Problem #69 says:

Euler’s Totient function, φ(n) [sometimes called the phi function], is used to determine the number of
numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than
nine and relatively prime to nine, φ(9)=6.

Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.

For this problem I created a class that held the number and it’s totient and then create them in parallel.
Then I loop through the mess looking for the instance with the highest check value. The Parallelism is handled by
the PSeq object from the F# power pack.

Solution in f# and requires .net 4 and the F# power pack from codeplex. Runs in ~48 seconds.

#light

open System
open System.Diagnostics
open Microsoft.FSharp.Linq
open Microsoft.FSharp.Collections

//set the limit and bound
let UpperLimit = 1000000.
let UpperBound = (Math.Sqrt(UpperLimit))

//generate numbers to check
let Primes = ref [2. .. (UpperLimit)] in

// loop through all numbers
// keep removing numbers that divide evenly
for div = 2 to int UpperBound do
  Primes := List.filter(fun num -> (num % (float div) <> 0. || (float div) >= num))
        !Primes
done;

let rec factorise (n:float) =
    if n = 1. then [] else
    let a = !Primes |> List.find (fun x -> n % x = 0.)
    a :: factorise (n / a)

let Totient (x:float) =
    (factorise x)
    |> Seq.distinct
    |> Seq.map (fun x -> 1. - (1./x))
    |> Seq.fold(fun acc a -> acc * a) x

type TestCase (Num:float) =
    let mutable _num = Num
    let _tot = Totient (float _num)
    member x.num
        with get() = _num
        and set(v) = _num <- v

    member x.Check
        with get() = x.num / _tot

Console.WriteLine("Starting Check")
let watch = new Stopwatch()
watch.Start()

let Answer =
    let mutable Ans = new TestCase(1.)
    let Tests =
        [2. .. 1000000.]
        |> PSeq.map (fun x -> new TestCase(x))
    for t in Tests do
        if t.Check > Ans.Check then
            Ans <- t
    Ans.num

Console.WriteLine(Answer)
watch.Stop()
Console.WriteLine(watch.Elapsed)
Console.ReadKey() |> ignore