Primitive Abundant Number

A primitive abundant number is a special type of abundant number. To understand it better, let’s break it down:

1. Abundant Number

• A number is called abundant if the sum of its proper divisors (excluding itself) is greater than the number itself.
• Example:
  • Proper divisors of 12: $1,2,3,4,6$
    • Sum: $1+2+3+4+6=16>12$
    • Thus, 12 is abundant.

2. Primitive Abundant Number

• A primitive abundant number is an abundant number that is not divisible by any other smaller abundant number.
• In simpler terms, removing any of its divisors must not make it a product of another smaller abundant number.
• It is the “simplest” form of an abundant number.

Example of Primitive Abundant Numbers:

1. 12
   • Sum of proper divisors: $1+2+3+4+6=16$
   • Abundant and not divisible by any smaller abundant number.
2. 18
   • Proper divisors: $1,2,3,6,9$
   • Sum: $1+2+3+6+9=21>18$
   • Abundant and not divisible by any smaller abundant number.
3. 20
   • Proper divisors: $1,2,4,5,10$
   • Sum: $1+2+4+5+10=22>20$
   • Abundant and primitive.

Non-Primitive Example:

• 24:
  • Proper divisors: $1,2,3,4,6,8,12$
  • Sum: $1+2+3+4+6+8+12=36>24$.
  • While 24 is abundant, it is divisible by the smaller abundant number 12. Hence, it is not primitive.

Properties of Primitive Abundant Numbers:

1. They form the foundation for all abundant numbers.
2. They cannot be expressed as a multiple of another smaller abundant number.

Primitive abundant numbers are rare and hold significance in number theory for their unique properties.