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A202361 Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32). 12
12102794130, 141702673770, 424052301750, 699699330330, 714303547230, 739544215410, 1623198312120, 2691533434590, 4207848555330, 4936074819480, 5887574660310, 6562654104930, 7205070907650, 8129061524010, 8362548652500, 9741706748970, 9967327212570 (list; graph; refs; listen; history; text; internal format)
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 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)).

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

LINKS

Norman Luhn, Table of n, a(n) for n = 1..44 (terms 1..27 from Dana Jacobsen).

Tony Forbes and 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. 44, 1-70, 1923.

Alexei Kourbatov, Maximal gaps between prime k-tuples

Alexei Kourbatov, Maximal gaps between prime k-tuples: 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.

Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.

Eric Weisstein's World of Mathematics, k-Tuple 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 10-tuples near p.

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.

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 A287747

Adjacent sequences:  A202358 A202359 A202360 * A202362 A202363 A202364

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Dec 18 2011

STATUS

approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)