From Wikipedia (2010-02-24):

In mathematics, the sieve of Atkin is a fast, modern algorithm for finding all prime numbers up to a specified integer. It is an optimized version of the ancient sieve of Eratosthenes, but does some preliminary work and then marks off multiples of primes squared, rather than multiples of primes. It was created by A. O. L. Atkin and Daniel J. Bernstein.

This is my translation of the pseudocode to f#:

// Generates a list of all primes below limit
let sieveOfAtkin limit =
    // initialize the sieve
    let sieve = Array.create (limit + 1) false
    // put in candidate primes: 
    // integers which have an odd number of
    // representations by certain quadratic forms
    let inline invCand n pred =
        if n < limit && pred then sieve.[n] <- not sieve.[n] 
    let sqrtLimit = int (sqrt (float limit))
    for x = 1 to sqrtLimit do
        for y = 1 to sqrtLimit do
            let xPow2 = x * x
            let yPow2 = y * y
            let n = 4 * xPow2 + yPow2 in invCand n (let m = n % 12 in m = 1 || m = 5)
            let n = 3 * xPow2 + yPow2 in invCand n (n % 12 = 7)
            let n = 3 * xPow2 - yPow2 in invCand n (x > y && n % 12 = 11)
    // eliminate composites by sieving
    let rec eliminate n =
        if n <= sqrtLimit 
        then if sieve.[n]
             then let nPow2 = n * n
                  for k in nPow2 .. nPow2 .. limit do
                      Array.set sieve k false
             eliminate (n + 2)
    eliminate 5
    // Generate list from the sieve (backwards)
    let rec generateList acc n =
        if n >= 5 then generateList (if sieve.[n] then n :: acc else acc) (n - 1)
        else acc
    2 :: 3 :: (generateList [] limit)

On my MacBook Pro 2.66 GHz Intel Core 2 Duo it generates all primes below 10.000.000 in 557 milliseconds:

> #time;;

--> Timing now on

> sieveOfAtkin 10000000;;
Real: 00:00:00.557, CPU: 00:00:00.471, GC gen0: 0
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; 101; 103; 107; 109; 113; 127; 131; 137; 139; 149; 151;
   157; 163; 167; 173; 179; 181; 191; 193; 197; 199; 211; 223; 227; 229; 233;
   239; 241; 251; 257; 263; 269; 271; 277; 281; 283; 293; 307; 311; 313; 317;
   331; 337; 347; 349; 353; 359; 367; 373; 379; 383; 389; 397; 401; 409; 419;
   421; 431; 433; 439; 443; 449; 457; 461; 463; 467; 479; 487; 491; 499; 503;
   509; 521; 523; 541; ...] 

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