Divisor Count of 1575
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Example of Use of Divisor Count Function
- $\map {\sigma_0} {1575} = 18$
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:
- $1575 = 3^2 \times 5^2 \times 7$
Thus:
\(\ds \map {\sigma_0} {1575}\) | \(=\) | \(\ds \map {\sigma_0} {3^2 \times 5^2 \times 7^1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 + 1} \paren {2 + 1} \paren {1 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 18\) |
The divisors of $1575$ can be enumerated as:
- $1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525, 1575$
$\blacksquare$