Divisor Count of 17,796,126,877,482,329,126,043

Example of Use of Divisor Count Function

$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043} = 32$

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:

$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043 = 3 \times 73 \times 2381 \times 63 \, 678 \, 479 \times 535 \, 956 \, 203$

Thus:

$\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043}$

$=$

$\ds \map {\sigma_0} {3^1 \times 73^1 \times 2381^1 \times 63 \, 678 \, 479^1 \times 535 \, 956 \, 203^1}$

$\ds $

$=$

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

$\ds $

$=$

$\ds 2^5$

$\ds $

$=$

$\ds 32$

$\blacksquare$