Equivalence of Definitions of Quasiperfect Number

Theorem

The following definitions of a quasiperfect number are equivalent:

Definition 1

Let $A \left({n}\right)$ denote the abundance of $n$.

$n$ is quasiperfect if and only if $A \left({n}\right) = 1$.

Definition 2

$n$ is quasiperfect 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 quasiperfect if and only if it is exactly one less 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:

$\map {\sigma_1} n$ is the sum of all the divisors of $n$.

Thus $\map {\sigma_1} n - n$ is the sum of the aliquot parts of $n$.

Hence the result.

$\blacksquare$