Def: a pair natural numbers $a$, $b$, $a\ne b$ are an [Amicable pair](http://en.wikipedia.org/wiki/Amicable_numbers) if

$\sum\_{d|a,a\ne d}d = b$ and $\sum\_{d|b, b\ne d}d = a$. 

Ok. So I'm trying to optimize a calculation for finding the number of amicable numbers below some integer $z$. What I'm trying to figure out, is if there is some ordering to the pairs? Ie. Is it true that if $(a,b)$ and $(c,d)$ are distinct amicable pairs, then either $a,b < c,d$ or $a,b > c,d$ ? (I suspect it is [true](http://djm.cc/amicable.txt), but have no proof)

Also, does anyone have a smart way of approaching this that doesn't require me to crawl back to the computer? Thanks!