- Weird number
mathematics, a weird number is a natural numberthat is abundant but not semiperfect. [cite journal
title =E2308 (in Problems and Solutions)
journal =The American Mathematical Monthly
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 subsetof 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
coauthors =Paul Erdős
title =On Weird and Pseudoperfect Numbers
journal =Mathematics of Computation
date =April 1974
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 is a positive integer, is a prime, and:is prime, then :is a weird number. [cite journal
title =A search for large weird numbers
journal =Journal of Recreational Mathematics
publisher =Baywood Publishing
date =1976 ] With this formula, he was able to find the large weird number :.
In Popular Culture
The ninth track on the album
Geogaddiby the band Boards of Canadais 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.
*MathWorld | urlname=WeirdNumber | title=Weird number
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