Divisor Count of 1638

Example of Use of Divisor Count Function

$\map {\sigma_0} {1638} = 24$

where $\sigma_0$ denotes the divisor count function.

$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

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:

$1638 = 2 \times 3^2 \times 7 \times 13$

Thus:

$\ds \map {\sigma_0} {1638}$

$=$

$\ds \map {\sigma_0} {2^1 \times 3^2 \times 7^1 \times 13^1}$

$\ds $

$=$

$\ds \paren {1 + 1} \paren {2 + 1} \paren {1 + 1} \paren {1 + 1}$

$\ds $

$=$

$\ds 24$

The divisors of $1638$ can be enumerated as:

$1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638$

$\blacksquare$