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

1,1

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.

Links

Table of n, a(n) for n=1..39.

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 + 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.

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 *)

Prog

(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)

Crossrefs

Cf. A007519, A269425, A269426.

Sequence in context: A364353 A317534 A240068 * A319928 A025102 A188671

Adjacent sequences: A269421 A269422 A269423 * A269425 A269426 A269427

Keyword

nonn

Author

Alexei Kourbatov, Feb 25 2016

Status

approved