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.

The term "weird number" also refers to a phenomenon in two's complement arithmetic.
Euler diagram of numbers under 100:
   Abundant
   Primitive abundant
   Highly abundant
   Superabundant and highly composite
   Colossally abundant and superior highly composite
   Weird
   Perfect
   Composite
   Deficient

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)

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

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: weird numbers that are not a multiple of other weird numbers (sequence A002975 in the OEIS). Among the 1765 weird numbers less than one million, there are 24 primitive weird numbers. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form 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 weird numbers, and Melfi has shown that the infiniteness of primitive weird numbers is a consequence of Cramér's conjecture.[7] Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.[8]

See also

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. JSTOR 2005938. MR 0347726. Zbl 0279.10005.
  5. 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
  6. Kravitz, Sidney (1976). "A search for large weird numbers". Journal of Recreational Mathematics. Baywood Publishing. 9 (2): 82–85. Zbl 0365.10003.
  7. 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.
  8. Amato, Gianluca; Hasler, Maximilian; Melfi, Giuseppe; Parton, Maurizio (2019). "Primitive abundant and weird numbers with many prime factors". Journal of Number Theory. Elsevier. 201: 436–459. arXiv:1802.07178. doi:10.1016/j.jnt.2019.02.027. S2CID 119136924.
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