%I #14 Nov 14 2022 23:05:10
%S 2,5,11,13,47,41,31,107,43,73,131,61,191,97,293,139,353,127,163,151,0,
%T 229,283,223,659,181,929,313,241,211,367,701,271,397,379,457,337,1031,
%U 1259,607,331,463,643,613,1409,733,911,1091,541,1997,421,727,709,673
%N Least prime p which generates exactly n primes of the form p+q1 where q < p is prime, or 0 if (conjecturally) no such p exists.
%C a(n) = 0 for n = 20, 165, 467, ... . Do there exist infinitely many such values of n?
%C These values of 0 are all conjectural.  _Robert Israel_, Apr 28 2021
%e a(1) = 5 since 5 is the least prime that generates exactly one prime 7=5+31 of the given form. Again a(3) = 13 since 13 generates exactly 3 primes 17=13+51, 19=13+71 and 23=13+111 of the given form.
%t Cn[n_] := Module[{c}, p = Prime[n]; c = 0; i = 1; While[i < n, If[PrimeQ[p + Prime[i]  1], c = c + 1] i++]; c]; t = {};
%t Do[p = 0; j = 0; While[++j < 2000 && p != 1, If[Cn[j] == k, AppendTo[t, Prime[j]]; p = 1, p = 0]]; If[p == 0, AppendTo[t, 0]], {k, 0, 200}]; t
%Y Cf. A072669, A224748.
%K nonn
%O 0,1
%A _Jayanta Basu_, Apr 18 2013
%E Definition clarified by _Robert Israel_, Apr 28 2021
