A269238 Record (maximal) gaps between primes of the form 10k + 7.

10, 20, 30, 60, 90, 130, 150, 180, 210, 240, 280, 380, 420, 430, 450, 470, 500, 530, 630, 640, 660, 780, 840, 920, 960, 970, 990, 1030, 1070, 1130, 1170, 1190, 1230, 1330, 1610, 1680, 2030

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

Table of n, a(n) for n=1..37.

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

Cf. A030432, A269239, A269240.

Sequence in context: A093037 A355695 A273016 * A279420 A274051 A345069

Adjacent sequences: A269235 A269236 A269237 * A269239 A269240 A269241

Keyword

nonn

Author

Alexei Kourbatov, Feb 20 2016

Status

approved