%N Primes of the form 6k + 1 preceding the first-occurrence gaps in A330853.
%C Subsequence of A002476. First differs from A268926 in that that sequence does not include 283; all terms of A268926 are in this sequence but many terms of this sequence are not in A268926.
%H Alexei Kourbatov, <a href="/A330854/b330854.txt">Table of n, a(n) for n = 1..135</a>
%H Alexei Kourbatov and Marek Wolf, <a href="https://arxiv.org/abs/2002.02115">On the first occurrences of gaps between primes in a residue class</a>, arXiv preprint arXiv:2002.02115 [math.NT], 2020.
%F a(n) = A330855(n) - A330853(n).
%e The first two primes of the form 6k + 1 are 7 and 13, so a(1) = 7. The next prime of that form is 19, and the gap 19 - 13 = 6 already occurred; so a new term is not added to the sequence. The next prime of the form 6k + 1 is 31, and the gap 31 - 19 = 12 is occurring for the first time; therefore a(2) = 19.
%e The gap between 241 and the next prime of the form 6k + 1 (271) is 30. So 241 is in the sequence.
%e Although the gap between 283 and 307 is only 24 (which is less than 30), the gap is of a size not previously encountered. So 283 is in the sequence.
%o (PARI) isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(s", "); isFirstOcc[g/6]=0); s=p)
%Y Cf. A002476, A014320, A058320, A330853 (gap sizes), A330855.
%A _Alexei Kourbatov_, Apr 27 2020