Amicable Numbers (part I)

I know of no practical use for Amicable numbers, but they’re rather fun. And rare. But let’s start with a definition. Let be a natural number (a positive integer), and let

with and , be the sum of the divisors of . We’re in fact interested in the sum of the proper divisors of , that is,

Now we’re ready to define amicable numbers!

Amicable numbers: Two numbers, are amicable, if, and only if, and .

Given , we can find and test if . But to do that efficiently, we need to compute (or ) very rapidly. The first expression above requires , but we can do much better. Let’s see how.

Compressed Series (Part I)

Numerical methods are generally rather hard to get right because of error propagation due to the limited precision of floats (or even doubles). This seems to be especially true with methods involving series, where a usually large number of ever diminishing terms must added to attain something that looks like convergence. Even fairly simple sequences such as

may be complex to evaluate. First, is cumbersome, and becomes small exceedingly rapidly.

Sohcahtoa!

Mathematics can ask you to remember things that have no obvious connection to common sense. Either because it’s arbitrary (the name of a function in respect to what it computes) or because you haven’t quite figured all the details out. In either cases, a few mnemonics are always useful!