Weird Numbers

To earn the title of a weird number, a number must meet two specific conditions: it must be “abundant,” but it must simultaneously refuse to be “semiperfect.” A weird number is a paradoxical mismatch: the number has an abundance of proper divisors (i.e., they add up to more than the number itself), yet it is not semiperfect (i.e., no combination or subset of those divisors can be added together to hit the exact value of the number).

To earn the title of a weird number, a number must meet two specific conditions: it must be “abundant,” but it must simultaneously refuse to be “semiperfect.”

Abundant Number

An abundant number, sometimes also called an excessive number, is a positive integer the sum of its positive proper divisors is greater than the number, e.g., the sum of the proper divisors of 12 is 1 + 2 + 3 + 4 + 6 = 16.

The first few abundant numbers are 12, 18, 20, 24, 30, 36, …

Example

12: 1 + 2 + 3 + 4 + 6 = 16 (Since 16 > 12, 12 is an abundant number, not a perfect number).

Semiperfect (Pseudoperfect) Number

A number n is semiperfect (pseudoperfect) if there is some combination of its proper divisors that adds up to n. If the number equals the sum of all of its proper divisors, it is a perfect number. A pseudoperfect number which is the sum of all its proper divisors is called a perfect number.

The first few pseudoperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, …

Examples

20: 1 + 4 + 5 + 10 = 20 (Perfect Number)

12: 1 + 2 + 3 + 4 + 6 = 16 (Since 16 > 12, 12 is an abundant number, not a perfect number). You must leave out at least one divisor to hit 12, therefore, 12 is pseudoperfect. There are two different subsets that work:

2 + 4 + 6 = 12 (leaving out 1 and 3)
1 + 2 + 3 + 6 = 12 (leaving out 4)

Weird Number

A “weird number” is a number that is abundant without being pseudoperfect. The pseudoperfect part of the definition means that finding weird numbers is a case of the subset sum problem. Since prime numbers are deficient, prime numbers are not weird. Similarly, since multiples of 6 are pseudoperfect, no weird number is a multiple of 6.

The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, …

Example

Abundant
- Positive divisors of 70: 1, 2, 5, 7, 10, 14, 35, and 70.
- The sum of divisors: 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74
Not Pseudoperfect
- 2 + 5 + 7 + 10 + 14 + 35 = 73
- 1 + 5 + 7 + 10 + 14 + 35 = 72
- 1 + 2 + 7 + 10 + 14 + 35 = 67

Looking at the image below, the classification of all numbers under 100 identifies 70 as the only weird number!

References

“Deficient Number.” Wikipedia, July 9, 2026. https://en.wikipedia.org/wiki/Deficient_number.

Frederick, Cole. “What’s up with Weird Numbers?” Science Spectrum, February 11, 2025. https://sciencespectrumu.com/whats-up-with-weird-numbers-78546fa75cb7.

[ ℰ ] North, Mansfield. “Perfect, Abundant, and Deficient Numbers Explained (With Practice Problems).” Mathnasium, September 8, 2026. https://www.mathnasium.com/math-centers/mansfieldnorth/news/perfect-abundant-deficient-numbers.

NovaPrism. “In Math, What Is a ‘Weird Number?’” Quora. Accessed November 12, 2026. https://qr.ae/pFBRaY.

“Semiperfect Number.” Wikipedia, September 4, 2025. https://en.wikipedia.org/wiki/Semiperfect_number.

Sloane, N. J. A. “A005101.” Abundant numbers (sum of divisors of m exceeds 2m). The On-Line Encyclopedia of Integer Sequences. November 12, 2026. https://oeis.org/A005101.

Sloane, N. J. A. “A005100.” Deficient numbers: numbers k such that sigma(k) < 2k. The On-Line Encyclopedia of Integer Sequences. Accessed November 14, 2026. https://oeis.org/A005100.

Sloane, N. J. A. “A000396.” Perfect numbers k: k is equal to the sum of the proper divisors of k. The On-Line Encyclopedia of Integer Sequences. Accessed November 14, 2026. https://oeis.org/A000396.

Sloane, N. J. A. “A091191.” Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor. The On-Line Encyclopedia of Integer Sequences. Accessed November 14, 2026. https://oeis.org/A091191.

Sloane, N. J. A. “A005835.” Pseudoperfect (or semiperfect) numbers k: some subset of the proper divisors of k sums to k. The On-Line Encyclopedia of Integer Sequences. Accessed November 12, 2026. https://oeis.org/A005835.

Sloane, N. J. A. “A006037.” Weird numbers: abundant (A005101) but not pseudoperfect (A005835). OEIS. Accessed November 14, 2026. https://oeis.org/A006037.

“Weird Number.” Wikipedia, September 9, 2026. https://en.wikipedia.org/wiki/Weird_number.

“Weird Number – Wolfram|Alpha.” Wolfram|Alpha. Accessed November 12, 2026. https://www.wolframalpha.com/input/?i=weird+number.

Weisstein, Eric W. “Abundant Number — from Wolfram MathWorld.” Wolfram MathWorld. Accessed November 12, 2026. https://mathworld.wolfram.com/AbundantNumber.html.

Weisstein, Eric W. “Deficient Number — from Wolfram MathWorld.” Wolfram MathWorld. Accessed November 12, 2026. https://mathworld.wolfram.com/DeficientNumber.html.

Weisstein, Eric W. “Pseudoperfect Number — from Wolfram MathWorld.” Wolfram MathWorld. Accessed November 12, 2026. https://mathworld.wolfram.com/PseudoperfectNumber.html.

Weisstein, Eric W. “Weird Number — from Wolfram MathWorld.” Wolfram MathWorld. Accessed November 12, 2026. https://mathworld.wolfram.com/WeirdNumber.html.

Notes

The Three Number Categories

Mathematicians group positive integers into three categories, deficient, abundant, and perfect, based on how the sum of their proper divisors compares to the number. What’s the difference between deficient, abundant, and perfect numbers?

The difference between these three groups comes down to one comparison. When we add up a number’s proper divisors, then check how that sum lines up against the number itself. That single check is a it takes to sort any whole number into one of the three groups.

Group	Comparison	Example
Perfect	Sum of proper divisors = the number	6 ⟶ 1 + 2 + 3 = 6
Abundant	Sum of proper divisors > the number	12 ⟶ 1 + 2 + 3 + 4 + 6 = 16
Deficient	Sum of proper divisors < the number (e.g., primes, powers of 2, and most integers)	8 ⟶ 1 + 2 + 4 = 7

Deficient (Defective) Number

Calling a prime number deficient means that the sum of its proper divisors is smaller than the number itself.

Proper Divisors: These are all the factors of a number except the number itself.
Prime Numbers: By definition, a prime number has only two factors: 1 and the number itself.
The Sum: When you remove the prime number itself, the only proper divisor left is 1.

Because 1 is always smaller than the prime number (for any prime (p > 1)), every prime number is automatically a deficient number.

The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, …

Examples

Prime Number 13

Factors of 13: 1 and 13
Proper divisors (excluding 13): 1
Comparison: 1 is less than 13, so 13 is deficient.

Even Number 8

Factors of 8: 1, 2, and 4
Proper divisors (excluding 8): 1, 2, and 4 (1 + 2 + 4 = 7)
Comparison: Their sum is less than 8, so 8 is deficient.

Perfect Number

A perfect number is a positive whole number that equals the exact sum of its proper divisors.

The first few perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, …

Example: Number 6

Find its proper divisors of 6: 1, 2, 3, and 6.
Remove the number itself leaves the proper divisors 1, 2, and 3.
We need to check if the proper divisors add up to the number.
1 + 2 + 3 = 6.
Because the sum of its proper divisors equals the original number 6 is a perfect number!

Primitive Abundant Number

A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number.

The first few primitive abundant numbers are 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, …

Proper Divisor

A positive proper divisor of a positive integer n is a positive divisor of n, excluding n itself. For example, 1, 2, and 3 are positive proper divisors of 6, but 6 itself is not.

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