\begin{definition}[Definition:Prime Exponent Function]
Let $n \in \N$ be a natural number.
Let the prime decomposition of $n$ be given as:
:$\ds n = \prod_{j \mathop = 1}^k \paren {\map p j}^{a_j}$
where $\map p j$ is the prime enumeration function.
Then the exponent $a_j$ of $\map p j$ in $n$ is denoted $\paren n_j$.
If $\map p j$ does not divide $n$, then $\paren n_j = 0$.
We also define:
:$\forall n \in \N: \paren n_0 = 0$
:$\forall j \in \N: \paren 0_j = 0$
:$\forall j \in \N: \paren 1_j = 0$
Category:Definitions/Mathematical Logic
\end{definition}