- Weird number
In

mathematics , a**weird number**is anatural number that is abundant but not semiperfect. [*cite journal*] In other words, the sum of the proper

last =Benkoski

first =Stan

authorlink =

coauthors =

title =E2308 (in Problems and Solutions)

journal =The American Mathematical Monthly

volume =79

issue =7

pages =774

date =Aug.-Sep. 1972

doi =10.2307/2316276divisor s (divisors including 1 but not itself) of the number is greater than the number, but nosubset 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

authorlink =

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 2

^{32}≈ 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:$R=frac\{2^kQ-(Q+1)\}\{(Q+1)-2^k\}$is prime, then :$n=2^\{k-1\}QR$is a weird number. [

*cite journal*] With this formula, he was able to find the large weird number :$n=2^\{56\}(2^\{61\}-1)153722867280912929approx2cdot10^\{52\}$.

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**In Popular Culture**The ninth track on the album

Geogaddi by the bandBoards 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 calledMusic70 .**References****External links***MathWorld | urlname=WeirdNumber | title=Weird number

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2010.*

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