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Sieve of Atkin

Modern and efficient algorithm to generate prime numbers up to certain number Uses Parallel Async to run computations in parallel

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// 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
val initSieve : topCandidate:int -> bool []

Full name: Script.initSieve
val topCandidate : int
val result : bool []
module Array

from Microsoft.FSharp.Collections
val zeroCreate : count:int -> 'T []

Full name: Microsoft.FSharp.Collections.Array.zeroCreate
type bool = System.Boolean

Full name: Microsoft.FSharp.Core.bool
val set : array:'T [] -> index:int -> value:'T -> unit

Full name: Microsoft.FSharp.Collections.Array.set
val removeSquares : sieve:bool [] -> topCandidate:int -> bool []

Full name: Script.removeSquares
val sieve : bool []
val squares : seq<int>
Multiple items
val seq : sequence:seq<'T> -> seq<'T>

Full name: Microsoft.FSharp.Core.Operators.seq

--------------------
type seq<'T> = System.Collections.Generic.IEnumerable<'T>

Full name: Microsoft.FSharp.Collections.seq<_>
module Seq

from Microsoft.FSharp.Collections
val filter : predicate:('T -> bool) -> source:seq<'T> -> seq<'T>

Full name: Microsoft.FSharp.Collections.Seq.filter
val n : int
val get : array:'T [] -> index:int -> 'T

Full name: Microsoft.FSharp.Collections.Array.get
val map : mapping:('T -> 'U) -> source:seq<'T> -> seq<'U>

Full name: Microsoft.FSharp.Collections.Seq.map
val takeWhile : predicate:('T -> bool) -> source:seq<'T> -> seq<'T>

Full name: Microsoft.FSharp.Collections.Seq.takeWhile
val n2 : int
val unfold : generator:('State -> ('T * 'State) option) -> state:'State -> seq<'T>

Full name: Microsoft.FSharp.Collections.Seq.unfold
val state : int
union case Option.Some: Value: 'T -> Option<'T>
val x : int
val iter : action:('T -> unit) -> source:seq<'T> -> unit

Full name: Microsoft.FSharp.Collections.Seq.iter
val pickPrimes : sieve:bool [] -> int []

Full name: Script.pickPrimes
val mapi : mapping:(int -> 'T -> 'U) -> array:'T [] -> 'U []

Full name: Microsoft.FSharp.Collections.Array.mapi
val i : int
val t : bool
union case Option.None: Option<'T>
val choose : chooser:('T -> 'U option) -> array:'T [] -> 'U []

Full name: Microsoft.FSharp.Collections.Array.choose
val t : int option
val doFirst : sieve:bool [] -> topCandidate:int -> unit

Full name: Script.doFirst
val set1 : Set<int>
Multiple items
module Set

from Microsoft.FSharp.Collections

--------------------
type Set<'T (requires comparison)> =
  interface IComparable
  interface IEnumerable
  interface IEnumerable<'T>
  interface ICollection<'T>
  new : elements:seq<'T> -> Set<'T>
  member Add : value:'T -> Set<'T>
  member Contains : value:'T -> bool
  override Equals : obj -> bool
  member IsProperSubsetOf : otherSet:Set<'T> -> bool
  member IsProperSupersetOf : otherSet:Set<'T> -> bool
  ...

Full name: Microsoft.FSharp.Collections.Set<_>

--------------------
new : elements:seq<'T> -> Set<'T>
val ofList : elements:'T list -> Set<'T> (requires comparison)

Full name: Microsoft.FSharp.Collections.Set.ofList
val mutable x : int
val mutable y : int
val mutable go : bool
val mutable x2 : int
val contains : element:'T -> set:Set<'T> -> bool (requires comparison)

Full name: Microsoft.FSharp.Collections.Set.contains
val not : value:bool -> bool

Full name: Microsoft.FSharp.Core.Operators.not
val doSecond : sieve:bool [] -> topCandidate:int -> unit

Full name: Script.doSecond
val set2 : Set<int>
val doThird : sieve:bool [] -> topCandidate:int -> unit

Full name: Script.doThird
val set3 : Set<int>
val ListAtkin : topCandidate:int -> int []

Full name: Script.ListAtkin
Multiple items
val int : value:'T -> int (requires member op_Explicit)

Full name: Microsoft.FSharp.Core.Operators.int

--------------------
type int = int32

Full name: Microsoft.FSharp.Core.int

--------------------
type int<'Measure> = int

Full name: Microsoft.FSharp.Core.int<_>
val async : AsyncBuilder

Full name: Microsoft.FSharp.Core.ExtraTopLevelOperators.async
Multiple items
type Async
static member AsBeginEnd : computation:('Arg -> Async<'T>) -> ('Arg * AsyncCallback * obj -> IAsyncResult) * (IAsyncResult -> 'T) * (IAsyncResult -> unit)
static member AwaitEvent : event:IEvent<'Del,'T> * ?cancelAction:(unit -> unit) -> Async<'T> (requires delegate and 'Del :> Delegate)
static member AwaitIAsyncResult : iar:IAsyncResult * ?millisecondsTimeout:int -> Async<bool>
static member AwaitTask : task:Task<'T> -> Async<'T>
static member AwaitWaitHandle : waitHandle:WaitHandle * ?millisecondsTimeout:int -> Async<bool>
static member CancelDefaultToken : unit -> unit
static member Catch : computation:Async<'T> -> Async<Choice<'T,exn>>
static member FromBeginEnd : beginAction:(AsyncCallback * obj -> IAsyncResult) * endAction:(IAsyncResult -> 'T) * ?cancelAction:(unit -> unit) -> Async<'T>
static member FromBeginEnd : arg:'Arg1 * beginAction:('Arg1 * AsyncCallback * obj -> IAsyncResult) * endAction:(IAsyncResult -> 'T) * ?cancelAction:(unit -> unit) -> Async<'T>
static member FromBeginEnd : arg1:'Arg1 * arg2:'Arg2 * beginAction:('Arg1 * 'Arg2 * AsyncCallback * obj -> IAsyncResult) * endAction:(IAsyncResult -> 'T) * ?cancelAction:(unit -> unit) -> Async<'T>
static member FromBeginEnd : arg1:'Arg1 * arg2:'Arg2 * arg3:'Arg3 * beginAction:('Arg1 * 'Arg2 * 'Arg3 * AsyncCallback * obj -> IAsyncResult) * endAction:(IAsyncResult -> 'T) * ?cancelAction:(unit -> unit) -> Async<'T>
static member FromContinuations : callback:(('T -> unit) * (exn -> unit) * (OperationCanceledException -> unit) -> unit) -> Async<'T>
static member Ignore : computation:Async<'T> -> Async<unit>
static member OnCancel : interruption:(unit -> unit) -> Async<IDisposable>
static member Parallel : computations:seq<Async<'T>> -> Async<'T []>
static member RunSynchronously : computation:Async<'T> * ?timeout:int * ?cancellationToken:CancellationToken -> 'T
static member Sleep : millisecondsDueTime:int -> Async<unit>
static member Start : computation:Async<unit> * ?cancellationToken:CancellationToken -> unit
static member StartAsTask : computation:Async<'T> * ?taskCreationOptions:TaskCreationOptions * ?cancellationToken:CancellationToken -> Task<'T>
static member StartChild : computation:Async<'T> * ?millisecondsTimeout:int -> Async<Async<'T>>
static member StartChildAsTask : computation:Async<'T> * ?taskCreationOptions:TaskCreationOptions -> Async<Task<'T>>
static member StartImmediate : computation:Async<unit> * ?cancellationToken:CancellationToken -> unit
static member StartWithContinuations : computation:Async<'T> * continuation:('T -> unit) * exceptionContinuation:(exn -> unit) * cancellationContinuation:(OperationCanceledException -> unit) * ?cancellationToken:CancellationToken -> unit
static member SwitchToContext : syncContext:SynchronizationContext -> Async<unit>
static member SwitchToNewThread : unit -> Async<unit>
static member SwitchToThreadPool : unit -> Async<unit>
static member TryCancelled : computation:Async<'T> * compensation:(OperationCanceledException -> unit) -> Async<'T>
static member CancellationToken : Async<CancellationToken>
static member DefaultCancellationToken : CancellationToken

Full name: Microsoft.FSharp.Control.Async

--------------------
type Async<'T>

Full name: Microsoft.FSharp.Control.Async<_>
static member Async.Parallel : computations:seq<Async<'T>> -> Async<'T []>
static member Async.RunSynchronously : computation:Async<'T> * ?timeout:int * ?cancellationToken:System.Threading.CancellationToken -> 'T
val ignore : value:'T -> unit

Full name: Microsoft.FSharp.Core.Operators.ignore
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More information

Link:http://fssnip.net/t2
Posted:3 years ago
Author:Sergey OCHKIN
Tags: sieve , atkin , prime