# Definition:Prime Exponent Function

## Definition

Let $n \in \N$ be a natural number.

Let the prime decomposition of $n$ be given as:

$\displaystyle n = \prod_{j \mathop = 1}^k \left({p \left({j}\right)}\right)^{a_j}$

where $p \left({j}\right)$ is the prime enumeration function.

Then the exponent $a_j$ of $p \left({j}\right)$ in $n$ is denoted $\left({n}\right)_j$.

If $p \left({j}\right)$ does not divide $n$, then $\left({n}\right)_j = 0$.

We also define:

$\forall n \in \N: \left({n}\right)_0 = 0$
$\forall j \in \N: \left({0}\right)_j = 0$
$\forall j \in \N: \left({1}\right)_j = 0$