Divisor Count of 17,796,126,877,482,329,126,053

Example of Use of Divisor Count Function

$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053} = 8$

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:

$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053 = 449 \times 11 \, 618 \, 801 \times 3 \, 411 \, 283 \, 698 \, 997$

Thus:

$\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053}$

$=$

$\ds \map {\sigma_0} {449^1 \times 11 \, 618 \, 801^1 \times 3 \, 411 \, 283 \, 698 \, 997^1}$

$\ds $

$=$

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

$\ds $

$=$

$\ds 2^3$

$\ds $

$=$

$\ds 8$

$\blacksquare$