A113404 Record gaps between prime quadruplets.

6, 90, 630, 660, 1170, 2190, 3780, 6420, 8940, 9030, 13260, 16470, 24150, 28800, 29610, 39990, 56580, 56910, 71610, 83460, 94530, 114450, 157830, 159060, 171180, 177360, 190500, 197910, 268050, 315840, 395520, 435240, 440910, 513570, 536010, 539310, 557340, 635130

Offset: 1

Author

Bernardo Boncompagni, Oct 28 2005

Keywords

• nonn

Comments

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=4 for quadruplets. 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 prime quadruplets are O(log^5(p)). - Alexei Kourbatov, Jan 04 2012

Examples

			The first prime quadruplets are (5,7,11,13) and (11,13,17,19), so a(1)=11-5=6. The next quadruplet is (101,103,107,109), so a(2)=101-11=90. The following quadruplet is 191,193,197,199 so 90 remains the record and no terms are added.
		

Links

• Alexei Kourbatov, Table of n, a(n) for n = 1..71
• T. Forbes, 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 (graphs, more terms)
• A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013. - From _N. J. A. Sloane_, Feb 09 2013
• 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 Weisstein's World of Mathematics, k-Tuple Conjecture.
• Eric Weisstein's World of Mathematics, Hardy-Littlewood Constants.
• Eric Weisstein's World of Mathematics, Prime Constellation.

Crossrefs

A229907 lists initial primes in quadruplets preceding the maximal gaps. A113403 lists the corresponding primes at the end of the maximal gaps. Cf. A008407, A007530.

Programs

Mathematica

DeleteDuplicates[Differences[#[[4]]&/@Select[Partition[Prime[Range[10^7]],4,1],Differences[#] == {2,4,2}&]],GreaterEqual] (* The program generates the first 29 terms of the sequence. *) (* Harvey P. Dale, Aug 04 2024 *)

Formula

From Alexei Kourbatov, Jan 04 2012: (Start)

(1) Upper bound: gaps between prime quadruplets (p, p+2, p+6, p+8) are smaller than 0.241*(log p)^5, 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)-0.55), where a = 0.241*(log p)^4 is the average gap between quadruplets 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 ~0.241 is reciprocal to the Hardy-Littlewood 4-tuple constant 4.15118... (End)

Extensions

Terms 159060 to 635130 added by Alexei Kourbatov, Jan 04 2012