Divisor Count Function/Examples/12

Example of Use of $\tau$ Function
The value of the $\tau$ function for the integer $12$ is $6$.

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

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:
 * $12 = 2^2 \times 3$

Thus:
 * $\tau \left({12}\right) = \tau \left({2^2 \times 3^1}\right) = \left({2 + 1}\right) \left({1 + 1}\right) = 6$

The divisors of $12$ can be enumerated as:
 * $1, 2, 3, 4, 6, 12$