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Def: a pair natural numbers $a$, $b$, $a\ne b$ are an Amicable pair 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, 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!

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up vote 2 down vote accepted

On the list given in the question, the 19th pair $(a,b) = (171856, 176336)$ interleaves with the 20th pair $(b,c) = (176272, 180848)$. There are certainly many more instances, but of course no one knows whether there are infinitely many.

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Wow I didn't even see that. Guess that answers that! Thanks! – gone Feb 3 '13 at 22:18

A smaller counter-example is (a,b) = (63020, 76084); (c,d) = (66928, 66,992), where a < c < d < b.

These are pairs 9 and 10 in

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