Divisor Counting Function/Examples/108

Example of Use of Divisor Counting Function

 * $\map {\sigma_0} {108} = 12$

where $\sigma_0$ denotes the divisor counting function.

Proof
From Divisor Counting Function from Prime Decomposition:
 * $\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

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:
 * $108 = 2^2 \times 3^3$

Thus:

The divisors of $108$ can be enumerated as:
 * $1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108$