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:

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

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

Hence the result.

$\blacksquare$