Regarding Wilson's Theorem

I was just wondering something

(p-1)! = -1 mod p

I was investigating the number (p-1)! + 1, which is divisible by p. I was wondering if n = ((p-1)! + 1)/p, then does the prime factorization of n consists of primes all of which are to the power of 1 (except for the case of p = 2, since n = 1)?

I couldn't really test much because of the factorial, it makes the numbers huge ... I tried till p = 37 ... after that I get problems with factoring the huge number generated by the factorial ...

Could someone test out a few more primes please ... or if someone is familiar with this idea, could you point me to more information about this ... thanks! :)