A260555 Primes p such that p = q^2 + 6*r^2 where q and r are also primes.

73, 79, 103, 193, 199, 223, 271, 313, 439, 463, 751, 823, 991, 1039, 1063, 1087, 1303, 1423, 1543, 1567, 1663, 1759, 1783, 1831, 1873, 1999, 2143, 2287, 2383, 2503, 2833, 3343, 3463, 3583, 3631, 3823, 3847, 3943, 4447, 4513, 4639, 4783, 5023, 5167, 5407

Offset: 1

Author

Colin Barker, Jul 29 2015

Keywords

• nonn

Comments

Green & Sawhney prove that this sequence is infinite. - Charles R Greathouse IV, Oct 08 2024

Examples

			73 is in the sequence because 73 = 7^2 + 6*2^2 and 73, 7 and 2 are all primes.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1500
• Ben Green and Mehtaab Sawhney, Primes of the form p^2 + nq^2, arXiv preprint (2024). arXiv:2410.04189 [math.NT]

Crossrefs

Cf. A260553, A260554, A260556, A260557.

Programs

Mathematica

Select[#1^2 + 6 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 60, 2], PrimeQ] // Sort (* Michael De Vlieger, Jul 29 2015 *)

PARI

list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\6), my(t=6*q^2); forprime(p=3, sqrtint(lim-t), my(r=t+p^2); if(isprime(r), listput(v, r)))); Set(v) \\ Charles R Greathouse IV, Oct 08 2024