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A201598
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Record (maximal) gaps between prime triplets (p, p+2, p+6).
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9
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6, 24, 60, 84, 114, 180, 210, 264, 390, 564, 630, 1050, 1200, 1530, 2016, 2844, 3426, 3756, 3864, 3936, 4074, 4110, 6090, 8250, 9240, 9270, 10344, 10506, 10734, 10920, 12930, 15204, 20190, 20286, 21216, 25746, 34920, 38820, 39390, 41754, 43020, 44310, 52500, 71346
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OFFSET
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1,1
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COMMENTS
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Prime triplets (p, p+2, 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(ln^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(ln^4(p)).
A201599 lists initial primes p in triplets (p, p+2, p+6) 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..44.
T. Forbes, Prime k-tuplets
Alexei Kourbatov, Maximal gaps between prime k-tuples
Eric W. Weisstein, k-Tuple Conjecture
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FORMULA
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Gaps between prime triplets (p, p+2, p+6) are smaller than 0.35*(ln p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(ln^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.
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EXAMPLE
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The gap of 6 between triplets starting at p=5 and p=11 is the very first gap, so a(1)=6. The gap of 6 between triplets starting at p=11 and p=17 is not a record, so it does not contribute to the sequence. The gap of 24 between triplets starting at p=17 and p=41 is a maximal gap - larger than any preceding gap; therefore a(2)=24.
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CROSSREFS
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Cf. A022004 (prime triplets p, p+2, p+6), A113274, A113404, A200503, A201596, A201062, A201073, A201051, A201251, A202281, A202361, A201599.
Sequence in context: A028595 A002653 A212904 * A211615 A195647 A086768
Adjacent sequences: A201595 A201596 A201597 * A201599 A201600 A201601
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KEYWORD
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nonn
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AUTHOR
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Alexei Kourbatov, Dec 03 2011
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STATUS
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approved
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