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, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... (sequence A006037 in the OEIS).

Properties

Unsolved problem in mathematics: Are there any odd weird numbers? (more unsolved problems in mathematics)

It is easy to see that an infinite number of weird numbers exist.^[3] For example, 70p is weird for all primes p ≥ 149, see the first phrase of the next section. In fact, the set of weird numbers has positive asymptotic density.^[4]

It is not known if any odd weird numbers exist; if so, they must be greater than 2³⁰ ≈ 1×10^9 [5] and greater than 10²¹.^[6]

Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2^k, and

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

also prime and greater than 2^k, then

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

is a weird number.^[7] With this formula, he found a large weird number

$n=2^{56}\cdot (2^{61}-1)\cdot 153722867280912929\ \approx \ 2\cdot 10^{52}$.

Primitive weird numbers

A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird.^[4] This leads to the definition of primitive weird numbers, i.e. weird numbers that are not multiple of other weird numbers (sequence A002975 in the OEIS). There are only 24 primitive weird numbers smaller than a million, compared to 1765 weird numbers up to that limit. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form ${\displaystyle 2^{k}pq}$ are primitive, but the existence of infinitely many k and Q which yield a prime R is not guaranteed. It is conjectured that there exist infinitely many primitive numbers, and Melfi has shown that the infiniteness of primitive weird numbers is a consequence of Cramér's conjecture.^[8]

References

1. Benkoski, Stan (August–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. Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 113–114. ISBN 1-4020-4215-9. Zbl 1151.11300.
4. Benkoski, Stan; Erdős, Paul (April 1974). "On Weird and Pseudoperfect Numbers". Mathematics of Computation. 28 (126): 617–623. doi:10.2307/2005938. MR 347726. Zbl 0279.10005.
5. Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". J. Number Theory. 44: 328–339. doi:10.1006/jnth.1993.1057. Zbl 0781.11015. The result is attributed to "M. Mossinghoff at University of Texas - Austin".
6. http://oeis.org/A006037 OEIS: Weird numbers; comments concerning odd weird numbers.
7. Kravitz, Sidney (1976). "A search for large weird numbers". Journal of Recreational Mathematics. Baywood Publishing. 9 (2): 82–85. Zbl 0365.10003.
8. Melfi, Giuseppe (2015). "On the conditional infiniteness of primitive weird numbers". Journal of Number Theory. Elsevier. 147: 508–514. doi:10.1016/j.jnt.2014.07.024.

External links

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