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
[edit]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
[edit]Infinitely many weird numbers exist.[3] For example, 70p is weird for all primes p ≥ 149. 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 1021.[5]
Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2k, and
also prime and greater than 2k, then
is a weird number.[6] With this formula, he found the large weird number
Primitive weird numbers
[edit]A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors
See also
[edit]References
[edit]- ^ Benkoski, Stan (August–September 1972). "E2308 (in Problems and Solutions)". The American Mathematical Monthly. 79 (7): 774. doi:10.2307/2316276. JSTOR 2316276.
- ^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248. Section B2.
- ^ 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.
- ^ a b Benkoski, Stan; Erdős, Paul (April 1974). "On Weird and Pseudoperfect Numbers". Mathematics of Computation. 28 (126): 617–623. doi:10.2307/2005938. JSTOR 2005938. MR 0347726. Zbl 0279.10005.
- ^ Sloane, N. J. A. (ed.). "Sequence A006037 (Weird numbers: abundant (A005101) but not pseudoperfect (A005835))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. -- comments concerning odd weird numbers
- ^ Kravitz, Sidney (1976). "A search for large weird numbers". Journal of Recreational Mathematics. 9 (2). Baywood Publishing: 82–85. Zbl 0365.10003.
- ^ Melfi, Giuseppe (2015). "On the conditional infiniteness of primitive weird numbers". Journal of Number Theory. 147. Elsevier: 508–514. doi:10.1016/j.jnt.2014.07.024.
- ^ Amato, Gianluca; Hasler, Maximilian; Melfi, Giuseppe; Parton, Maurizio (2019). "Primitive abundant and weird numbers with many prime factors". Journal of Number Theory. 201. Elsevier: 436–459. arXiv:1802.07178. doi:10.1016/j.jnt.2019.02.027. S2CID 119136924.