A269513 Record (maximal) gaps between primes of the form 8k + 5.

8, 16, 40, 48, 56, 64, 72, 80, 88, 96, 112, 128, 144, 192, 216, 224, 264, 288, 296, 360, 368, 440, 456, 480, 608, 616, 672, 752, 760, 856, 912, 920, 960, 1128, 1176, 1216, 1424, 1432, 1440, 1464, 1480, 1552, 1728, 1872

Offset

1,1

Comments

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 5 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(A269515(n)) almost always.

A269514 lists the primes preceding the maximal gaps.

A269515 lists the corresponding primes at the end of the maximal gaps.

Links

Table of n, a(n) for n=1..44.

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 + 5 are 5 and 13, so a(1)=13-5=8. The next prime of this form is 29 and the gap 29-13=16 is a new record, so a(2)=16.

Prog

(PARI) re=0; s=5; forprime(p=13, 1e8, if(p%8!=5, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

Crossrefs

Cf. A007521, A269514, A269515.

Sequence in context: A057584 A063526 A156331 * A024700 A108576 A052207

Adjacent sequences: A269510 A269511 A269512 * A269514 A269515 A269516

Keyword

nonn

Author

Alexei Kourbatov, Feb 28 2016

Status

approved