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 because its only proper divisor is 1.
• Many abundant numbers are even, and every multiple of 12 is abundant.
• Deficient numbers have a natural density of about 75.2%, while abundant numbers have a density of about 24.8%.