Generate Deficient Numbers

What is a Deficient Number?

A deficient number (or defective number) is a positive integer for which the sum of its proper divisors is strictly less than the number itself. Proper divisors are all positive integers that divide the number evenly, excluding the number itself.

Mathematically, if $s(n)$ is the sum of proper divisors of $n$, then $n$ is deficient if:

Using the divisor sum function $\sigma(n)$ (the sum of all divisors including $n$), $n$ is deficient if $\sigma(n) < 2n$.

A Simple Example: Why 8 is Deficient

Let's trace the proper divisors of 8:

• Divisors of 8 are: 1, 2, 4, and 8.
• Proper divisors (excluding 8) are: 1, 2, and 4.
• Summing the proper divisors: $1 + 2 + 4 = 7$.

Since 7 is strictly less than 8 ($7 < 8$), 8 is classified as a deficient number. The amount by which the sum falls short of the number itself (in this case, $8 - 7 = 1$) is called the deficiency of the number.

Fascinating Properties of Deficient Numbers

• Abundance of Primes: All prime numbers are deficient, because their only proper divisor is 1, and $1 < p$ for any prime $p \ge 2$.
• Prime Powers: All prime powers ($p^k$) are deficient.
• Divisors of Deficient/Perfect Numbers: All proper divisors of a deficient number or a perfect number are deficient.
• The Majority: There are infinitely many deficient numbers, and they make up the majority of positive integers (around 75.2% of all integers are deficient).