

A202281


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


10



33081664140, 50040961320, 211797665730, 278538937950, 314694286830, 446820068310, 589320949140, 1135263664920, 1154348695500, 1280949740070, 1340804150070, 1458168320490, 1539906870810, 1858581264540, 2590180927950, 3182865274050, 4949076176310, 5719502339670
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OFFSET

1,1


COMMENTS

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).
Average gaps between prime ktuples are O(log^k(p)), with k=10 for decuplets, by the HardyLittlewood ktuple 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)).
A202282 lists initial primes in decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding the maximal gaps.


LINKS

Dana Jacobsen, Table of n, a(n) for n = 1..33
T. Forbes, Prime ktuplets
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. 44, 170, 1923.
Alexei Kourbatov, Maximal gaps between prime ktuples
A. Kourbatov, Maximal gaps between prime ktuples: 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.
Eric W. Weisstein, kTuple Conjecture


FORMULA

(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.
(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 10tuples near p.
Both formulas (1) and (2) are derived from the HardyLittlewood ktuple conjecture via probabilitybased heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof.


EXAMPLE

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.


PROG

(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


CROSSREFS

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
Sequence in context: A234392 A212942 A233794 * A271818 A034655 A099600
Adjacent sequences: A202278 A202279 A202280 * A202282 A202283 A202284


KEYWORD

nonn


AUTHOR

Alexei Kourbatov, Dec 15 2011


STATUS

approved



