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

1,1

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

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A002522, A005574, A048161.

Sequence in context: A213457 A060595 A303440 * A086619 A294373 A194026

Adjacent sequences: A382666 A382667 A382668 * A382670 A382671 A382672

Keyword

nonn

Author

Ya-Ping Lu, Apr 24 2025

Status

approved