Equivalence of Definitions of Quasiperfect Number

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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:

$\sigma \left({n}\right) = 2 n + 1$

where $\sigma \left({n}\right)$ denotes the $\sigma$ 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:

$A \left({n}\right) = \sigma \left({n}\right) - 2 n$

By definition of $\sigma$ function:

$\sigma \left({n}\right)$ is the sum of all the divisors of $n$.

Thus $\sigma \left({n}\right) - n$ is the sum of the aliquot parts of $n$.

Hence the result.

$\blacksquare$