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%I #18 Aug 04 2022 17:29:29
%S 7,13,37,103,307,457,613,2137,2377,2797,3463,4783,5737,9433,14557,
%T 24103,45817,52177,126487,317587,580687,715873,2719663,6227563,
%U 8114857,10085623,10137493,18773137,21297553,25291363,43472497,52645423,69718147,80002627,89776327
%N Initial prime in prime triples (p, p+4, p+6) preceding the maximal gaps in A201596.
%C Prime triples (p, p+4, p+6) are one of the two types of densest permissible constellations of 3 primes. Maximal gaps between triples of this type are listed in A201596; see more comments there.
%H Alexei Kourbatov, <a href="/A201597/b201597.txt">Table of n, a(n) for n = 1..79</a>
%H Tony Forbes, <a href="http://anthony.d.forbes.googlepages.com/ktuplets.htm">Prime k-tuplets</a>
%H Alexei Kourbatov, <a href="http://www.javascripter.net/math/primes/maximalgapsbetweenprimetriplets.htm">Maximal gaps between prime triples</a>
%H Alexei Kourbatov and Marek Wolf, <a href="http://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%e The gap of 6 between triples starting at p=7 and p=13 is the very first gap, so a(1)=7. 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)=13. The gap of 30 between triples at p=37 and p=67 is again a maximal gap, so a(3)=37. The next gap is smaller, so it does not contribute to the sequence.
%t DeleteDuplicates[{#[[1]],#[[2]]-#[[1]]}&/@Partition[Select[Prime[Range[ 5206000]],AllTrue[#+{4,6},PrimeQ]&],2,1],GreaterEqual[#1[[2]],#2[[2]]]&] [[All,1]] (* _Harvey P. Dale_, Aug 04 2022 *)
%Y Cf. A022005 (prime triples p, p+4, p+6), A201596.
%K nonn
%O 1,1
%A _Alexei Kourbatov_, Dec 03 2011
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