Weird number

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In mathematics, 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 first weird number is 70, because its proper divisors are 1, 2, 5, 7, 10, 14, and 35; which sum to 74, but no subset of these sums to 70. The number 12 is abundant but not a weird number 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). It has been shown that an infinite number of weird numbers exist, and the sequence of weird numbers has also been proven to have positive asymptotic density.

It is not known if any odd weird numbers exist; if any do, they must be greater than 10¹⁸ (as noted by Bob Hearn in a July 2005 posting to the SeqFans mailing list).