A201051 Record (maximal) gaps between prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).

165690, 903000, 10831800, 13773480, 22813770, 31090080, 43751820, 60881310, 86746170, 118516860, 239951250, 281573040, 359932650, 384903750, 518385000, 902801550, 1027007520, 1086331680, 1329198570, 2176467090

Offset: 1

Author

Alexei Kourbatov, Nov 28 2011

Keywords

• nonn
• hard

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.

Examples

			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.
		

Links

• Alexei Kourbatov, Table of n, a(n) for n = 1..36
• Tony Forbes and Norman Luhn, Prime k-tuplets
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
• Alexei Kourbatov, Maximal gaps between prime k-tuples
• Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2
• Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
• Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
• Norman Luhn, Record Gaps Between Prime Septuplets, up to 10^17
• Eric Weisstein's World of Mathematics, k-Tuple Conjecture

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.

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.