# Weird number

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Weird number

In mathematics, a weird number is a natural number that is abundant but not semiperfect. [cite journal
last =Benkoski
first =Stan
coauthors =
title =E2308 (in Problems and Solutions)
journal =The American Mathematical Monthly
volume =79
issue =7
pages =774
date =Aug.-Sep. 1972
doi =10.2307/2316276
] 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 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, ... OEIS|id=A006037. It has been shown that an infinite number of weird numbers exist, and the sequence of weird numbers has been proven to have positive asymptotic density.cite journal
last =Benkoski
first =Stan
coauthors =Paul Erdős
title =On Weird and Pseudoperfect Numbers
journal =Mathematics of Computation
volume =28
issue =126
pages =617–623
date =April 1974
doi =10.2307/2005938
]

It is not known if any odd weird numbers exist; if any do, they must be greater than 232 ≈ 4e|9. [CN Friedman, "Sums of Divisors and Egyptian Fractions", "Journal of Number Theory" (1993). The result is attributed to "M. Mossinghoff at University of Texas - Austin".]

Stanley Kravitz has shown that if $k$ is a positive integer, $Q$ is a prime, and:$R=frac\left\{2^kQ-\left(Q+1\right)\right\}\left\{\left(Q+1\right)-2^k\right\}$is prime, then :$n=2^\left\{k-1\right\}QR$is a weird number. [cite journal
last =Kravitz
first =Stanley
title =A search for large weird numbers
journal =Journal of Recreational Mathematics
volume =9
issue =2
pages =82–85
publisher =Baywood Publishing
location =
date =1976
] With this formula, he was able to find the large weird number :$n=2^\left\{56\right\}\left(2^\left\{61\right\}-1\right)153722867280912929approx2cdot10^\left\{52\right\}$.

In Popular Culture

The ninth track on the album Geogaddi by the band Boards of Canada is called "The Smallest Weird Number" [http://bocpages.org/wiki/The_Smallest_Weird_Number bocpages wiki, retrieved on 12 September 2008.] . The band's private music label is called Music70.

References

*MathWorld | urlname=WeirdNumber | title=Weird number

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