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A202361
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Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).
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12
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12102794130, 141702673770, 424052301750, 699699330330, 714303547230, 739544215410, 1623198312120, 2691533434590, 4207848555330, 4936074819480, 5887574660310, 6562654104930, 7205070907650, 8129061524010, 8362548652500, 9741706748970, 9967327212570
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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(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.
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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