

A201251


Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).


11



83160, 195930, 341880, 5414220, 9270030, 18980220, 25622520, 36077370, 51597630, 92184750, 125523090, 140407470, 141896370, 336026460, 403369470, 435390270, 442452570, 627852330, 754383210, 1008582120, 1021464990, 1073692620, 1088148810, 1145336850
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OFFSET

1,1


COMMENTS

Prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes (A022009 and A022010). Average gaps between prime ktuples can be deduced from the HardyLittlewood ktuple 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)).
A201252 lists initial primes in septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) preceding the maximal gaps. A233038 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, 170, 1923.


LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..52
T. Forbes, Prime ktuplets
Alexei Kourbatov, Maximal gaps between prime ktuples
A. Kourbatov, Maximal gaps between prime ktuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013 and J. Int. Seq. 16 (2013) #13.5.2
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, kTuple Conjecture


FORMULA

Gaps between prime septuplets (p, p+2, p+8, p+12, p+14, 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 83160 between septuplets starting at p=5639 and p=88799 is the very first gap, so a(1)=83160. The gap of 195930 between septuplets starting at p=88799 and p=284729 is a maximal gap  larger than any preceding gap; therefore a(2)=195930. The next gap of 341880 is again a maximal gap, so a(3)=341880. The next gap is smaller, so it does not contribute to the sequence.


CROSSREFS

Cf. A022010 (prime septuplets p, p+2, p+8, p+12, p+14, p+18, p+20), A113274, A113404, A200503, A201062, A201073, A201596, A201598, A202281, A202361, A201051, A002386, A233038.
Sequence in context: A251112 A298437 A190385 * A235097 A125779 A233925
Adjacent sequences: A201248 A201249 A201250 * A201252 A201253 A201254


KEYWORD

nonn,hard


AUTHOR

Alexei Kourbatov, Nov 28 2011


STATUS

approved



