cp's OEIS Frontend

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

%I A260554 #18 Jul 06 2025 23:56:04
%S A260554 41,61,109,137,149,157,269,317,389,397,461,509,557,653,701,773,797,
%T A260554 857,877,941,977,997,1013,1061,1181,1277,1453,1493,1613,1637,1697,
%U A260554 1733,1877,1949,1973,1997,2141,2237,2309,2333,2357,2477,2693,2837,2909,2957,3373
%N A260554 Primes p such that p = q^2 + 4*r^2 where q and r are also primes.
%C A260554 Green and Sawhney prove that this sequence is infinite, verifying a conjecture of Friedlander and Iwaniec. - _Charles R Greathouse IV_, Oct 08 2024
%C A260554 Empirically end digit of terms asymptotically tends to 3 or 7. - _Bill McEachen_, Jul 02 2025
%H A260554 Colin Barker, <a href="/A260554/b260554.txt">Table of n, a(n) for n = 1..1400</a>
%H A260554 John Friedlander and Henryk Iwaniec, <a href="https://arxiv.org/abs/1811.05507">Coordinate distribution of Gaussian primes</a>, J. Eur. Math. Soc. (JEMS) 24 (2022), pp. 737-772. arXiv:1811.05507 [math.NT]
%H A260554 Ben Green and Mehtaab Sawhney, <a href="https://arxiv.org/abs/2410.04189">Primes of the form p^2 + nq^2</a>, arXiv preprint (2024). arXiv:2410.04189 [math.NT]
%e A260554 149 is in the sequence because 149 = 7^2 + 4*5^2 and 149, 7 and 5 are all primes.
%t A260554 Select[#1^2 + 4 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 60, 2], PrimeQ] // Sort (* _Michael De Vlieger_, Jul 29 2015 *)
%o A260554 (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
%Y A260554 Cf. A260553, A260555, A260556, A260557.
%Y A260554 Supersequence of A182476.
%K A260554 nonn
%O A260554 1,1
%A A260554 _Colin Barker_, Jul 29 2015