A269519 Record (maximal) gaps between primes of the form 8k + 7.

16, 24, 40, 48, 96, 184, 200, 216, 288, 296, 312, 344, 384, 456, 504, 560, 624, 744, 760, 776, 800, 824, 840, 864, 880, 896, 952, 984, 1008, 1056, 1152, 1208, 1312, 1384, 1448, 1464, 1472, 1720, 1872

Offset: 1

Author

Alexei Kourbatov, Feb 28 2016

Keywords

• nonn

Comments

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 7 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(A269521(n)) almost always.

A269520 lists the primes preceding the maximal gaps.

A269521 lists the corresponding primes at the end of the maximal gaps.

Examples

			The first two primes of the form 8k + 7 are 7 and 23, so a(1)=23-7=16. The next primes of this form are 31, 47; the gaps 31-23 and 47-31 are not records so nothing is added to the sequence. The next prime of this form is 71 and the gap 71-47=24 is a new record, so a(2)=24.
		

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.

Crossrefs

Cf. A007522, A269520, A269521.

Programs

PARI

re=0; s=7; forprime(p=23, 1e8, if(p%8!=7, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)