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%I #31 Apr 23 2021 07:36:10
%S 2,3,5,7,13,11,19,17,29,23,37,43,31,47,41,61,53,67,71,59,73,89,79,83,
%T 103,101,107,109,97,127,131,113,139,137,151,157,149,163,173,181,191,
%U 167,193,211,179,199,223,197,227,233,239,229,241,257,251,277,293,283,313,281,271,307,269,337,263,311,347,317,359,353,367,331,373,379,349,389,383,401,397,421,409,443,431,419,449,433,439,467,457,463,491,487,479
%N Fajtlowicz q-primes.
%C S. Fajtlowicz defined two related sequences of primes, p(n) and q(n), as follows:
%C 1. q(1)=2 and p(1)=7.
%C 2. q(n+1) is the smallest prime dividing p(n)+2.
%C 3. p(n+1) is the smallest prime p larger than p(n) such that p+2 is not prime and not divisible by any of q(1),q(2),...,q(n+1).
%C Paul Erdős and C. Larson conjecture that all primes occur in the sequence of q's.
%C The values of p and q were computed by Bethany Turner.
%D Siemion Fajtlowicz, Written on the Wall: Conjectures of Graffiti, #784 (1994).
%H Amiram Eldar, <a href="/A185956/b185956.txt">Table of n, a(n) for n = 1..500</a> (terms 1..218 from R. J. Mathar)
%H Siemion Fajtlowicz, <a href="https://web.archive.org/web/20090302092953/https://www.math.uh.edu/~clarson/graffiti.html">Graffity & automated conjecture making</a> (2009), click on "conjectures up to No. 894", see page 136.
%p A185956 := proc(n)
%p option remember;
%p if n = 1 then
%p 2;
%p else
%p A020639(2+A185955(n-1)) ;
%p end if;
%p end proc:
%p seq(A185956(n),n=1..20) ; # _R. J. Mathar_, Jul 28 2019
%t lpf[n_] := FactorInteger[n][[1, 1]]; q[1] = 2; p[1] = 7; q[n_] := q[n] = lpf[p[n - 1] + 2]; p[n_] := Module[{pn = NextPrime[p[n - 1]]}, While[PrimeQ[pn + 2] || AnyTrue[Array[q, n], Divisible[pn + 2, #] &], pn = NextPrime[pn]]; pn]; Array[q, 50] (* _Amiram Eldar_, Apr 23 2021 *)
%Y Cf. A185955.
%K nonn
%O 1,1
%A _Craig Eric Larson_, Feb 07 2011
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