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%I #25 Apr 23 2021 07:35:57
%S 7,23,47,167,251,359,389,839,941,1367,1847,1889,2207,2417,3719,3761,
%T 4889,5039,6311,7079,7919,8609,9377,10607,11411,11447,13841,15227,
%U 16127,17159,18869,19319,20411,24611,25589,26669,29501,29927,36017,36479,37907,43037,44519,44927,45569,49727,50627,52889,54287,57119,62057,65309,66047,70529,85037,85847,95369,97967,99191
%N Fajtlowicz p-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 proved that the series of reciprocal of the p-primes converges.
%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="/A185955/b185955.txt">Table of n, a(n) for n = 1..500</a> (terms 1..201 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 A185955 := proc(n)
%p option remember;
%p local a, admit,k ;
%p if n = 1 then
%p 7;
%p else
%p a := ithprime(n) ;
%p while true do
%p if not isprime(a+2) then
%p admit := true ;
%p for k from 1 to n do
%p if modp(a+2, A185956(k)) =0 then
%p admit := false;
%p break;
%p end if;
%p end do:
%p if admit then
%p return a;
%p end if ;
%p end if;
%p a := nextprime(a) ;
%p end do:
%p end if;
%p end proc ;
%p seq(A185955(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[p, 50] (* _Amiram Eldar_, Apr 23 2021 *)
%Y Cf. A185956.
%K nonn
%O 1,1
%A _Craig Eric Larson_, Feb 07 2011
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