Divisor Count of 64,000
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Example of Use of Divisor Count Function
- $\map {\sigma_0} {64 \, 000} = 40$
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:
- $64 \, 000 = 2^9 \times 5^3$
Thus:
\(\ds \map {\sigma_0} {64 \, 000}\) | \(=\) | \(\ds \map {\sigma_0} {2^9 \times 5^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {9 + 1} \paren {3 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 40\) |
The divisors of $64 \, 000$ can be enumerated as:
- $1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100,$
- $125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 640, 800, 1000, 1280,$
- $1600, 2000, 2560, 3200, 4000, 6400, 8000, 12800, 16000, 32000, 64000$
$\blacksquare$