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%I #39 Jun 15 2020 20:13:15
%S 6,12,18,30,36,42,54,60,84,126,150,162,168,246,258,318,342,354,372,
%T 408,468,534,552,600,654,762,768,798,864,894,942,960,1068,1224,1302,
%U 1320,1344
%N Record (maximal) gaps between primes of the form 6k - 1.
%C Dirichlet's theorem on arithmetic progressions and the GRH suggest that average gaps between primes of the form 6k - 1 below x are about phi(6)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(6)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(6)=2.
%C Conjecture: a(n) < phi(6)*log^2(A268930(n)) almost always.
%C Conjecture: phi(6)*n^2/6 < a(n) < phi(6)*n^2 almost always. - _Alexei Kourbatov_, Nov 27 2019
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1610.03340">On the distribution of maximal gaps between primes in residue classes</a>, arXiv:1610.03340 [math.NT], 2016.
%H Alexei Kourbatov, <a href="https://arxiv.org/abs/1709.05508">On the nth record gap between primes in an arithmetic progression</a>, arXiv:1709.05508 [math.NT], 2017; <a href="https://doi.org/10.12988/imf.2018.712103">Int. Math. Forum, 13 (2018), 65-78</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.
%F a(n) = A268930(n) - A268929(n). - _Alexei Kourbatov_, Jun 15 2020.
%e The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=12.
%t re = 0; s = 5; Reap[For[p = 11, p < 10^8, p = NextPrime[p], If[Mod[p, 6] != 5, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* _Jean-François Alcover_, Dec 12 2018, from PARI *)
%o (PARI) re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
%Y Cf. A007528, A268929 (primes preceding the maximal gaps), A268930 (primes at the end of the maximal gaps), A334543, A334544.
%K nonn,more
%O 1,1
%A _Alexei Kourbatov_, Feb 15 2016
%E Terms a(31)..a(37) from _Alexei Kourbatov_, Jun 15 2020