Divisor Count of 17,796,126,877,482,329,126,044

Example of Use of Divisor Count Function

$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044} = 48$

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:

$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044 = 2^2 \times 7 \times 4327 \times 456 \, 293 \times 321 \, 911 \, 699 \, 243$

Thus:

$\ds $

$\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044}$

$\ds $

$=$

$\ds \map {\sigma_0} {2^2 \times 7^1 \times 4327^1 \times 456 \, 293^1 \times 321 \, 911 \, 699 \, 243^1}$

$\ds $

$=$

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

$\ds $

$=$

$\ds 3 \times 2^4$

$\ds $

$=$

$\ds 48$

$\blacksquare$