Primitive abundant number

Euler diagram of numbers under 100:
   Abundant
   Primitive abundant
   Highly abundant
   Superabundant and highly composite
   Colossally abundant and superior highly composite
   Weird
   Perfect
   Composite
   Deficient

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:

20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS)

The smallest odd primitive abundant number is 945.

A variant definition is abundant numbers having no abundant proper divisor (sequence A091191 in the OEIS). It starts:

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114


  1. ^ Weisstein, Eric W. "Primitive Abundant Number". MathWorld.
  2. ^ 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.