A269420 Record (maximal) gaps between primes of the form 8k + 3.

8, 24, 32, 40, 48, 72, 120, 144, 152, 176, 216, 264, 320, 400, 520, 592, 600, 824, 856, 872, 936, 992, 1064, 1072, 1112, 1336, 1392, 1408, 1584, 1720, 2080

Offset

1,1

Comments

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 3 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.

Conjecture: a(n) < phi(8)*log^2(A269422(n)) almost always.

A269421 lists the primes preceding the maximal gaps.

A269422 lists the corresponding primes at the end of the maximal gaps.

Links

Table of n, a(n) for n=1..31.

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 8k + 3 are 3 and 11, so a(1)=11-3=8. The next prime of this form is 19; the gap 19-11 is not a record so nothing is added to the sequence. The next prime of this form is 43 and the gap 43-19=24 is a new record, so a(2)=24.

Prog

(PARI) re=0; s=3; forprime(p=11, 1e8, if(p%8!=3, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

Crossrefs

Cf. A007520, A269421, A269422.

Sequence in context: A028644 A227175 A340930 * A056196 A044069 A028628

Adjacent sequences: A269417 A269418 A269419 * A269421 A269422 A269423

Keyword

nonn

Author

Alexei Kourbatov, Feb 25 2016

Status

approved