

A201596


Record (maximal) gaps between prime triples (p, p+4, p+6).


11



6, 24, 30, 90, 150, 156, 210, 240, 306, 366, 384, 444, 810, 834, 1086, 1200, 1326, 2316, 3876, 4230, 4350, 8244, 8880, 9450, 10686, 10950, 11784, 12816, 13554, 15504, 15576, 16254, 16506, 16596, 19446, 19944, 21516, 38340, 39990, 41556, 45786, 47190, 48246, 59856
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OFFSET

1,1


COMMENTS

Prime triples (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the HardyLittlewood ktuple conjecture, average gaps between prime ktuples are O(log^k(p)), with k=3 for triples. 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 between triples are O(log^4(p)).
A201597 lists initial primes p in triples (p, p+4, p+6) preceding the maximal gaps. A233435 lists the corresponding primes p at the end of the maximal gaps.


LINKS



FORMULA

Gaps between prime triples (p, p+4, p+6) are smaller than 0.35*(log p)^4, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^4(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.


EXAMPLE

The gap of 6 between triples starting at p=7 and p=13 is the very first gap, so a(1)=6. The gap of 24 between triples starting at p=13 and p=37 is a maximal gap  larger than any preceding gap; therefore a(2)=24. The gap of 30 between triples at p=37 and p=67 is again a maximal gap, so a(3)=30. The next gap is smaller, so it does not contribute to the sequence.


MATHEMATICA

DeleteDuplicates[Differences[Select[Partition[Prime[Range[5*10^6]], 3, 1], Differences[#]=={4, 2}&][[;; , 1]]], GreaterEqual] (* Harvey P. Dale, Feb 26 2023 *)


CROSSREFS

Cf. A022005 (prime triples p, p+4, p+6), A113274, A113404, A200503, A201598, A201062, A201073, A201051, A201251, A202281, A202361, A201597, A233435.


KEYWORD

nonn


AUTHOR



STATUS

approved



