Equivalence of Definitions of Deficient Number
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Theorem
The following definitions of a deficient number are equivalent:
Definition 1
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is deficient if and only if $A \left({n}\right) < 0$.
Definition 2
Let $\map {\sigma_1} n$ be the divisor sum function of $n$.
$n$ is deficient if and only if:
- $\dfrac {\map {\sigma_1} n} n < 2$
Definition 3
$n$ is deficient if and only if it is greater than its aliquot sum.
Proof
By definition of abundance:
- $\map A n = \map {\sigma_1} n - 2 n$
By definition of divisor sum function:
Thus $\map {\sigma_1} n - n$ is the aliquot sum of $n$.
Hence the result.
$\blacksquare$