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).

Example

For example, 20 is a primitive abundant number because:

1. 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.
2. 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 ... (sequence A071395 in the OEIS)

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.

References