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%I #16 Jan 17 2019 15:44:02
%S 10,20,30,60,90,130,150,180,210,240,280,380,420,430,450,470,500,530,
%T 630,640,660,780,840,920,960,970,990,1030,1070,1130,1170,1190,1230,
%U 1330,1610,1680,2030
%N Record (maximal) gaps between primes of the form 10k + 7.
%C Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 10k + 7 below x are about phi(10)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(10)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(10)=4.
%C Conjecture: a(n) < phi(10)*log^2(A269240(n)) almost always.
%C A269239 lists the primes preceding the maximal gaps.
%C A269240 lists the corresponding primes at the end of the maximal gaps.
%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="https://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%e The first two primes of the form 10k + 7 are 7 and 17, so a(1)=17-7=10. The next prime of this form is 37 and the gap 37-17=20 is a new record, so a(2)=20.
%o (PARI) re=0; s=7; forprime(p=17, 1e8, if(p%10!=7, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
%Y Cf. A030432, A269239, A269240.
%K nonn
%O 1,1
%A _Alexei Kourbatov_, Feb 20 2016