Definition:Abundance
Definition
Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $\map {\sigma_1} n$ be the divisor sum function of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then the abundance of $n$ is defined as $\map A n = \map {\sigma_1} n - 2 n$.
Abundant Number
Let $\map A n$ denote the abundance of $n$.
$n$ is abundant if and only if $\map A n > 0$.
Perfect Number
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is perfect if and only if $A \left({n}\right) = 0$.
Quasiperfect Number
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is quasiperfect if and only if $A \left({n}\right) = 1$.
Almost Perfect Number
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is almost perfect if and only if $A \left({n}\right) = -1$.
Deficient Number
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is deficient if and only if $A \left({n}\right) < 0$.
Also known as
There may possibly be a vanishingly tiny usage which refers to abundance as the excedent function.
Also see
- Results about abundance can be found here.
Historical Note
The concepts of abundant and deficient appear to have originated with the Neo-Pythagorean school, in particular Nicomachus of Gerasa, who wrote fancifully on the subject in his Ἀριθμητικὴ εἰσαγωγή (Introduction to Arithmetic).