Definition:Abundant Number

Definition

Let $n \in \Z_{\ge 0}$ be a positive integer.

Definition 1

Let $\map A n$ denote the abundance of $n$.

$n$ is abundant if and only if $\map A n > 0$.

Definition 2

Let $\map {\sigma_1} n$ be the divisor sum function of $n$.

$n$ is abundant if and only if $\dfrac {\map {\sigma_1} n} n > 2$.

Definition 3

$n$ is abundant if and only if it is smaller than its aliquot sum.

Sequence

The sequence of abundant numbers begins:

$12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, \ldots$

Examples

Example: $12$

The aliquot parts of $12$ are:

$1, 2, 3, 4, 6$

We have that:

$1 + 2 + 3 + 4 + 6 = 16 > 12$

demonstrating that $12$ is an abundant number.

Also see

• Equivalence of Definitions of Abundant Number

• Definition:Primitive Abundant Number

• Definition:Deficient Number
• Definition:Perfect Number

• Definition:Abundance
• Definition:Abundancy Index

• Definition:Superabundant Number

• Results about abundant numbers can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): abundant number
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): abundant number