A073443 Numbers k such that k! - k - 1 is prime.

3, 4, 10, 12, 346

Offset: 1

Author

Rick L. Shepherd, Jul 31 2002

Keywords

• nonn
• hard
• more

Comments

Clearly n <> 2 (mod 3). For n>3, n!-n, n!-n+1, ..., n!-3, n!-2 is a sequence of n-1 consecutive composite numbers. Additional terms are greater than 2000.

a(5) > 7500. - Michael S. Branicky, Mar 04 2021

a(5) > 24000. - Michael S. Branicky, Nov 13 2024

Crossrefs

Cf. A073444 (corresponding primes), A002982 (n!-1 is prime), A073308 (n!+n+1 is prime).

Programs

Mathematica

Select[Range[3, 346], PrimeQ[#! - # - 1] &] (* Arkadiusz Wesolowski, Jan 04 2012 *)

PARI

for(n=3,2000,if(isprime(n!-n-1),print1(n,",")))

Python

from math import factorial
from sympy import isprime
def ok(n): return isprime(factorial(n) - n - 1)
print([m for m in range(3, 500) if ok(m)]) # Michael S. Branicky, Mar 04 2021

Extensions

Offset corrected by Arkadiusz Wesolowski, Jan 04 2012