Divisor Count of 997,920

Example of Use of Divisor Count Function

$\map {\sigma_0} {997 \, 920} = 240$

where $\sigma_0$ denotes the divisor count 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:

$997 \, 920 = 2^5 \times 3^4 \times 5 \times 7 \times 11$

Thus:

\(\ds \map {\sigma_0} {997 \, 920}\)

\(=\)

\(\ds \map {\sigma_0} {2^5 \times 3^4 \times 5^1 \times 7^1 \times 11^1}\)

\(\ds \)

\(=\)

\(\ds \paren {5 + 1} \paren {4 + 1} \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}\)

\(\ds \)

\(=\)

\(\ds \paren {2 \times 3} \times 5 \times 2 \times 2 \times 2\)

\(\ds \)

\(=\)

\(\ds 2^4 \times 3 \times 5\)

\(\ds \)

\(=\)

\(\ds 240\)

$\blacksquare$