cp's OEIS Frontend

A201073 Record (maximal) gaps between prime 5-tuples (p, p+2, p+6, p+8, p+12).

%I A201073 #45 Feb 16 2025 08:33:16
%S A201073 6,90,1380,14580,21510,88830,97020,107100,112140,301890,401820,577710,
%T A201073 689850,846210,857010,986160,1655130,2035740,2266320,2467290,2614710,
%U A201073 3305310,3530220,3880050,3885420,5290440,5713800,6049890
%N A201073 Record (maximal) gaps between prime 5-tuples (p, p+2, p+6, p+8, p+12).
%C A201073 Prime quintuplets (p, p+2, p+6, p+8, 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 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^6(p)).
%C A201073 A201074 lists initial primes in quintuplets (p, p+2, p+6, p+8, p+12) preceding the maximal gaps. A233432 lists the corresponding primes at the end of the maximal gaps.
%H A201073 Alexei Kourbatov, <a href="/A201073/b201073.txt">Table of n, a(n) for n = 1..64</a>
%H A201073 Tony Forbes, <a href="http://anthony.d.forbes.googlepages.com/ktuplets.htm">Prime k-tuplets</a>
%H A201073 G. H. Hardy and J. E. Littlewood, <a href="https://dx.doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
%H A201073 Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenprimequintuplets.htm">Maximal gaps between prime 5-tuples</a> (graphs/data up to 10^15)
%H A201073 A. Kourbatov, <a href="http://arxiv.org/abs/1301.2242">Maximal gaps between prime k-tuples: a statistical approach</a>, arXiv preprint arXiv:1301.2242, 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kourbatov/kourbatov3.html">J. Int. Seq. 16 (2013) #13.5.2</a>
%H A201073 Alexei Kourbatov, <a href="http://arxiv.org/abs/1309.4053">Tables of record gaps between prime constellations</a>, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
%H A201073 Alexei Kourbatov, <a href="http://arxiv.org/abs/1401.6959">The distribution of maximal prime gaps in Cramer's probabilistic model of primes</a>, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
%H A201073 Alexei Kourbatov and Marek Wolf, <a href="http://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%H A201073 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%F A201073 (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.
%F A201073 (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.
%F A201073 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.
%F A201073 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...
%e A201073 The initial four gaps of 6, 90, 1380, 14580 (between quintuplets starting at p=5, 11, 101, 1481, 16061) form an increasing sequence of records. Therefore a(1)=6, a(2)=90, a(3)=1380, and a(4)=14580. The next gap (after 16061) is smaller, so a new term is not added.
%Y A201073 Cf. A022006 (prime 5-tuples p, p+2, p+6, p+8, p+12), A113274, A113404, A200503, A201596, A201598, A201051, A201251, A202281, A202361, A201062, A201074, A002386, A233432.
%K A201073 nonn
%O A201073 1,1
%A A201073 _Alexei Kourbatov_, Nov 26 2011