Primitive abundant number

In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.[1][2]

For example, 20 is a primitive abundant number because:

  1. The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.
  2. 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:

OEIS)

The smallest odd primitive abundant number is 945.

A variant definition is abundant numbers having no abundant proper divisor (sequence perfect numbers including all prime multiples of 6. It starts:

12, 18, 20, 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 O \left( \frac{n}{\log^2(n)} \right)\, .[3]

References


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