%I #16 Jan 17 2019 18:32:53
%S 8,16,40,48,56,64,72,80,88,96,112,128,144,192,216,224,264,288,296,360,
%T 368,440,456,480,608,616,672,752,760,856,912,920,960,1128,1176,1216,
%U 1424,1432,1440,1464,1480,1552,1728,1872
%N Record (maximal) gaps between primes of the form 8k + 5.
%C Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 5 below x are about phi(8)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(8)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(8)=4.
%C Conjecture: a(n) < phi(8)*log^2(A269515(n)) almost always.
%C A269514 lists the primes preceding the maximal gaps.
%C A269515 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 8k + 5 are 5 and 13, so a(1)=13-5=8. The next prime of this form is 29 and the gap 29-13=16 is a new record, so a(2)=16.
%o (PARI) re=0; s=5; forprime(p=13, 1e8, if(p%8!=5, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
%Y Cf. A007521, A269514, A269515.
%K nonn
%O 1,1
%A _Alexei Kourbatov_, Feb 28 2016