Divisor Count of 192
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Example of Use of Divisor Count Function
- $\map {\sigma_0} {192} = 14$
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:
- $192 = 2^6 \times 3$
Thus:
\(\ds \map {\sigma_0} {192}\) | \(=\) | \(\ds \map {\sigma_0} {2^6 \times 3^1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {6 + 1} \paren {1 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 14\) |
The divisors of $192$ can be enumerated as:
- $1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192$
This sequence is A018328 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
$\blacksquare$