cp's OEIS Frontend

A269424 Record (maximal) gaps between primes of the form 8k + 1.

%I A269424 #20 Jan 17 2019 18:33:05
%S A269424 24,32,56,64,88,112,120,136,160,216,232,240,264,304,384,480,488,528,
%T A269424 544,576,624,640,720,760,816,888,960,1032,1064,1200,1296,1320,1432,
%U A269424 1464,1520,1560,1608,1832,1848
%N A269424 Record (maximal) gaps between primes of the form 8k + 1.
%C A269424 Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 1 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 A269424 Conjecture: a(n) < phi(8)*log^2(A269426(n)) almost always.
%C A269424 A269425 lists the primes preceding the maximal gaps.
%C A269424 A269426 lists the corresponding primes at the end of the maximal gaps.
%H A269424 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 A269424 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 A269424 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 A269424 The first two primes of the form 8k + 1 are 17 and 41, so a(1)=41-17=24. The next prime of this form is 73 and the gap 73-41=32 is a new record, so a(2)=32.
%t A269424 re = 0; s = 17; Reap[For[p = 41, p < 10^8, p = NextPrime[p], If[Mod[p, 8] == 1, g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]]][[2, 1]] (* _Jean-François Alcover_, Oct 17 2016, adapted from PARI *)
%o A269424 (PARI) re=0; s=17; forprime(p=41, 1e8, if(p%8!=1, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
%Y A269424 Cf. A007519, A269425, A269426.
%K A269424 nonn
%O A269424 1,1
%A A269424 _Alexei Kourbatov_, Feb 25 2016