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A201051
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Record (maximal) gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20)
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9
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165690, 903000, 10831800, 13773480, 22813770, 31090080, 43751820, 60881310, 86746170, 118516860, 239951250, 281573040, 359932650, 384903750, 518385000, 902801550, 1027007520, 1086331680, 1329198570, 2176467090
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OFFSET
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1,1
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COMMENTS
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Prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes (A022009 and A022010). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(ln^k(p)), with k=7 for septuplets. 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^8(p)).
A201249 lists initial primes p in septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) 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..20.
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 septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) are smaller than 0.02*(ln p)^8, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(ln^8(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 165690 between septuplets starting at p=11 and p=165701 is the very first gap, so a(1)=165690. The gap of 903000 between septuplets starting at p=165701 and p=1068701 is a maximal gap - larger than any preceding gap; therefore a(2)=903000. The next gap of 10831800 is again a maximal gap, so a(3)=10831800. The next gap is smaller, so it does not contribute to the sequence.
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CROSSREFS
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Cf. A022009 (prime septuplets p, p+2, p+6, p+8, p+12, p+18, p+20), A113274, A113404, A200503, A201062, A201073, A201596, A201598, A201251, A202281, A202361, A201249, A002386.
Sequence in context: A196675 A034603 A224582 * A183834 A203274 A061741
Adjacent sequences: A201048 A201049 A201050 * A201052 A201053 A201054
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KEYWORD
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nonn,hard
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AUTHOR
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Alexei Kourbatov, Nov 28 2011
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STATUS
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approved
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