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A268928 Record (maximal) gaps between primes of the form 6k - 1. 4
6, 12, 18, 30, 36, 42, 54, 60, 84, 126, 150, 162, 168, 246, 258, 318, 342, 354, 372, 408, 468, 534, 552, 600, 654, 762, 768, 798, 864, 894, 942, 960, 1068, 1224, 1302, 1320, 1344 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Dirichlet's theorem on arithmetic progressions and the GRH suggest that average gaps between primes of the form 6k - 1 below x are about phi(6)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(6)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(6)=2.

Conjecture: a(n) < phi(6)*log^2(A268930(n)) almost always.

Conjecture: phi(6)*n^2/6 < a(n) < phi(6)*n^2 almost always. - Alexei Kourbatov, Nov 27 2019

LINKS

Table of n, a(n) for n=1..37.

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.

FORMULA

a(n) = A268930(n) - A268929(n). - Alexei Kourbatov, Jun 15 2020.

EXAMPLE

The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=12.

MATHEMATICA

re = 0; s = 5; Reap[For[p = 11, p < 10^8, p = NextPrime[p], If[Mod[p, 6] != 5, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* Jean-Fran├žois Alcover, Dec 12 2018, from PARI *)

PROG

(PARI) re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

CROSSREFS

Cf. A007528, A268929 (primes preceding the maximal gaps), A268930 (primes at the end of the maximal gaps), A334543, A334544.

Sequence in context: A330853 A268657 A232742 * A162864 A198682 A268925

Adjacent sequences:  A268925 A268926 A268927 * A268929 A268930 A268931

KEYWORD

nonn,more

AUTHOR

Alexei Kourbatov, Feb 15 2016

EXTENSIONS

Terms a(31)..a(37) from Alexei Kourbatov, Jun 15 2020

STATUS

approved

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Last modified May 17 20:44 EDT 2021. Contains 343990 sequences. (Running on oeis4.)