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In [[mathematics]] a '''primitive abundant number''' is an [[abundant number]] whose [[proper divisor]]s are all [[deficient number]]s.<ref>{{MathWorld|title=Primitive Abundant Number|urlname=PrimitiveAbundantNumber}}</ref><ref>Erdős adopts a wider definition that requires a primitive abundant number to be not deficient, but not necessarily abundant (Erdős, Surányi and Guiduli. ''Topics in the Theory of Numbers'' p214. Springer 2003.). The Erdős definition allows [[perfect number]]s to be primitive abundant numbers too.</ref>

For example, 20 is a primitive abundant number because:
:#The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.
:#The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number.

The first few primitive abundant numbers are:

:[[20 (number)|20]], [[70 (number)|70]], [[88 (number)|88]], [[104 (number)|104]], 272, 304, 368, 464, 550, 572 ... {{OEIS|id=A071395}}

The smallest odd primitive abundant number is 945.

A variant definition is abundant numbers having no abundant proper divisor {{OEIS|id=A091191}}, this therefore also includes prime multiples of [[perfect number]]s including all prime multiples of 6. It starts: 

: [[12 (number)|12]], [[18 (number)|18]], [[20 (number)|20]], [[30 (number)|30]], 42, 56, 66, 70, 78, 88, 102, 104, 114 
 
==Properties==

Every multiple of a primitive abundant number is an abundant number.

Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number.

Every primitive abundant number is either a [[primitive semiperfect number]] or a [[weird number]].

There are an infinite number of primitive abundant numbers.

The number of primitive abundant numbers less than or equal to ''n'' is <math>O \left( \frac{n}{\log^2(n)} \right)\, .</math><ref>Paul Erdős, ''Journal of the London Mathematical Society'' 9 (1934) 278&ndash;282.</ref>

== References ==
{{reflist}}


{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Divisor function]]
[[Category:Integer sequences]]