Divisor Count of 14

Example of Use of Divisor Count Function

$\map {\sigma_0} {14} = 4$

where $\sigma_0$ denotes the divisor count function.

Proof

From Divisor Count 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:

$14 = 2 \times 7$

Thus:

\(\ds \map {\sigma_0} {14}\)

\(=\)

\(\ds \map {\sigma_0} {2^1 \times 7^1}\)

\(\ds \)

\(=\)

\(\ds \paren {1 + 1} \paren {1 + 1}\)

\(\ds \)

\(=\)

\(\ds 4\)

The divisors of $14$ can be enumerated as:

$1, 2, 7, 14$

$\blacksquare$