|
|
A201251
|
|
Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).
|
|
11
|
|
|
83160, 195930, 341880, 5414220, 9270030, 18980220, 25622520, 36077370, 51597630, 92184750, 125523090, 140407470, 141896370, 336026460, 403369470, 435390270, 442452570, 627852330, 754383210, 1008582120, 1021464990, 1073692620, 1088148810, 1145336850
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes (A022009 and A022010). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=7 for septuplets. 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^8(p)).
A201252 lists initial primes in septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) preceding the maximal gaps. A233038 lists the corresponding primes at the end of the maximal gaps.
|
|
LINKS
|
|
|
FORMULA
|
Gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are smaller than 0.02*(log p)^8, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^8(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.
|
|
EXAMPLE
|
The gap of 83160 between septuplets starting at p=5639 and p=88799 is the very first gap, so a(1)=83160. The gap of 195930 between septuplets starting at p=88799 and p=284729 is a maximal gap - larger than any preceding gap; therefore a(2)=195930. The next gap of 341880 is again a maximal gap, so a(3)=341880. The next gap is smaller, so it does not contribute to the sequence.
|
|
CROSSREFS
|
Cf. A022010 (prime septuplets p, p+2, p+8, p+12, p+14, p+18, p+20), A113274, A113404, A200503, A201062, A201073, A201596, A201598, A202281, A202361, A201051, A002386, A233038.
|
|
KEYWORD
|
nonn,hard
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|