A297306 Primes p such that q = 4*p+1 and r = (2*p+1)/3 are also primes.

7, 43, 79, 163, 673, 853, 919, 1063, 1429, 1549, 1663, 2143, 2683, 3229, 3499, 4993, 5119, 5653, 5779, 6229, 6343, 7333, 7459, 7669, 8353, 8539, 8719, 9829, 10009, 10243, 10303, 11383, 11689, 12583, 13399, 14149, 14653, 14923, 15649, 16603, 17053, 17389, 17749

Offset: 1

Author

David S. Newman, Jan 04 2018

Keywords

• nonn

Comments

This sequence was suggested by Moshe Shmuel Newman. It has its source in his study of finite groups.

Examples

			Prime p = 7 is in the sequence because q = 4*7+1 = 29 and r = (2*7+1)/3 = 5 are also primes.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040.

Intersection of A023212 and A104163.

Programs

Maple

a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p); if irem(p, 3)=1 and
         isprime(4*p+1) and isprime((2*p+1)/3) then break fi
      od; p
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 07 2018

Mathematica

a[n_] := a[n] = Module[{p}, p = If[n == 1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[Mod[p, 3] == 1 && PrimeQ[4p+1] && PrimeQ[(2p+1)/3], Break[]]]; p];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

PARI

isok(p) = isprime(p) && isprime(4*p+1) && iferr(isprime((2*p+1)/3), E, 0); \\ Michel Marcus, Nov 27 2020

Extensions

More terms from Alois P. Heinz, Jan 07 2018