A355644 Primes p such that p^2-1 does not have a divisor d with d + (p^2-1)/d prime.

2, 3, 467, 487, 787, 887, 1279, 2063, 2557, 2657, 2903, 3323, 3413, 3547, 3583, 4273, 4373, 4517, 4567, 4801, 5233, 5393, 5443, 6047, 6823, 6911, 7507, 9133, 9137, 9721, 9973, 10103, 10313, 10937, 12227, 12763, 13183, 13627, 14407, 15073, 15083, 15187, 15359, 15787, 16903, 17047, 17911, 18013, 18587

Offset: 1

Author

J. M. Bergot and Robert Israel, Jul 11 2022

Keywords

• nonn

Comments

Primes p such that p^2-1 is not in A355643.

Examples

			a(2) = 3 is a term because it is prime, the divisors of 3^2-1 = 8 are 1, 2, 4 and 8, and none of 1+8/1 = 9, 2+8/2 = 6, 4+8/4 = 6, 8+8/8 = 9 are prime.
		

Links

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

Crossrefs

Cf. A355643.

Programs

Maple

filter:= proc(p) local n,F,t;
    n:= p^2-1;
  F:= select(t -> t^2 <=n, numtheory:-divisors(n));
  not ormap(isprime, map(t -> t+n/t, F))
end proc:
select(filter, [seq(ithprime(i),i=1..3000)]);

Mathematica

q[n_] := AllTrue[Divisors[n], !PrimeQ[# + n/#] &]; Select[Prime[Range[2000]], q[#^2 - 1] &] (* Amiram Eldar, Jul 11 2022 *)

PARI

isok(p) = isprime(p) && fordiv(p^2-1, d, if (isprime(d + (p^2-1)/d), return(0))); return(1); \\ Michel Marcus, Jul 11 2022

Python

from sympy import divisors, isprime
def ok(n):
    if not isprime(n): return False
    t = n**2 - 1
    return not any(isprime(d+t//d) for d in divisors(t, generator=True))
print([k for k in range(19000) if ok(k)]) # Michael S. Branicky, Jul 11 2022