%N Record (maximal) gaps between primes of the form 10k + 9.
%C Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 10k + 9 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(A269263(n)) almost always.
%C A269262 lists the primes preceding the maximal gaps.
%C A269263 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 + 9 are 19 and 29, so a(1)=29-19=10. The next prime of this form is 59 and the gap 59-29=30 is a new record, so a(2)=30.
%o (PARI) re=0; s=19; forprime(p=29, 1e8, if(p%10!=9, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
%Y Cf. A030433, A269262, A269263.
%A _Alexei Kourbatov_, Feb 20 2016