The Wilson quotient W(p) is defined as: If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer.
If p divides W(p), it is called a Wilson prime.
The integer values of W(p) are (sequence A007619 in the OEIS): It is known that[1] where
is the k-th Bernoulli number.
Note that the first relation comes from the second one by subtraction, after substituting