Divisor Count of 40,310

Example of Use of Divisor Count Function

$\map {\sigma_0} {40 \, 310} = 8$

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:

$40 \, 310 = 2 \times 5 \times 29 \times 139$

Thus:

$\ds \map {\sigma_0} {40 \, 310}$

$=$

$\ds \map {\sigma_0} {2^1 \times 5^1 \times 29^1 \times 139^1}$

$\ds $

$=$

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

$\ds $

$=$

$\ds 16$

The divisors of $40 \, 310$ can be enumerated as:

$1, 2, 5, 10, 29, 58, 139, 145, 278, 290, 695, 1390, 4031, 8062, 20 \, 155, 40 \, 310$

$\blacksquare$