OFFSET
1,1
COMMENTS
Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 10k + 7 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(A269240(n)) almost always.
A269239 lists the primes preceding the maximal gaps.
A269240 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 + 7 are 7 and 17, so a(1)=17-7=10. The next prime of this form is 37 and the gap 37-17=20 is a new record, so a(2)=20.
PROG
(PARI) re=0; s=7; forprime(p=17, 1e8, if(p%10!=7, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Feb 20 2016
STATUS
approved