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

where ${\displaystyle B_{k}}$ is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting ${\displaystyle t=1}$ and ${\displaystyle t=2}$.