Equivalence of Definitions of Almost Perfect Number
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Theorem
The following definitions of the concept of Almost Perfect Number are equivalent:
Definition 1
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is almost perfect if and only if $A \left({n}\right) = -1$.
Definition 2
$n$ is almost perfect if and only if:
- $\map {\sigma_1} n = 2 n - 1$
where $\map {\sigma_1} n$ denotes the divisor sum function of $n$.
Definition 3
$n$ is almost perfect if and only if it is exactly one greater than the sum of its aliquot parts.
Proof
By definition of abundance:
- $\map A n = \map {\sigma_1} n - 2 n$
By definition of divisor sum function:
Thus $\map {\sigma_1} - n$ is the sum of the aliquot parts of $n$.
Hence the result.
$\blacksquare$