Divisor Count of 831,600

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Example of Use of Divisor Count Function

$\map {\sigma_0} {831 \, 600} = 240$

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:

$831 \, 600 = 2^4 \times 3^3 \times 5^2 \times 7 \times 11$

Thus:

\(\ds \map {\sigma_0} {831 \, 600}\) \(=\) \(\ds \map {\sigma_0} {2^4 \times 3^3 \times 5^2 \times 7^1 \times 11^1}\)
\(\ds \) \(=\) \(\ds \paren {4 + 1} \paren {3 + 1} \paren {2 + 1} \paren {1 + 1} \paren {1 + 1}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3 \times 5\)
\(\ds \) \(=\) \(\ds 240\)

$\blacksquare$