There are no odd weird numbers < 10^18

[Bob Hearn]

Hi all,

I'm new here... I understand there's been some discussion of weird  
numbers here lately (A006037). I've been running big searches for odd  
weirds. My latest run finished this morning. A month ago I had  
exhaustively searched to 10^17; this last run searched 10^17 - 10^18.  
The search took a little over a week, on a dual 2.5 GHz G5. So, there  
are no odd weirds < 10^18. The search steps through factorizations of  
primitive abundants, cutting off the search based on the bounds.  
(Hugo van der Sanden has been working on a similar program.) The  
highest bound I'd seen previously was 2^32 ~= 4 * 10^9, although Hugo  
mentions that an email from Jud McCranie says:

:I have an old email from a friend in which he states that there are  
no odd
:weird numbers < about 2^40, but there is no reference.

So that would be about 10^12. The density of primitive abundants  
decreases by about half for every extra power of 10, so a search to  
10^19 would take my program over a month, unless it could be sped up  
some more. But I'm doubtful I can wring much more out of it.

I have various other search results about odd weirds, not very well  
organized. E.g., there are higher bounds for odd weirds with small  
abundances. Soon I'll put together a page describing all the results.

One more thing... if 9 were prime, then 6,237 = 3^2 * 7 * 9 * 11  
would be weird. The next such odd "bogus weird" may be 1,772,632,575  
= 3 * 5^2 * 7 * 21 * 103 * 1,561; at any rate, there are no more <  
10^9. I'm not sure whether this means anything.

Bob Hearn

---------------------------------------------
Robert A. Hearn
rah at ai.mit.edu
http://www.swiss.ai.mit.edu/~bob/