A202281 - OEIS

%I #58 Feb 16 2025 08:33:16

%S 33081664140,50040961320,211797665730,278538937950,314694286830,

%T 446820068310,589320949140,1135263664920,1154348695500,1280949740070,

%U 1340804150070,1458168320490,1539906870810,1858581264540,2590180927950,3182865274050,4949076176310,5719502339670

%N Record (maximal) gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32).

%C Prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are one of the two types of densest permissible constellations of 10 primes (A027569 and A027570).

%C Average gaps between prime k-tuples are O(log^k(p)), with k=10 for decuplets, by the Hardy-Littlewood k-tuple conjecture. 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^11(p)).

%C A202282 lists initial primes in decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding the maximal gaps.

%H Norman Luhn, <a href="/A202281/b202281.txt">Table of n, a(n) for n = 1..54</a> (terms 1..32 from Dana Jacobsen).

%H Tony Forbes and Norman Luhn, <a href="http://pzktupel.de/ktuplets.htm">Prime k-tuplets</a>

%H G. H. Hardy and J. E. Littlewood, <a href="http://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 Math. 44, 1-70, 1923.

%H Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenktuples.htm">Maximal gaps between prime k-tuples</a>

%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1301.2242">Maximal gaps between prime k-tuples: a statistical approach</a>, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kourbatov/kourbatov3.html">J. Int. Seq. 16 (2013) #13.5.2</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>

%F (1) Upper bound: gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are smaller than 0.00059*(log p)^11, where p is the prime at the end of the gap.

%F (2) Estimate for the actual size of maximal gaps near p: max gap = a(log(p/a)-0.2), where a = 0.00059(log p)^10 is the average gap between 10-tuples near p.

%F 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.

%e The gap of 33081664140 after the first decuplet starting at p=11 is the term a(1). The next three gaps of 50040961320, 211797665730, 278538937950 form an increasing sequence, each setting a new record; therefore each of these gaps is in the sequence, as a(2), a(3), a(4). The next gap is not a record, so it is not in this sequence.

%o (Perl) use ntheory ":all"; my($i,$l,$max)=(-1,0,0); for (sieve_prime_cluster(1,1e13,2,6,8,12,18,20,26,30,32)) { my $gap=$_-$l; if ($gap>$max) { say "$i $gap" if ++$i > 0; $max=$gap; } $l=$_; } # _Dana Jacobsen_, Oct 08 2015

%Y Cf. A027569 (prime decuplets p+0,2,6,8,12,18,20,26,30,32), A202282, A202361, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251

%K nonn

%O 1,1

%A _Alexei Kourbatov_, Dec 15 2011