A274525 Prime numbers p such that p - 2, p^2 - p - 1, p^2 - p + 1 are prime numbers.

7, 139, 1789, 2731, 4159, 5641, 13339, 13399, 19429, 21739, 22369, 32059, 32911, 33601, 42571, 45319, 54541, 55339, 65449, 68821, 106189, 108499, 111871, 132859, 136399, 138079, 141511, 142981, 148201, 149629, 152041, 152839, 173431, 174049, 178249

Offset: 1

Author

Pierre CAMI, Jun 27 2016

Keywords

• nonn

Examples

			5 - 2 = 3 prime, 5 prime, 5*(5-1) - 1 = 19 prime, 5*(5-1) + 1 = 21 composite, so 5 is not in the sequence.
7 - 2 = 5 prime, 7 prime, 7*(7-1) - 1 = 41 prime, 7*(7-1) + 1 = 43 prime so 7 is in the sequence.
		

Links

• Pierre CAMI, Table of n, a(n) for n = 1..30000

Crossrefs

Cf. A228968.

Programs

Mathematica

Select[Prime[Range[100]], Union[PrimeQ[{# - 2, #^2 - # - 1, #^2 - # + 1}]] == {True} &] (* Alonso del Arte, Jun 27 2016 *)
Select[Prime[Range[17000]],AllTrue[{#-2,#^2-#-1,#^2-#+1},PrimeQ]&] (* Harvey P. Dale, Jun 20 2024 *)

PARI

lista(nn) = forprime(p=2, nn, if (isprime(p-2) && isprime(p^2-p-1) && isprime(p^2-p+1), print1(p, ", "))); \\ Michel Marcus, Jul 07 2016

Python

from sympy import isprime, primerange
def aupto(n):
    t = []
    for p in primerange(2, n+1):
        if isprime(p-2) and isprime(p**2 - p - 1) and isprime(p**2 - p + 1):
            t.append(p)
    return t # Paul Muljadi, Jun 21 2024