|
|
A201598
|
|
Record (maximal) gaps between prime triples (p, p+2, p+6).
|
|
11
|
|
|
6, 24, 60, 84, 114, 180, 210, 264, 390, 564, 630, 1050, 1200, 1530, 2016, 2844, 3426, 3756, 3864, 3936, 4074, 4110, 6090, 8250, 9240, 9270, 10344, 10506, 10734, 10920, 12930, 15204, 20190, 20286, 21216, 25746, 34920, 38820, 39390, 41754, 43020, 44310, 52500, 71346
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Prime triples (p, p+2, p+6) are one of the two types of densest permissible constellations of 3 primes (A022004 and A022005). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples 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)).
A201599 lists initial primes p in triples (p, p+2, p+6) preceding the maximal gaps. A233434 lists the corresponding primes p at the end of the maximal gaps.
|
|
LINKS
|
|
|
FORMULA
|
Gaps between prime triples (p, p+2, 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=5 and p=11 is the very first gap, so a(1)=6. The gap of 6 between triples starting at p=11 and p=17 is not a record, so it does not contribute to the sequence. The gap of 24 between triples starting at p=17 and p=41 is a maximal gap - larger than any preceding gap; therefore a(2)=24.
|
|
CROSSREFS
|
Cf. A022004 (prime triples p, p+2, p+6), A113274, A113404, A200503, A201596, A201062, A201073, A201051, A201251, A202281, A202361, A201599, A233434.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|