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A321510 Primes p for which there exists a prime q < p such that 3*q == 1 (mod p).

%I A321510 #23 Dec 24 2018 08:28:51
%S A321510 5,7,19,43,61,79,109,151,163,223,271,349,421,439,523,601,613,631,673,
%T A321510 691,811,853,919,991,1009,1051,1063,1153,1213,1231,1279,1321,1429,
%U A321510 1531,1549,1663,1693,1789,1801,1873,1933,1951,2113,2143,2179,2221,2239,2503,2539,2683,2791,2833,2851
%N A321510 Primes p for which there exists a prime q < p such that 3*q == 1 (mod p).
%C A321510 A104163 with 5 prepended (see example). For any prime p in A104163 q = (2*p+1)/3, then q < p and 3*q == 1 (mod p).
%F A321510 a(n+1) = A104163(n); n >= 1.
%e A321510 For p = 11, the only number t < 11 such that 3*t == 1 (mod 11) is t = 4, which is not prime, therefore 11 is not a term.
%e A321510 For p = 5, q = 2 (prime); 2*3 = 6 == 1 (mod 5) therefore 5 is a term.
%p A321510 for n from 3 to 300 do
%p A321510 Y := ithprime(n);
%p A321510 Z := 1/3 mod Y;
%p A321510 if isprime(Z) then print(Y);
%p A321510 end if:
%p A321510 end do:
%t A321510 aQ[p_]:=Module[{ans=False, q=2}, While[q<p, If[Mod[3*q, p]==1, ans=True; Break[]]; q=NextPrime[q]]; ans]; Select[Prime[Range[350]], aQ] (* _Amiram Eldar_, Nov 12 2018 *)
%t A321510 Join[{5}, Select[Prime[Range[400]], PrimeQ[((2 # + 1)) / 3] &]] (* _Vincenzo Librandi_, Nov 17 2018 *)
%o A321510 (PARI) isok(p) = if (isprime(p), forprime(q=1, p-1, if ((3*q % p) == 1, return (1)))); \\ _Michel Marcus_, Nov 14 2018
%Y A321510 Cf. A104163 (essentially the same sequence), A005383.
%K A321510 nonn
%O A321510 1,1
%A A321510 _David James Sycamore_, Nov 11 2018