In number theory, a **weird number** is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

**Examples**

The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but *not* weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2+4+6 = 12.

The first few weird numbers are

- 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, … (sequence A006037 in OEIS).
Properties

It has been shown that an infinite number of weird numbers exist; in fact, the sequence of weird numbers has positive asymptotic density.^{
}

It is not known if any odd weird numbers exist; if any do, they must be greater than 2^{32} ≈ 4×10^{9}.^{
}

Sidney Kravitz has shown that for *k* a positive integer, *Q* a prime exceeding 2^{k}, and

- ;

also prime and greater than 2^{k}, then

is a weird number. With this formula, he found a large weird number

- .

If *n* is weird, and *p* is a prime greater than the sum of divisors σ(*n*), then *pn* is also weird.