|
%I #14 Jun 02 2020 22:11:55
%S 6,12,18,30,24,36,42,54,48,60,84,66,78,72,126,90,102,108,114,96,120,
%T 150,138,162,132,144,168,246,156,180,186,240,204,192,216,198,210,174,
%U 258,252,222,234,228,318,282,264,276,342,306,294,312,270,354,372
%N First occurrences of gaps between primes 6k - 1: gap sizes.
%C Contains A268928 as a subsequence. First differs from A268928 at a(5)=24.
%C Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
%C Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.
%H Alexei Kourbatov, <a href="/A334543/b334543.txt">Table of n, a(n) for n = 1..160</a>
%H Alexei Kourbatov and Marek Wolf, <a href="https://arxiv.org/abs/2002.02115">On the first occurrences of gaps between primes in a residue class</a>, arXiv preprint arXiv:2002.02115 [math.NT], 2020.
%F a(n) = A334545(n) - A334544(n).
%e 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 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 has not occurred before, so a(2)=12.
%o (PARI) isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)
%Y Cf. A007528, A014320, A058320, A268928, A330853, A334544, A334545.
%K nonn
%O 1,1
%A _Alexei Kourbatov_, May 05 2020
|
