

A202361


Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).


10



12102794130, 141702673770, 424052301750, 699699330330, 714303547230, 739544215410, 1623198312120, 2691533434590, 4207848555330, 4936074819480, 5887574660310, 6562654104930, 7205070907650, 8129061524010, 8362548652500, 9741706748970, 9967327212570
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OFFSET

1,1


COMMENTS

Prime decuplets (p+0,2,6,12,14,20,24,26,30,32) 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)).
A202362 lists initial primes in decuplets (p+0,2,6,12,14,20,24,26,30,32) preceding the maximal gaps.


LINKS

Dana Jacobsen, Table of n, a(n) for n = 1..27
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
Alexei Kourbatov, Maximal gaps between prime ktuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 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.
Eric W. Weisstein, kTuple Conjecture


FORMULA

(1) Upper bound: gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32) 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 12102794130 between the very first decuplets starting at p=9853497737 and p=21956291867 is the initial term a(1)=12102794130.
The next gap after the decuplet starting at p=21956291867 is smaller, so it is not in this sequence.
The next gap of 141702673770 between the decuplets at p=22741837817 and p=164444511587 is a new record; therefore the next term is a(2)=141702673770.


PROG

(Perl) use ntheory ":all"; my($i, $l, $max)=(1, 0, 0); for (sieve_prime_cluster(1, 1e13, 2, 6, 12, 14, 20, 24, 26, 30, 32)) { my $gap=$_$l; if ($gap>$max) { say "$i $gap" if ++$i > 0; $max=$gap; } $l=$_; } # Dana Jacobsen, Oct 09 2015


CROSSREFS

Cf. A027570 (prime decuplets p+0,2,6,12,14,20,24,26,30,32), A202362, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251, A202281.
Sequence in context: A172587 A172581 A213534 * A015430 A135496 A113642
Adjacent sequences: A202358 A202359 A202360 * A202362 A202363 A202364


KEYWORD

nonn


AUTHOR

Alexei Kourbatov, Dec 18 2011


STATUS

approved



