Deficient and Abundant Numbers

Let $s(n)$ be the sum of all proper divisors of a positive integer $n$.

• If $s(n) < n$, then $n$ is a deficient number.
• If $s(n) > n$, then $n$ is an abundant number.

Examples

• $n = 8$
  • Proper divisors: 1, 2, 4
  • Sum: $1 + 2 + 4 = 7 < 8$ → deficient number
• $n = 12$
  • Proper divisors: 1, 2, 3, 4, 6
  • Sum: $16 > 12$ → abundant number

Notes

• Most natural numbers are deficient.
• The smallest abundant number is 12.
• Every prime number is deficient (its only proper divisor is 1).
• Many even numbers, when sufficiently large, are abundant.
• The distribution of deficient and abundant numbers is still not fully understood.