Divisor Count of 40,314

Example of Use of Divisor Count Function

$\map {\sigma_0} {40 \, 314} = 8$

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:

$40 \, 314 = 2 \times 3 \times 6719$

Thus:

\(\ds \map {\sigma_0} {40 \, 314}\)

\(=\)

\(\ds \map {\sigma_0} {2^1 \times 3^1 \times 6719^1}\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 8\)

The divisors of $40 \, 314$ can be enumerated as:

$1, 2, 3, 6, 6719, 13 \, 438, 20 \, 157, 40 \, 314$

$\blacksquare$