Definition:Almost Perfect Number

Definition

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

Definition 1

Let $A \left({n}\right)$ denote the abundance of $n$.

$n$ is almost perfect if and only if $A \left({n}\right) = -1$.

Definition 2

$n$ is almost perfect if and only if:

$\map {\sigma_1} n = 2 n - 1$

where $\map {\sigma_1} n$ denotes the divisor sum function of $n$.

Definition 3

$n$ is almost perfect if and only if it is exactly one greater than the sum of its aliquot parts.

Thus an almost perfect number can be described as "deficient, but only just".

Also known as

An almost perfect number is also known as:

a least deficient number

a slightly defective number.

Some sources hyphenate: almost-perfect.

Sequence

The sequence of almost perfect numbers begins:

$1, 2, 4, 8, 16, 32, 64, 128, \ldots$

Examples

$16$

$16$ is an almost perfect number.

Also see

• Equivalence of Definitions of Almost Perfect Number

• Definition:Perfect Number
• Definition:Quasiperfect Number
• Definition:Abundance

• Power of 2 is Almost Perfect

• Results about almost perfect numbers can be found here.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $32$