Divisor Count of 4,100,625

Example of Use of Divisor Count Function

$\map {\sigma_0} {4 \, 100 \, 625} = 45$

where $\sigma_0$ denotes the divisor cunt function.

Proof

From Divisor Count Function from Prime Decomposition:

$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

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:

$4 \, 100 \, 625 = 5^4 \times 3^8$

Thus:

\(\ds \map {\sigma_0} {4 \, 100 \, 625}\)

\(=\)

\(\ds \map {\sigma_0} {5^4 \times 3^8}\)

\(\ds \)

\(=\)

\(\ds \paren {4 + 1} \times \paren {8 + 1}\)

\(\ds \)

\(=\)

\(\ds 5 \times 9\)

\(\ds \)

\(=\)

\(\ds 45\)

$\blacksquare$