A330853 First occurrences of gaps between primes 6k+1: gap sizes.

6, 12, 18, 30, 24, 54, 42, 36, 48, 60, 78, 66, 72, 84, 90, 96, 114, 102, 162, 108, 126, 120, 132, 150, 138, 144, 174, 168, 156, 192, 204, 180, 198, 252, 270, 216, 222, 186, 228, 210, 240, 282, 246, 234, 276, 264, 258, 312, 330, 318, 288, 306, 294, 336, 300, 378

Offset: 1

Author

Alexei Kourbatov, Apr 27 2020

Keywords

• nonn

Comments

Contains A268925 as a subsequence.

Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.

Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

Examples

			The first primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. The next prime 6k+1 is 19, and the gap 19-13=6 already occurred, so a new term is not added to the sequence. The next prime 6k+1 is 31, and the gap 31-19=12 is occurring for the first time; therefore a(2)=12.
		

Links

• Alexei Kourbatov, Table of n, a(n) for n = 1..135
• Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Crossrefs

Cf. A002476, A014320, A058320, A330854 (primes 6k+1 preceding the first-occurrence gaps), A330855 (primes 6k+1 at the end of the first-occurrence gaps).

Programs

PARI

isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

Formula

a(n) = A330855(n) - A330854(n).