Weird number: Definition from Answers.com

In mathematics, a weird number is a natural number that is abundant but not semiperfect. ^[1] ^[2] 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, ... (sequence A006037 in OEIS). 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.^[3]

It is not known if any odd weird numbers exist; if any do, they must be greater than 2³² ≈ 4 × 10⁹.^[4]

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 Q R

is a weird number. ^[5] With this formula, he was able to find the large weird number

$n=2^{56}(2^{61}-1)153722867280912929\approx2\cdot10^{52}$ .

In popular culture

The ninth track on the album Geogaddi by the band Boards of Canada is called "The Smallest Weird Number"^[6]. The band's private music label is called Music70.

References

^ Benkoski, Stan (Aug.-September 1972). "E2308 (in Problems and Solutions)". The American Mathematical Monthly 79 (7): 774. doi:10.2307/2316276.
^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248. Section B2.
^ Benkoski, Stan; Paul Erdős (April 1974). "On Weird and Pseudoperfect Numbers". Mathematics of Computation 28 (126): 617–623. doi:10.2307/2005938.
^ 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".
^ Kravitz, Stanley (1976). "A search for large weird numbers". Journal of Recreational Mathematics (Baywood Publishing) 9 (2): 82–85.
^ http://bocpages.org/wiki/The_Smallest_Weird_Number bocpages wiki, retrieved on 12 September 2008.

External links

Weisstein, Eric W., "Weird number" from MathWorld.

Mathematics portal

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