Wilson quotient

The Wilson quotient $W_n$ for a given positive integer $n$ is the rational number ${\displaystyle \frac{\mathrm{\Gamma }(n)+1}{n}}$, where $\mathrm{\Gamma }(x)$ is Euler’s Gamma function (since we’re dealing with integer inputs here, in effect this is merely a quicker way to write $(n-1)!$).

From Wilson’s theorem it follows that the Wilson quotient is an integer only if $n$ is not composite. When $n$ is composite, the numerator of the Wilson quotient is $(n-1)!+1$ and the denominator is $n$. For example, if $n=7$ we have numerator 721 with denominator 7, and since these have 7 as their greatest common divisor, in lowest terms the Wilson quotient of 7 is 103 (with 1 as tacit numerator). But for $n=8$ we have

$$W_8=\frac{5041}{8}.$$