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%I #53 Feb 21 2023 13:22:52
%S 83160,195930,341880,5414220,9270030,18980220,25622520,36077370,
%T 51597630,92184750,125523090,140407470,141896370,336026460,403369470,
%U 435390270,442452570,627852330,754383210,1008582120,1021464990,1073692620,1088148810,1145336850
%N Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).
%C 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)).
%C 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.
%H Alexei Kourbatov, <a href="/A201251/b201251.txt">Table of n, a(n) for n = 1..52</a>
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
%H Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenktuples.htm">Maximal gaps between prime k-tuples</a>.
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1301.2242">Maximal gaps between prime k-tuples: a statistical approach</a>, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kourbatov/kourbatov3.html">J. Int. Seq. 16 (2013) #13.5.2</a>.
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1309.4053">Tables of record gaps between prime constellations</a>, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1401.6959">The distribution of maximal prime gaps in Cramer's probabilistic model of primes</a>, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
%H Norman Luhn, <a href="https://pzktupel.de/ktpatt_hl.php">Patterns of prime k-tuplets & the Hardy-Littlewood constants</a>.
%H Norman Luhn, <a href="https://pzktupel.de/RecordGaps/GAP07.php">Record Gaps Between Prime Septuplets, up to 10^17</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%F 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.
%e 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.
%Y 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.
%K nonn,hard
%O 1,1
%A _Alexei Kourbatov_, Nov 28 2011
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