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

1,1

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.

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 + 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.

Prog

(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)

Crossrefs

Cf. A007522, A269520, A269521.

Sequence in context: A349241 A120142 A110228 * A175161 A045875 A046434

Adjacent sequences: A269516 A269517 A269518 * A269520 A269521 A269522

Keyword

nonn

Author

Alexei Kourbatov, Feb 28 2016

Status

approved