A319304 Integers q for which f(q) = ((((q - 1)! + 1) / q) + 1) / (q + 1) is a prime number.

7, 17, 31, 67, 89

Offset: 1

Author

Rashid Naimi, Sep 16 2018

Keywords

• nonn
• hard
• more

Comments

f(89) is a 131-digit prime.

According to Wilson's theorem, f(q) can be an integer only if q is prime.

a(6) > 30000. - Michael S. Branicky, Apr 19 2025

Crossrefs

Cf. A193447, A319224.

Programs

Magma

[n: n in  [1..100] | IsPrime(n) and IsPrime((((Factorial(n-1)+1) div n)+1) div (n+1))]; // Vincenzo Librandi, Sep 21 2018

Mathematica

Select[Prime[Range[100]], PrimeQ[((((# - 1)! + 1) / #) + 1) / (# + 1)] &] (* Vincenzo Librandi, Sep 21 2018 *)

PARI

forprime(q=7, 89, my(p = ((((q - 1)! + 1) / q) + 1) / (q + 1)); if(ispseudoprime(p), print1(q, ", ")))

Extensions

1621 and 1699, which do not belong here, removed by Rashid Naimi, Mar 21 2019