I’ve remarked before that to Mathematicians, numbers aren’t, … well, they’re not just numbers. Numbers have personalities. Whilst some are deficient, there are others which are aspiring, and a few who are actually perfect; half of them are rather odd, some downright weird. Of more concern are the odious and evil numbers, especially those that consider themselves untouchable^{1}.

But today we’ll look at a more friendly subject: the amicable numbers. What is an amicable number? Let’s get mathematical and define a function d(n) to be the sum of all proper divisors of n (a proper divisor of n is any divisor of n less than n). Then an amicable number a is one which has a friend b where d(a) = b and d(b) = a (note that we don’t allow a to be narcissistic and count itself as a friend). If you do the math (or these days, more realistically, the google search) you’ll find that the first pair of amicable numbers is 220 and 284.

Amicable numbers have been studied since Greek times: the Pythagoreans in particular credited these special numbers with mystical powers. They’ve even by tried out as a kind of mathematical aphrodisiac:

El Madshriti, an Arab of the 11th century, experimented with the erotic effects of amicable numbers by giving a beautiful woman the smaller number 220 to eat in the form of a cookie, and himself eating the larger 284!

My sudden interest in amicable numbers was driven by strictly mathematical desire - to solve Project Euler 21^{2}. The problem simply asks us to find the sum of all amicable numbers less than 10000. And in fact, the solution to the problem pretty much falls out of its definition.

First, I’ll borrow a function from a few problems back that lists all the divisors of a number:

public static class NumericExtensions { public static IEnumerable<int> Divisors(this int number) { return (from factor in 1.To((int)Math.Sqrt(number)) where number.IsDivisibleBy(factor) select new [] { factor, number / factor }).Concat(); } }

With that, the answer is straightforward:

[EulerProblem(21, Title = "Evaluate the sum of all amicable pairs under 10000.")] public class Problem21 { public long Solve() { Func<int, int> d = n => n.Divisors().Sum() - n; return ( from a in 1.To(10000) let b = d(a) where a != b && d(b) == a select a ) .Sum(); } }

As always, full code is available on the Project Euler Code Gallery page.

##### Footnotes

- If you want to learn more about number personalities, see this catalogue.
- Thanks to Bela Istok who sent me a nice functional solution to the problem, which prompted me to think about how I would solve it myself.