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Weird number
From Wikipedia, the free encyclopedia
| Divisibility-based sets of integers |
| Form of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Deficient number |
| Weird number |
| Amicable number |
| Sociable number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
- The term "weird number" also refers to a phenomenon in two's complement arithmetic.
In mathematics, 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.
The first weird number is 70, because its proper divisors are 1, 2, 5, 7, 10, 14, and 35; which sum to 74, but no subset of these sums to 70. The number 12 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). It has been shown that an infinite number of weird numbers exist, and the sequence of weird numbers has also been proven to have positive asymptotic density.
It is not known if any odd weird numbers exist; if any do, they must be greater than 1018 (as noted by Bob Hearn in a July 2005 posting to the SeqFans mailing list).
- A track titled "The Smallest Weird Number" appears on the album Geogaddi by Boards of Canada. This is a reference to their record label, Music70. There are other references to the number 70 in their music, often under the name "sixtyten" rather than "seventy".
- Eric W. Weisstein, Weird number at MathWorld.