A201062 Record (maximal) gaps between prime 5-tuples (p, p+4, p+6, p+10, p+12).

90, 1770, 2190, 10080, 24360, 35910, 156750, 208620, 304920, 306390, 328020, 422190, 526350, 639330, 706860, 866460, 1030770, 1111620, 1147440, 1151100, 1447530, 1769670, 1793070, 2024610, 2320170, 2335080, 2403570

Offset: 1

Author

Alexei Kourbatov, Nov 26 2011

Keywords

• nonn

Comments

Prime quintuplets (p, p+4, p+6, p+10, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=5 for quintuplets. If a gap is larger than all preceding gaps, 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 quintuplets are O(log^6(p)).

A201063 lists initial primes in quintuplets (p, p+4, p+6, p+10, p+12) preceding the maximal gaps. A233433 lists the corresponding primes at the end of the maximal gaps.

Examples

			The gap of 90 between quintuplets starting at p=7 and p=97 is the very first gap, so a(1)=90. The gap of 1770 between quintuplets starting at p=97 and p=1867 is a maximal gap - larger than any preceding gap; therefore a(2)=1770. The gap after p=1867 is smaller, so a new term is not added.
		

Links

• Alexei Kourbatov, Table of n, a(n) for n = 1..71
• Tony Forbes, 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 Mathematica, Vol. 44, pp. 1-70, 1923.
• Alexei Kourbatov, Maximal gaps between prime 5-tuples (graphs/data up to 10^15)
• A. 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.
• Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
• Eric W. Weisstein, k-Tuple Conjecture

Crossrefs

Cf. A022007 (prime 5-tuples p, p+4, p+6, p+10, p+12), A113274, A113404, A200503, A201596, A201598, A201073, A201051, A201251, A202281, A202361, A201063, A002386, A233433.

Programs

Mathematica

DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],5,1],Differences[#]=={4,2,4,2}&][[;;,1]]],GreaterEqual] (* The program generates the first 18 terms of the sequence. *) (* Harvey P. Dale, Apr 20 2025 *)

Formula

(1) Upper bound: gaps between prime 5-tuples are smaller than 0.0987*(log p)^6, where p is the prime at the end of the gap.

(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap ~ a(log(p/a)-0.4), where a = 0.0987*(log p)^5 is the average gap between quintuplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.

Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.

Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.0987 is reciprocal to the Hardy-Littlewood 5-tuple constant 10.1317...