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Record (maximal) gaps between primes of the form 10k + 3.
2

%I #16 Jan 17 2019 15:44:20

%S 10,20,50,70,80,90,100,110,120,130,150,300,360,420,500,510,540,550,

%T 610,630,650,690,780,810,820,840,870,890,960,990,1280,1370,1380,1470,

%U 1690

%N Record (maximal) gaps between primes of the form 10k + 3.

%C Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 10k + 3 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(A269236(n)) almost always.

%C A269235 lists the primes preceding the maximal gaps.

%C A269236 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 + 3 are 3 and 13, so a(1)=13-3=10. The next prime of this form is 23; the gap 23-13 is not a record so nothing is added to the sequence. The next prime of this form is 43 and the gap 43-23=20 is a new record, so a(2)=20.

%o (PARI) re=0; s=3; forprime(p=13, 1e8, if(p%10!=3, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

%Y Cf. A030431, A269235, A269236.

%K nonn

%O 1,1

%A _Alexei Kourbatov_, Feb 20 2016