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

1,1

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.

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.

Formula

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

Example

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.

Prog

(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)

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).

Sequence in context: A088345 A057826 A334543 * A268657 A232742 A268928

Adjacent sequences: A330850 A330851 A330852 * A330854 A330855 A330856

Keyword

nonn

Author

Alexei Kourbatov, Apr 27 2020

Status

approved