A260554 Primes p such that p = q^2 + 4*r^2 where q and r are also primes.

41, 61, 109, 137, 149, 157, 269, 317, 389, 397, 461, 509, 557, 653, 701, 773, 797, 857, 877, 941, 977, 997, 1013, 1061, 1181, 1277, 1453, 1493, 1613, 1637, 1697, 1733, 1877, 1949, 1973, 1997, 2141, 2237, 2309, 2333, 2357, 2477, 2693, 2837, 2909, 2957, 3373

Offset: 1

Author

Colin Barker, Jul 29 2015

Keywords

• nonn

Comments

Green and Sawhney prove that this sequence is infinite, verifying a conjecture of Friedlander and Iwaniec. - Charles R Greathouse IV, Oct 08 2024

Empirically end digit of terms asymptotically tends to 3 or 7. - Bill McEachen, Jul 02 2025

Examples

			149 is in the sequence because 149 = 7^2 + 4*5^2 and 149, 7 and 5 are all primes.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1400
• John Friedlander and Henryk Iwaniec, Coordinate distribution of Gaussian primes, J. Eur. Math. Soc. (JEMS) 24 (2022), pp. 737-772. arXiv:1811.05507 [math.NT]
• Ben Green and Mehtaab Sawhney, Primes of the form p^2 + nq^2, arXiv preprint (2024). arXiv:2410.04189 [math.NT]

Crossrefs

Cf. A260553, A260555, A260556, A260557.

Supersequence of A182476.

Programs

Mathematica

Select[#1^2 + 4 #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)\4), my(t=4*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