Primitive abundant number

In math, a primitive abundant number is a special kind of abundant number. Its proper divisors, however, must all be deficient numbers (numbers whose sum of proper divisors are less than 2 times that number).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 numbers to be primitive abundant numbers too.
Example
For example, 20 is a primitive abundant number because:

 20 is an abundant number. This is because the sum of its divisors is 1 + 2 + 4 + 5 + 10 + 20 > 40. This makes 20 an abundant number.
 The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8. All of these numbers are a deficient number. This makes 20 a primitive abundant number.

The first few primitive abundant numbers are 20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... 

The smallest odd primitive abundant number is 945.

Another definition of a primitive abundant number would be abundant numbers having no abundant proper divisors, which could also include perfect-numbered divisors (i.e. multiples of 6, 28, 496, 8128, etc...)

The first few primitive abundant numbers which include perfect numbered divisors are 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114... (sequence A091191 in the OEIS)

 Properties 

Every multiple of a primitive abundant number is abundant .

Every abundant number is a multiple of either 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 is an infinite amount of primitive abundant numbers.Paul Erdős, Journal of the London Mathematical Society 9 (1934) 278–282.

 References 

Category:Divisor function
Category:Integer sequences