Divisor Count of 1312
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Example of Use of Divisor Count Function
- $\map {\sigma_0} {1312} = 12$
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:
- $1312 = 2^5 \times 41$
Thus:
\(\ds \map {\sigma_0} {1312}\) | \(=\) | \(\ds \map {\sigma_0} {2^5 \times 41^1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {5 + 1} \paren {1 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12\) |
The divisors of $1312$ can be enumerated as:
- $1, 2, 4, 8, 16, 32, 41, 82, 164, 328, 656, 1312$
$\blacksquare$