A268925 Record (maximal) gaps between primes of the form 6k + 1.

6, 12, 18, 30, 54, 60, 78, 84, 90, 96, 114, 162, 174, 192, 204, 252, 270, 282, 312, 330, 336, 378, 462, 486, 522, 528, 534, 600, 606, 612, 642, 666, 780, 810, 894, 1002

Offset: 1

Author

Alexei Kourbatov, Feb 15 2016

Keywords

• nonn
• more

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(A268927(n)) almost always.

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

Examples

			The first two primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. The next prime of this form is 19; the gap 19-13 is not a record so nothing is added to the sequence. The next prime of this form is 31; the gap 31-19=12 is a new record, so a(2)=12.
		

Links

• 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.

Crossrefs

Cf. A002476, A268926 (primes preceding the maximal gaps), A268927 (primes at the end of maximal gaps), A330853, A330854.

Programs

Mathematica

re = 0; s = 7; Reap[For[p = 13, p < 10^8, p = NextPrime[p], If[Mod[p, 6] != 1, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* Jean-François Alcover, Dec 12 2018, from PARI *)
records[n_]:=Module[{ri=n,m=0,rcs={},len},len=Length[ri];While[len>0,If[ First[ri]>m,m=First[ri];AppendTo[rcs,m]];ri=Rest[ri];len--]; rcs]; records[ Differences[Select[6*Range[0,3*10^6]+1,PrimeQ]]] (* the program generates the first 30 terms of the sequence. *) (* Harvey P. Dale, Dec 19 2021 *)

PARI

re=0; s=7; forprime(p=13, 1e8, if(p%6!=1, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

Formula

a(n) = A268927(n) - A268926(n). - Alexei Kourbatov, Jun 21 2020