Definition:Deficient 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 deficient if and only if $\map A n < 0$.

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.

Sequence

The sequence of deficient numbers begins:

$2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, \ldots$

Also known as

A deficient number is also known as a defective number.

Also see

• Equivalence of Definitions of Deficient Number

• Definition:Abundant Number
• Definition:Perfect Number

• Definition:Abundance
• Definition:Abundancy Index

• Results about deficient numbers can be found here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): deficient number (defective number)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): deficient number (defective number)