Definition:Abundant Number/Definition 3

Definition

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

$n$ 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

• Results about abundant numbers can be found here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): abundant number
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): perfect number
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): perfect number
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): abundant number
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): abundant number