OFFSET
1,1
COMMENTS
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.
Conjecture: a(n) < phi(10)*log^2(A269236(n)) almost always.
A269235 lists the primes preceding the maximal gaps.
A269236 lists the corresponding primes at the end of the maximal gaps.
LINKS
Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
EXAMPLE
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.
PROG
(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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Feb 20 2016
STATUS
approved