Number of Abelian Groups

Theorem
Let $n \in \Z_{\ge 1}$ be a (strictly) positive integer.

Let:
 * $n = \ds \prod_{i \mathop = 1}^s p_i^{m_i}$

where the $p_i$ are distinct primes.

Let $\map {\nu_a} n$ denote the number of abelian groups of order $n$.

Then:
 * $\map {\nu_a} n = \ds \prod_{i \mathop = 1}^s \map {\nu_a} {p_i^{m_i} }$

where:
 * $\map {\nu_a} {p_i^{m_i} }$ is the number of integer partitions of $m_i$.