#### Problem

Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).

If d(a) = b and d(b) = a, where a != b, then a and b are an amicable pair and each of a and b are called amicable numbers.

For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.

Evaluate the sum of all the amicable numbers under 10000.

#### Solution

open System let findDivisors(n) = let upperBound = int32(Math.Sqrt(double(n))) [1..upperBound] |> Seq.filter (fun x -> n % x = 0) |> Seq.collect (fun x -> [x; n/x]) |> Seq.filter (fun x -> x <> n) let d(n) = findDivisors(n) |> Seq.sum let dList = [ for n = 1 to 9999 do yield (n, d(n)) ] let answer = dList |> List.filter (fun (a, da) -> dList |> List.exists (fun (b, db) -> b = da && a = db && a <> b)) |> List.sumBy (fun (n, dn) -> n)

First I defined a *findDivisors* function which returns all the divisors of a given number excluding itself as is the case in the example given in the brief. Then I defined the function *d* which returns the sum of the given number’s proper divisors.

To get to the answer, I generated a list of tuples for the numbers from 1 to 9999 in the form of *(n, d(n))* and from there I iterated through the list and for each tuple check whether there exists another tuple which matches the conditions required for the two to be considered an amicable pair.

You might have noticed in the function applied to List.filter to identify amicable pairs that I have decomposed the tuple into (a, da), as I have mentioned before this is a form of the pattern matching in F#. The same pattern matching technique is also used when I calculate the sum of the amicable pairs.

Hi, I’m **Yan**. I’m an **AWS Serverless Hero** and the author of **Production-Ready Serverless**.

I specialise in rapidly transitioning teams to serverless and building production-ready services on AWS.

Are you struggling with serverless or need guidance on best practices? Do you want someone to review your architecture and help you avoid costly mistakes down the line? Whatever the case, I’m here to help.

Check out my new course, **Complete Guide to AWS Step Functions**. In this course, we’ll cover everything you need to know to use AWS Step Functions service effectively. Including basic concepts, HTTP and event triggers, activities, callbacks, nested workflows, design patterns and best practices.

Further reading

Here is a complete list of all my posts on serverless and AWS Lambda. In the meantime, here are a few of my most popular blog posts.

- Lambda optimization tip – enable HTTP keep-alive
- You are thinking about serverless costs all wrong
- Many faced threats to Serverless security
- We can do better than percentile latencies
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- Yubl’s road to Serverless
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- AWS Lambda – compare coldstart time with different languages, memory and code sizes
- Guys, we’re doing pagination wrong