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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
List of unsolved problems in mathematics
Are there any odd weird numbers?

It has been shown that an infinite number of weird numbers exist;[3] in fact, the sequence of weird numbers has positive asymptotic density.[4]

It is not known if any odd weird numbers exist; if any do, they must be greater than 232 ≈ 4×109[5] or 1×1017[6]

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

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

also prime and greater than 2k , 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)\cdot153722867280912929\ \approx\ 2\cdot10^{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. 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[8]

References

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.
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.