Divisor Count of 4030

Example of Use of Divisor Count Function

$\map {\sigma_0} {4030} = 16$

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:

$4030 = 2 \times 5 \times 13 \times 31$

Thus:

$\ds \map {\sigma_0} {4030}$

$=$

$\ds \map {\sigma_0} {2^1 \times 5^1 \times 13^1 \times 31^1}$

$\ds $

$=$

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

$\ds $

$=$

$\ds 16$

The divisors of $4030$ can be enumerated as:

$1, 2, 5, 10, 13, 26, 31, 62, 65, 130, 155, 310, 403, 806, 2015, 4030$

$\blacksquare$