# Definition:Superabundant Number

## Definition

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

Then $n$ is superabundant if and only if:

$\forall m \in \Z_{>0}, m < n: \dfrac {\map {\sigma_1} m} m < \dfrac {\map {\sigma_1} n} n$

where $\sigma_1$ denotes the divisor sum function.

That is, if and only if $n$ has a higher abundancy index than any smaller positive integer.

## Sequence

The sequence of superabundant numbers begins:

$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, \ldots$

## Examples

### 4

The abundancy index of $4$ is:

$\dfrac {\map {\sigma_1} 4} 4 = \dfrac 7 4 = 1 \cdotp 75$

### 6

The abundancy index of $6$ is:

$\dfrac {\map {\sigma_1} 6} 6 = \dfrac {12} 6 = 2$

### 12

The abundancy index of $12$ is:

$\dfrac {\map {\sigma_1} {12} } {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$

### 24

The abundancy index of $24$ is:

$\dfrac {\map {\sigma_1} {24} } {24} = \dfrac {60} {24} = 2 \cdotp 5$

### 36

The abundancy index of $36$ is:

$\dfrac {\map {\sigma_1} {36} } {36} = \dfrac {91} {36} = 2 \cdotp 52 \dot 7$

### 48

The abundancy index of $48$ is:

$\dfrac {\map {\sigma_1} {48} } {48} = \dfrac {124} {48} = 2 \cdotp 58 \dot 3$

### 60

The abundancy index of $60$ is:

$\dfrac {\map {\sigma_1} {60} } {60} = \dfrac {168} {60} = 2 \cdotp 8$

### 120

The abundancy index of $120$ is:

$\dfrac {\map {\sigma_1} {120} } {120} = \dfrac {360} {120} = 3$

### 180

The abundancy index of $180$ is:

$\dfrac {\map {\sigma_1} {180} } {180} = \dfrac {546} {180} = 3 \cdotp 0 \dot 3$

### 240

The abundancy index of $240$ is:

$\dfrac {\map {\sigma_1} {240} } {240} = \dfrac {744} {240} = 3 \cdotp 1$

## Also see

• Results about superabundant numbers can be found here.