Definition:Primitive Abundant Number

From ProofWiki
Jump to navigation Jump to search

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:

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

This sequence is A091191 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also see


Sources