986160, 1655130, 2035740, 2266320, 2467290, 2614710, 3305310, 3530220, 3880050, 3885420, 5290440, 5713800, 6049890
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OFFSET
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1,1
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COMMENTS
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Prime quintuplets (p, p+2, p+6, p+8, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(ln^k(p)), with k=5 for quintuplets. 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^6(p)).
A201074 lists initial primes in quintuplets (p, p+2, p+6, p+8, p+12) 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..28.
T. Forbes, Prime k-tuplets
Alexei Kourbatov, Maximal gaps between prime quintuplets (graphs/data up to 10^15)
Eric W. Weisstein, k-Tuple Conjecture
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FORMULA
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(1) Upper bound: gaps between prime quintuplets are smaller than 0.0987*(ln p)^6, 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)-0.4), where a = 0.0987*(ln p)^5 is the average gap between quintuplets 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 ~0.0987 is reciprocal to the Hardy-Littlewood 5-tuple constant 10.1317...
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EXAMPLE
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The initial four gaps of 6, 90, 1380, 14580 (between quintuplets starting at p=5, 11, 101, 1481, 16061) form an increasing sequence of records. Therefore a(1)=6, a(2)=90, a(3)=1380, and a(4)=14580. The next gap (after 16061) is smaller, so a new term is not added.
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CROSSREFS
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Cf. A022006 (prime quintuplets p, p+2, p+6, p+8, p+12), A113274, A113404, A200503, A201596, A201598, A201051, A201251, A202281, A202361, A201062, A201074, A002386.
Sequence in context: A002432 A091800 A037959 * A006480 A138462 A002896
Adjacent sequences: A201070 A201071 A201072 * A201074 A201075 A201076
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KEYWORD
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nonn
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AUTHOR
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Alexei Kourbatov, Nov 26 2011
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STATUS
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approved
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