Product of Proper Divisors
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Theorem
Let $n$ be an integer such that $n \ge 1$.
Let $\map P n$ denote the product of the proper divisors of $n$.
Then:
- $\map P n = n^{\map {\sigma_0} n / 2 - 1}$
where $\map {\sigma_0} n$ denotes the divisor count Function of $n$.
Proof
Let $\map D n$ denote the product of all the divisors of $n$.
From Product of Divisors:
- $\map D n = n^{\map {\sigma_0} n / 2}$
The proper divisors of $n$ are defined as being the divisors of $n$ excluding $n$ itself.
Thus:
- $\map P n = \dfrac {\map D n} n = \dfrac {n^{\map {\sigma_0} n / 2} } n = n^{\map {\sigma_0} n / 2 - 1}$
$\blacksquare$
Examples
Product of Proper Divisors of $12$
The product of the proper divisors of $12$ is $144 = 12^2$.
Product of Proper Divisors of $20$
The product of the proper divisors of $20$ is $400 = 20^2$.