Divisor Count of 420
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Example of Use of Divisor Count Function
- $\map {\sigma_0} {420} = 24$
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:
- $420 = 2^2 \times 3 \times 5 \times 7$
Thus:
\(\ds \map {\sigma_0} {420}\) | \(=\) | \(\ds \map {\sigma_0} {2^2 \times 3^1 \times 5^1 \times 7^1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 + 1} \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 24\) |
The divisors of $420$ can be enumerated as:
- $1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420$
This sequence is A018444 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
$\blacksquare$