Odd Weird Search/en

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This project is a number-theoretic project which searches for odd weird numbers.

What are weird numbers.

For a number N, if the sum of all its proper divisor is greater than itself, then it is called an abundant number. For example, 8 is not an abundant number, since its proper divisors are 1, 2 and 4, and 1+2+4=7<8. 12 is an abundant number, since 1+2+3+4+6=16>12.

An abundant number is a weird number if no subset of its proper divisors sums to itself. For example, 12 is not a weird number, since 1+2+3+6=12. 70 is a weird number. Its proper divisors are 1,2,5,7,10,14,35, which sums to 74. If there is a subset of proper divisors that sums to 70, its complement will have a sum of 4, which is clearly impossible, therefore 70 is a weird number.

Here is the OEIS page for weird numbers. It also contains a list of known weird numbers. It is worth noting that every number in the list is even.

In fact, no odd weird number is known. Previous effort searches up to 1017. The great mathematician Paul Erdos offered a 10$ prize for finding an odd weird number, and a 25$ for a proof of non-existence. We may conclude that Erdos thought an odd weird number should be more probable to exist, and the problem is interesting enough for Erdos to put money in it, since he is known for have money prizes for unsolved mathematical problems.

The project continues this effort of searching for odd weird numbers up to 1020 or 3*1020, or even 1021. The CPU time to test up to 1020 would be roughly 26.3 years, including the factor for duplicating results for verification.

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