# Definition:Primitive Abundant Number

## Definition

A primitive abundant number is an abundant number whose aliquot parts are all deficient.

### Sequence of Primitive Abundant Numbers

The sequence of primitive abundant numbers begins:

$20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, \ldots$

## Examples

### 20

$20$ is a primitive abundant number:

$1 + 2 + 4 + 5 + 10 = 22 > 20$

### 70

$70$ is a primitive abundant number:

$1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 > 70$

### 88

$88$ is a primitive abundant number:

$1 + 2 + 4 + 8 + 11 + 22 + 44 = 92 > 88$

### 104

$104$ is a primitive abundant number:

$1 + 2 + 4 + 8 + 13 + 26 + 52 = 106 > 104$

### 272

$272$ is a primitive abundant number:

$1 + 2 + 4 + 8 + 16 + 17 + 34 + 68 + 136 = 286 > 272$

and so on.

## Also defined as

Some sources define a primitive abundant number as an abundant number which has no abundant aliquot parts.

The difference between the definitions here is that perfect numbers are allowed as divisors.

Under this definition, the sequence of primitive abundant numbers begins:

$12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, \ldots$