A382669 Even numbers m such that both p = m^2 + 1 and q = (p^2 + 1)/2 are primes.

2, 10, 150, 160, 230, 270, 400, 890, 910, 920, 1060, 1430, 1550, 1970, 2700, 2960, 3280, 3290, 3520, 3660, 4140, 4330, 4510, 4700, 4780, 4850, 4920, 5180, 5360, 5500, 5560, 5620, 5880, 5960, 6220, 6460, 6980, 7160, 7190, 7520, 7550, 7820, 9630, 9760, 9900

Offset: 1

Author

Ya-Ping Lu, Apr 24 2025

Keywords

• nonn

Comments

Except 2, all terms are divisible by 10, and p-1 and q-1 are divisible by 100.

Numbers m such that p = m^2+1 and p + m^4/2 are both prime. - Chai Wah Wu, May 01 2025

Examples

			10 is a term because both 10^2 + 1 = 101 and (101^2 + 1)/2 = 5101 are primes.
		

Links

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

Crossrefs

Cf. A002522, A005574, A048161.

Programs

Maple

filter:= proc(m) local p;
  p:= m^2 + 1;
  isprime(p) and isprime((p^2+1)/2)
end proc:
select(filter, [2,seq(i,i=10..10000,10)]); # Robert Israel, May 02 2025

Mathematica

Select[2*Range[5000], PrimeQ[#^2 + 1] && PrimeQ[#^4/2 + #^2 + 1] &] (* Amiram Eldar, Apr 24 2025 *)

Python

from sympy import isprime
for n in range(2, 10000, 2): x = n*n + 1; ct = 0; print(n, end = ', ') if isprime(x) and isprime((x*x + 1)//2) else 0

Python

from itertools import count, islice
from sympy import isprime
def A382669_gen(): # generator of terms
    yield 2
    yield from filter(lambda m: isprime(p:=m**2+1) and isprime(p+(m**4>>1)),(10*k for k in count(1)))
A382669_list = list(islice(A382669_gen(),45)) # Chai Wah Wu, May 02 2025