# Sieve of Eratosthenes

A function to generate prime numbers is often useful. This sieve of eratosthenes is one of the simplest ways to find prime numbers, and it is much more efficient than the brute force method.

Here is a nice description of how to perform the sieve, and here is a previous post I wrote about it. This time, I rewrote it in OCaml, and got a little farther by only generating the odd prime numbers.

First, here is a helper function to take an array of integers, and return a new array that contains only the non-zero values. This will be useful later in all of the sieve functions.

(* return a new array of the non-zero items *) let filter_non_zeros ar = let count = Array.fold_left (fun count x -> if x <> 0 then count + 1 else count) 0 ar in let result = Array.make count 0 in let rec go idx jdx = if jdx = count then result else if ar.(idx) = 0 then go (idx + 1) jdx else (result.(jdx) <- ar.(idx); go (idx + 1) (jdx + 1)) in go 0 0

Next, the straightforward sieve, first written to use recursion instead of loops. Loops are not necessary in languages like OCaml, but sometimes they are nicer than the recursion, depending on the problem.

(* Sieve of eratosthenes, using all recursion for the looping *) let sieve n = let result = Array.init (n+1) (fun i -> i) in let stop_index = int_of_float (sqrt (float_of_int n) ) + 1 in result.(1) <- 0; let rec loop idx = if idx = stop_index then () else if result.(idx) = 0 then (* not prime *) loop (idx + 1) else let rec clear j = if j <= n then (result.(j) <- 0; clear (j + idx)); in clear (idx * idx); loop (idx + 1) in loop 2; filter_non_zeros result

This function creates an array of potential prime numbers. Then, it walks through the array, looking for the next prime. When a new prime is found, the function clears all of the multiples of the prime in the rest of the array.

Next is a function that moves the inner most recursion into a loop.

(* Sieve of Eratosthenes, moving the inner loop into a loop *) let sieve2 n = let result = Array.init (n+1) (fun i -> i) in let stop_index = int_of_float (sqrt (float_of_int n) ) + 1 in result.(1) <- 0; let rec loop idx = if idx = stop_index then () else if result.(idx) = 0 then (* not prime *) loop (idx + 1) else (let j = ref (idx * idx) in while !j <= n do (result.(!j) <- 0; j := !j + idx); done; loop (idx + 1)) in loop 2; filter_non_zeros result

That function takes the same time as the first one, and is just a difference in style.

Next is a function that does away with the recursion altogether.

(* Sieve of Eratosthenes, using all loops instead of recursion *) let sieve3 n = let result = Array.init (n+1) (fun i -> i) in let stop_index = int_of_float (sqrt (float_of_int n) ) + 1 in result.(1) <- 0; let idx = ref 2 in while !idx <= stop_index do if result.(!idx) <> 0 then (let p = !idx in let j = ref (p * p) in while !j <= n do result.(!j) <- 0; j := !j + p; done); idx := !idx + 1; done; filter_non_zeros result

Finally, the last method only generates the odd primes. We can ignore even numbers right from the start, and halve the memory and timing requirements. This requires a little trickery to do all of the indexing correctly, but it isn't too bad.

(* This one ignores the even numbers entirely, so it is about twice as fast now *) let sieve4 n = let m = (n / 2) + 1 in let result = Array.init m (fun i->2*i+1) in (* Remeber that 2 is a prime *) result.(0) <- 2; let stop_index = int_of_float (sqrt (float_of_int n) /. 2.) + 1 in let idx = ref 1 in while !idx <= stop_index do if result.(!idx) <> 0 then (* 2 * idx + 1 is the next prime. Now remove its multiples. We want to start at p*p, and step by p. for example, If idx is 1, then p is 3. To get result(i) = 9, we need i to be 4. If idx is 2, and p = 5, then to get result(i) = 25, we need i to be 12. So, we can move out by idx * p, then step by p *) (let p = result.(!idx) in let j = ref (!idx * (p + 1)) in while !j < m do result.(!j) <- 0; j := !j + p; done); idx := !idx + 1; done; filter_non_zeros result

And finally, a test driver.

let main sieve_fcn = let n = 15485863 in let m = Array.length (sieve_fcn n) in Printf.printf "number of primes less than %d: %d\n" n m; ;; main sieve4;;