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A201051 Record (maximal) gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20). 11
165690, 903000, 10831800, 13773480, 22813770, 31090080, 43751820, 60881310, 86746170, 118516860, 239951250, 281573040, 359932650, 384903750, 518385000, 902801550, 1027007520, 1086331680, 1329198570, 2176467090 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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(log^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(log^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. A233425 lists the corresponding primes at the end of the maximal gaps.

REFERENCES

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.

LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..36

Tony Forbes, Prime k-tuplets

Alexei Kourbatov, Maximal gaps between prime k-tuples

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

Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.

Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959, 2014

Eric W. Weisstein, k-Tuple Conjecture

FORMULA

Gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20) are smaller than 0.02*(log p)^8, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^8(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

EXAMPLE

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.

CROSSREFS

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, A233425.

Sequence in context: A233706 A233701 A224582 * A233425 A183834 A203274

Adjacent sequences:  A201048 A201049 A201050 * A201052 A201053 A201054

KEYWORD

nonn,hard

AUTHOR

Alexei Kourbatov, Nov 28 2011

STATUS

approved

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Last modified October 21 23:13 EDT 2014. Contains 248381 sequences.