Divisor Count of 9240

Example of Use of Divisor Count Function

$\map {\sigma_0} {9240} = 64$

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:

$9240 = 2^3 \times 3 \times 5 \times 7 \times 11$

Thus:

$\ds \map {\sigma_0} {9240}$

$=$

$\ds \map {\sigma_0} {2^3 \times 3^1 \times 5^1 \times 7^1 \times 11^1}$

$\ds $

$=$

$\ds \paren {3 + 1} \paren {1 + 1} \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}$

$\ds $

$=$

$\ds 64$

The divisors of $9240$ can be enumerated as:

$1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 20, 21, 22, 24, 28, 30, 33, 35, 40, 42,$

$44, 55, 56, 60, 66, 70, 77, 84, 88, 105, 110, 120, 132, 140, 154, 165, 168, 210, 220, 231,$

$264, 280, 308, 330, 385, 420, 440, 462, 616, 660, 770, 840, 924, 1155, 1320, 1540, 1848, 2310, 3080, 4620, 9240$

$\blacksquare$