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A330853
First occurrences of gaps between primes 6k+1: gap sizes.
10
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 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
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Apr 27 2020
STATUS
approved