# There are no odd weird numbers < 10^18

Bob Hearn rah at ai.mit.edu
Mon Jul 11 01:32:52 CEST 2005

```I'm pretty sure searching all 8-factor primitive abundants is not
feasible; the number of primitive abundants with n factors is finite,
but grows very fast (might make a good series!). I set my program
trying to count them. It started by setting the first 7 factors to 3
* 5 * 7 * 11 * 389 * 29,959 * 128,194,589. This is lexicographically
first; lowering any factor will lead to a number that is abundant
before using the 8th factor. But now the 8th factor can range from
128,194,589 up to 5.6 * 10^14 - already far more than the ~10^8 7-
factor primitive abundants. After that, we can increment 128,194,589
to the next prime, and try again for the last factor. And so on.

If there were some way of ruling out a potential weird number based
on a factor prefix rather than the entire factorization, then that
would be different...

Bob

On Jul 10, 2005, at 1:28 PM, hv at crypt.org wrote:

> Bob Hearn <rah at ai.mit.edu> wrote:
> :So, there are no odd weirds < 10^18.
> [...]
> :(Hugo van der Sanden has been working on a similar program.)
>
> My program works on number of factors rather than size of product;
> it confirms there are no odd weirds with fewer than 8 factors, and
> I hope to improve on that (but don't expect the next result any
> time soon).
>
> Hugo
>
>

```