cp's OEIS Frontend

A269519 Record (maximal) gaps between primes of the form 8k + 7.

%I A269519 #16 Jan 17 2019 18:32:41
%S A269519 16,24,40,48,96,184,200,216,288,296,312,344,384,456,504,560,624,744,
%T A269519 760,776,800,824,840,864,880,896,952,984,1008,1056,1152,1208,1312,
%U A269519 1384,1448,1464,1472,1720,1872
%N A269519 Record (maximal) gaps between primes of the form 8k + 7.
%C A269519 Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 7 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 A269519 Conjecture: a(n) < phi(8)*log^2(A269521(n)) almost always.
%C A269519 A269520 lists the primes preceding the maximal gaps.
%C A269519 A269521 lists the corresponding primes at the end of the maximal gaps.
%H A269519 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 A269519 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 A269519 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 A269519 The first two primes of the form 8k + 7 are 7 and 23, so a(1)=23-7=16. The next primes of this form are 31, 47; the gaps 31-23 and 47-31 are not records so nothing is added to the sequence. The next prime of this form is 71 and the gap 71-47=24 is a new record, so a(2)=24.
%o A269519 (PARI) re=0; s=7; forprime(p=23, 1e8, if(p%8!=7, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
%Y A269519 Cf. A007522, A269520, A269521.
%K A269519 nonn
%O A269519 1,1
%A A269519 _Alexei Kourbatov_, Feb 28 2016