Weird Number

Innumber theory, a weird number is anatural number that isabundant but notsemiperfect.In other words, the sum of the properdivisors (divisors including 1 but not itself) of the number is greater than the number, but nosubset of those divisors sums to the number itself.

Examples [ edit ]

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, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... (sequence A006037 in theOEIS).

Properties [ edit ]

Unsolved problem in mathematics :

Are there any odd weird numbers?

(more unsolved problems in mathematics)

Infinitely many weird numbers exist.For example, 70 p is weird for allprimes p ≥ 149. In fact, the set of weird numbers has positiveasymptotic density.

It is not known if any odd weird numbers exist. If so, they must be greater than 10 21 .

Sidney Kravitz has shown that for k a positive integer, Q aprime 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

Primitive weird numbers [ edit ]

A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ( n ), then pn is also weird.This leads to the definition of primitive weird numbers , i.e. weird numbers that are not multiple of other weird numbers (sequence A002975 in theOEIS). There are only 24 primitive weird numbers smaller than a million, compared to 1765 weird numbers up to that limit. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form are primitive, but the existence of infinitely many k and Q which yield a prime R is not guaranteed. It is conjectured that there exist infinitely many primitive numbers, andMelfi has shown that the infiniteness of primitive weird numbers is a consequence ofCramér's conjecture.Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.

See also [ edit ]

• Untouchable number

References [ edit ]