Divisor Count of 45

Example of Use of Divisor Counting Function

 * $\map \tau {45} = 6$

where $\tau$ denotes the divisor counting (tau) function.

Proof
From Tau Function from Prime Decomposition:
 * $\ds \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

where:
 * $r$ denotes the number of distinct prime factors in the prime decomposition of $n$
 * $k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.

We have that:
 * $45 = 3^2 \times 5$

Thus:

The divisors of $45$ can be enumerated as:
 * $1, 3, 5, 9, 15, 45$