\begin{definition}[Definition:Primorial/Positive Integer]
Let $n$ be a positive integer.
Then:
:$\ds n\# := \prod_{i \mathop = 1}^{\map \pi n} p_i = p_{\map \pi n}\#$
where $\map \pi n$ is the prime counting function.
That is, $n\#$ is defined as the product of all primes less than or equal to $n$.
Thus:
:$n\# = \begin {cases}
1 & : n \le 1 \\
n \paren {\paren {n - 1}\#} & : \text {$n$ prime} \\
\paren {n - 1}\# & : \text {$n$ composite}
\end {cases}$
\end{definition}