A260553 Primes p such that p = q^2 + 2*r^2 where q and r are also primes.

17, 43, 59, 67, 107, 139, 251, 307, 347, 379, 547, 587, 859, 1699, 1867, 1931, 3371, 3499, 3739, 4507, 5059, 5347, 6907, 6971, 7451, 10091, 10627, 10667, 11467, 12491, 18787, 20411, 21227, 22907, 29947, 32059, 32779, 37547, 38651, 39619, 49307, 49747, 53147

Offset: 1

Author

Colin Barker, Jul 29 2015

Keywords

• nonn

Examples

			43 is in the sequence because 43 = 5^2 + 2*3^2 and 43, 5 and 3 are all primes.
		

Links

• Colin Barker and Chai Wah Wu, Table of n, a(n) for n = 1..1873 n = 1..150 from Colin Barker.

Crossrefs

Main entry for this sequence is A201613.

Cf. A260554, A260555, A260556, A260557.

Programs

Mathematica

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

PARI

lista(nn)=forprime(p=2, nn, forprime(r=2, sqrtint(p\2), if (issquare(q2 = p-2*r^2) && isprime(sqrtint(q2)), print1(p, ", ")););); \\ Michel Marcus, Jul 29 2015

PARI

list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\2), my(t=2*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 09 2024

Python

from sympy import prime, isprime
n = 5000
A260553_list, plimit = [], prime(n)**2+8
for i in range(1, n):
    q = 2*prime(i)**2
    for j in range(1, n):
        p = q + prime(j)**2
        if p < plimit and isprime(p):
            A260553_list.append(p)
A260553_list = sorted(A260553_list) # Chai Wah Wu, Jul 30 2015