A269234 Record (maximal) gaps between primes of the form 10k + 3.

10, 20, 50, 70, 80, 90, 100, 110, 120, 130, 150, 300, 360, 420, 500, 510, 540, 550, 610, 630, 650, 690, 780, 810, 820, 840, 870, 890, 960, 990, 1280, 1370, 1380, 1470, 1690

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

Table of n, a(n) for n=1..35.

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

Cf. A030431, A269235, A269236.

Sequence in context: A007927 A284991 A327692 * A160517 A072081 A034087

Adjacent sequences: A269231 A269232 A269233 * A269235 A269236 A269237

Keyword

nonn

Author

Alexei Kourbatov, Feb 20 2016

Status

approved