Learning F# - printing prime numbers - algorithm

Question

Yesterday I started looking at F# during some spare time. I thought I would start with the standard problem of printing out all the prime numbers up to 100. Heres what I came up with...
#light
open System
let mutable divisable = false
let mutable j = 2
for i = 2 to 100 do
j <- 2
while j < i do
if i % j = 0 then divisable <- true
j <- j + 1
if divisable = false then Console.WriteLine(i)
divisable <- false
The thing is I feel like I have approached this from a C/C# perspective and not embraced the true functional language aspect.
I was wondering what other people could come up with - and whether anyone has any tips/pointers/suggestions. I feel good F# content is hard to come by on the web at the moment, and the last functional language I touched was HOPE about 5 years ago in university.

Answer 1

Here is a simple implementation of the Sieve of Eratosthenes in F#:
let rec sieve = function
| (p::xs) -> p :: sieve [ for x in xs do if x % p > 0 then yield x ]
| [] -> []
let primes = sieve [2..50]
printfn "%A" primes // [2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47]
This implementation won't work for very large lists but it illustrates the elegance of a functional solution.

Answer 2

Using a Sieve function like Eratosthenes is a good way to go. Functional languages work really well with lists, so I would start with that in mind for struture.
On another note, functional languages work well constructed out of functions (heh). For a functional language "feel" I would build a Sieve function and then call it to print out the primes. You could even split it up--one function builds the list and does all the work and one goes through and does all the printing, neatly separating functionality.
There's a couple of interesting versions here.
And there are well known implementations in other similar languages. Here's one in OCAML that beats one in C.

Answer 3

Here are my two cents:
let rec primes =
seq {
yield 2
yield! (Seq.unfold (fun i -> Some(i, i + 2)) 3)
|> Seq.filter (fun p ->
primes
|> Seq.takeWhile (fun i -> i * i <= p)
|> Seq.forall (fun i -> p % i <> 0))
}
for i in primes do
printf "%d " i
Or maybe this clearer version of the same thing as isprime is defined as a separate function:
let rec isprime x =
primes
|> Seq.takeWhile (fun i -> i*i <= x)
|> Seq.forall (fun i -> x%i <> 0)
and primes =
seq {
yield 2
yield! (Seq.unfold (fun i -> Some(i,i+2)) 3)
|> Seq.filter isprime
}

Answer 4

You definitely do not want to learn from this example, but I wrote an F# implementation of a NewSqueak sieve based on message passing:
type 'a seqMsg =
| Die
| Next of AsyncReplyChannel<'a>
type primes() =
let counter(init) =
MailboxProcessor.Start(fun inbox ->
let rec loop n =
async { let! msg = inbox.Receive()
match msg with
| Die -> return ()
| Next(reply) ->
reply.Reply(n)
return! loop(n + 1) }
loop init)
let filter(c : MailboxProcessor<'a seqMsg>, pred) =
MailboxProcessor.Start(fun inbox ->
let rec loop() =
async {
let! msg = inbox.Receive()
match msg with
| Die ->
c.Post(Die)
return()
| Next(reply) ->
let rec filter' n =
if pred n then async { return n }
else
async {let! m = c.AsyncPostAndReply(Next)
return! filter' m }
let! testItem = c.AsyncPostAndReply(Next)
let! filteredItem = filter' testItem
reply.Reply(filteredItem)
return! loop()
}
loop()
)
let processor = MailboxProcessor.Start(fun inbox ->
let rec loop (oldFilter : MailboxProcessor<int seqMsg>) prime =
async {
let! msg = inbox.Receive()
match msg with
| Die ->
oldFilter.Post(Die)
return()
| Next(reply) ->
reply.Reply(prime)
let newFilter = filter(oldFilter, (fun x -> x % prime <> 0))
let! newPrime = oldFilter.AsyncPostAndReply(Next)
return! loop newFilter newPrime
}
loop (counter(3)) 2)
member this.Next() = processor.PostAndReply( (fun reply -> Next(reply)), timeout = 2000)
interface System.IDisposable with
member this.Dispose() = processor.Post(Die)
static member upto max =
[ use p = new primes()
let lastPrime = ref (p.Next())
while !lastPrime <= max do
yield !lastPrime
lastPrime := p.Next() ]
Does it work?
> let p = new primes();;
val p : primes
> p.Next();;
val it : int = 2
> p.Next();;
val it : int = 3
> p.Next();;
val it : int = 5
> p.Next();;
val it : int = 7
> p.Next();;
val it : int = 11
> p.Next();;
val it : int = 13
> p.Next();;
val it : int = 17
> primes.upto 100;;
val it : int list
= [2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
73; 79; 83; 89; 97]
Sweet! :)

Answer 5

Simple but inefficient suggestion:
Create a function to test whether a single number is prime
Create a list for numbers from 2 to 100
Filter the list by the function
Compose the result with another function to print out the results
To make this efficient you really want to test for a number being prime by checking whether or not it's divisible by any lower primes, which will require memoisation. Probably best to wait until you've got the simple version working first :)
Let me know if that's not enough of a hint and I'll come up with a full example - thought it may not be until tonight...

Answer 6

Here is my old post at HubFS about using recursive seq's to implement prime number generator.
For case you want fast implementation, there is nice OCaml code by Markus Mottl
P.S. if you want to iterate prime number up to 10^20 you really want to port primegen by D. J. Bernstein to F#/OCaml :)

Answer 7

While solving the same problem, I have implemented Sieve of Atkins in F#. It is one of the most efficient modern algorithms.
// Create sieve
let initSieve topCandidate =
let result = Array.zeroCreate<bool> (topCandidate + 1)
Array.set result 2 true
Array.set result 3 true
Array.set result 5 true
result
// Remove squares of primes
let removeSquares sieve topCandidate =
let squares =
seq { 7 .. topCandidate}
|> Seq.filter (fun n -> Array.get sieve n)
|> Seq.map (fun n -> n * n)
|> Seq.takeWhile (fun n -> n <= topCandidate)
for n2 in squares do
n2
|> Seq.unfold (fun state -> Some(state, state + n2))
|> Seq.takeWhile (fun x -> x <= topCandidate)
|> Seq.iter (fun x -> Array.set sieve x false)
sieve
// Pick the primes and return as an Array
let pickPrimes sieve =
sieve
|> Array.mapi (fun i t -> if t then Some i else None)
|> Array.choose (fun t -> t)
// Flip solutions of the first equation
let doFirst sieve topCandidate =
let set1 = Set.ofList [1; 13; 17; 29; 37; 41; 49; 53]
let mutable x = 1
let mutable y = 1
let mutable go = true
let mutable x2 = 4 * x * x
while go do
let n = x2 + y*y
if n <= topCandidate then
if Set.contains (n % 60) set1 then
Array.get sieve n |> not |> Array.set sieve n
y <- y + 2
else
y <- 1
x <- x + 1
x2 <- 4 * x * x
if topCandidate < x2 + 1 then
go <- false
// Flip solutions of the second equation
let doSecond sieve topCandidate =
let set2 = Set.ofList [7; 19; 31; 43]
let mutable x = 1
let mutable y = 2
let mutable go = true
let mutable x2 = 3 * x * x
while go do
let n = x2 + y*y
if n <= topCandidate then
if Set.contains (n % 60) set2 then
Array.get sieve n |> not |> Array.set sieve n
y <- y + 2
else
y <- 2
x <- x + 2
x2 <- 3 * x * x
if topCandidate < x2 + 4 then
go <- false
// Flip solutions of the third equation
let doThird sieve topCandidate =
let set3 = Set.ofList [11; 23; 47; 59]
let mutable x = 2
let mutable y = x - 1
let mutable go = true
let mutable x2 = 3 * x * x
while go do
let n = x2 - y*y
if n <= topCandidate && 0 < y then
if Set.contains (n % 60) set3 then
Array.get sieve n |> not |> Array.set sieve n
y <- y - 2
else
x <- x + 1
y <- x - 1
x2 <- 3 * x * x
if topCandidate < x2 - y*y then
go <- false
// Sieve of Atkin
let ListAtkin (topCandidate : int) =
let sieve = initSieve topCandidate
[async { doFirst sieve topCandidate }
async { doSecond sieve topCandidate }
async { doThird sieve topCandidate }]
|> Async.Parallel
|> Async.RunSynchronously
|> ignore
removeSquares sieve topCandidate |> pickPrimes
I know some don't recommend to use Parallel Async, but it did increase the speed ~20% on my 2 core (4 with hyperthreading) i5. Which is about the same increase I got using TPL.
I have tried rewriting it in functional way, getting read of loops and mutable variables, but performance degraded 3-4 times, so decided to keep this version.