A269424 Record (maximal) gaps between primes of the form 8k + 1.

24, 32, 56, 64, 88, 112, 120, 136, 160, 216, 232, 240, 264, 304, 384, 480, 488, 528, 544, 576, 624, 640, 720, 760, 816, 888, 960, 1032, 1064, 1200, 1296, 1320, 1432, 1464, 1520, 1560, 1608, 1832, 1848

Offset: 1

Author

Alexei Kourbatov, Feb 25 2016

Keywords

• nonn

Comments

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 1 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(A269426(n)) almost always.

A269425 lists the primes preceding the maximal gaps.

A269426 lists the corresponding primes at the end of the maximal gaps.

Examples

			The first two primes of the form 8k + 1 are 17 and 41, so a(1)=41-17=24. The next prime of this form is 73 and the gap 73-41=32 is a new record, so a(2)=32.
		

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. A007519, A269425, A269426.

Programs

Mathematica

re = 0; s = 17; Reap[For[p = 41, p < 10^8, p = NextPrime[p], If[Mod[p, 8] == 1, g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]]][[2, 1]] (* Jean-François Alcover, Oct 17 2016, adapted from PARI *)

PARI

re=0; s=17; forprime(p=41, 1e8, if(p%8!=1, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)