# Weird number

In number theory, 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.

## Examples

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).

## Properties

It has been shown that an infinite number of weird numbers exist; in fact, the sequence of weird numbers has 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

$R=\frac{2^kQ-(Q+1)}{(Q+1)-2^k}$;

if R is prime, then

$n=2^{k-1}QR$

is a weird number.[5] With this formula, he found a large weird number

$n=2^{56}\cdot(2^{61}-1)\cdot153722867280912929\ \approx\ 2\cdot10^{52}$.

## In popular culture

The Boards of Canada album Geogaddi contains a song titled "The Smallest Weird Number," a reference to another song, "Sixtyten" (equalling 70), as well as their record label, Music70. Despite the song having a running time of 1:17, the audio cuts out at exactly 1:10 (70 seconds).

## References

1. ^ Benkoski, Stan (Aug.-September 1972). "E2308 (in Problems and Solutions)". The American Mathematical Monthly 79 (7): 774. doi:10.2307/2316276. JSTOR 2316276.
2. ^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248. Section B2.
3. ^ Benkoski, Stan; Paul Erdős (April 1974). "On Weird and Pseudoperfect Numbers". Mathematics of Computation 28 (126): 617–623. doi:10.2307/2005938. MR 347726. Zbl 0279.10005.
4. ^ 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".
5. ^ Kravitz, Stanley (1976). "A search for large weird numbers". Journal of Recreational Mathematics (Baywood Publishing) 9 (2): 82–85. Zbl 0365.10003.