Divisor Count of 6561
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Example of Use of Divisor Count Function
- $\map {\sigma_0} {6561} = 9$
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:
- $6561 = 3^8$
Thus:
\(\ds \map {\sigma_0} {6561}\) | \(=\) | \(\ds \map {\sigma_0} {3^8}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9\) |
The divisors of $6561$ can be enumerated as:
- $1, 3, 9, 27, 81, 243, 729, 2187, 6561$
$\blacksquare$