Weird Number — Definition, Formula & Examples

A weird number is a natural number that is abundant (its proper divisors sum to more than itself) but is not semiperfect — meaning no subset of its proper divisors adds up exactly to the number.

A natural number $n$ is called weird if $\sigma(n) - n > n$ , where $\sigma(n)$ is the sum of all divisors of $n$ , and there exists no subset $S$ of the proper divisors of $n$ such that $\sum_{d \in S} d = n$ .

How It Works

To check whether a number is weird, you first confirm it is abundant by verifying that its proper divisors sum to more than the number itself. Then you check every possible subset of those proper divisors to see if any subset sums to exactly the number. If no such subset exists, the number is weird. This subset-sum check is what makes weird numbers rare and computationally interesting — the number of subsets grows exponentially with the number of divisors.

Worked Example

Problem: Verify that 70 is a weird number.

Step 1: Find proper divisors: The proper divisors of 70 are 1, 2, 5, 7, 10, 14, and 35.

Step 2: Check abundance: Add the proper divisors and compare to 70.

1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 > 70

Step 3: Check all subsets for a sum of 70: You need a subset of {1, 2, 5, 7, 10, 14, 35} that sums to exactly 70. The largest subset sum excluding 35 is 1+2+5+7+10+14 = 39, which is too small. Any subset including 35 needs the remaining elements to sum to 35, but the best you can do from {1, 2, 5, 7, 10, 14} is 39 (all of them, giving 35+39 = 74, too large). Systematically checking all subsets containing 35 confirms no combination hits exactly 70.

Answer: 70 is abundant (divisor sum 74 > 70) and no subset of its proper divisors sums to 70, so 70 is a weird number — in fact, the smallest weird number.

Why It Matters

Weird numbers sit at the intersection of additive and multiplicative number theory. Studying them connects to open questions: for instance, it is still unknown whether any odd weird numbers exist. The subset-sum problem underlying their definition also appears in cryptography and computer science.

Common Mistakes

Mistake: Assuming every abundant number is weird.

Correction: Most abundant numbers are semiperfect — some subset of their proper divisors does sum to the number. For example, 12 is abundant (divisor sum 16) but not weird, because 1 + 2 + 3 + 6 = 12. Weird numbers are the rare abundant numbers that are not semiperfect.