A352604 Primes p such that p^2+3*p+1 and p^2+p-1 are also prime.

2, 3, 5, 19, 53, 59, 163, 263, 349, 373, 419, 449, 499, 1013, 1093, 1259, 1303, 1423, 1489, 1493, 1669, 1759, 2069, 2729, 2879, 3463, 3943, 4159, 4243, 4283, 4493, 4603, 4793, 4969, 5113, 5303, 5563, 6323, 6599, 6803, 6829, 6883, 7369, 7523, 7529, 7963, 8039, 8713, 8969, 9043, 9173, 9293, 9623

Offset: 1

Author

J. M. Bergot and Robert Israel, Mar 22 2022

Keywords

• nonn

Comments

Primes p such that (p-1)*p+(p-1)+p and p*(p+1)+p+(p+1) are also prime.

Examples

			a(3) = 5 is a term because 5, 5^2+3*5+1 = 41 and 5^2+5-1 = 29 are all prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A053184 and A153590.

Programs

Maple

select(t -> isprime(t^2+3*t+1) and isprime(t^2+t-1), [seq(ithprime(i),i=1..10000)]);

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p = 2
    while True:
        if isprime(p**2 + 3*p + 1) and isprime(p**2 + p - 1):
            yield p
        p = nextprime(p)
print(list(islice(agen(), 53))) # Michael S. Branicky, Mar 22 2022