Divisor Count Function from Prime Decomposition

Theorem
Let $n$ be an integer such that $n \ge 2$.

Let the prime decomposition of $n$ be:
 * $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$

Let $\map \tau n$ be the tau function of $n$.

Then:
 * $\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$