

A269424


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


2



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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), 6578.
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)=4117=24. The next prime of this form is 73 and the gap 7341=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]] (* JeanFrançois Alcover, Oct 17 2016, adapted from PARI *)


PROG

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


CROSSREFS

Cf. A007519, A269425, A269426.
Sequence in context: A102374 A317534 A240068 * A319928 A025102 A188671
Adjacent sequences: A269421 A269422 A269423 * A269425 A269426 A269427


KEYWORD

nonn


AUTHOR

Alexei Kourbatov, Feb 25 2016


STATUS

approved



