In number theory, a weird number is a natural number that is abundant but not semiperfect. 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
Are there any odd weird numbers?
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. With this formula, he found a 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. This leads to the definition of primitive weird numbers, i.e. weird numbers that are not multiple of other weird numbers. Indeed the construction of Kravitz allows to build primitive weird numbers. 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 conjecture
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- 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.
- 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".
- http://oeis.org/A006037 OEIS - Odd weird numbers
- Kravitz, Sidney (1976). "A search for large weird numbers". Journal of Recreational Mathematics (Baywood Publishing) 9 (2): 82–85. Zbl 0365.10003.
- 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.