In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, a weird number is a natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

 that is abundant but not semiperfect

Semiperfect number

In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number....

.
In other words, the sum of the proper divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

s (divisors including 1 but not itself) of the number is greater than the number, but no subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 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, ... .

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 2³² ≈ 4.

Stanley Kravitz has shown that if k is a positive integer, Q is a prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, and
is prime, then
is a weird number.
With this formula, he was able to find a large weird number.