Divisor Count of 18

Example of Use of $\tau$ Function

 * $\map \tau {18} = 6$

where $\tau$ denotes the $\tau$ Function.

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

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:
 * $18 = 2 \times 3^2$

Thus:
 * $\map \tau {18} = \map \tau {2^1 \times 3^2} = \paren {1 + 1} \paren {2 + 1} = 6$

The divisors of $18$ can be enumerated as:
 * $1, 2, 3, 6, 9, 18$