Almost prime

A k-Almost-prime is a natural number $n$ that is the product of $k$ (possibly identical) primes.

Example

1-almost-primes, where $k=1$ , are the prime numbers themselves.
2-almost-primes, where $k=2$ , are the semiprimes.

Task

Write a function/method/subroutine/... that generates k-almost primes and use it to create a table here of the first ten members of k-Almost primes for $1<=K<=5$ .

Related tasks

Ada[edit]

This imports the package Prime_Numbers from Prime decomposition#Ada.

with Prime_Numbers, Ada.Text_IO; 
 
procedure Test_Kth_Prime is
 
   package Integer_Numbers is new 
     Prime_Numbers (Natural, 0, 1, 2); 
   use Integer_Numbers;
 
   Out_Length: constant Positive := 10; -- 10 k-th almost primes
   N: Positive; -- the "current number" to be checked
 
begin
   for K in 1 .. 5 loop
      Ada.Text_IO.Put("K =" & Integer'Image(K) &":  ");
      N := 2;
      for I in 1 .. Out_Length loop
	 while Decompose(N)'Length /= K loop
	    N := N + 1;
	 end loop; -- now N is Kth almost prime;
	 Ada.Text_IO.Put(Integer'Image(Integer(N)));
	 N := N + 1;
      end loop;
      Ada.Text_IO.New_Line;
   end loop;
end Test_Kth_Prime;

Output:

K = 1:   2 3 5 7 11 13 17 19 23 29
K = 2:   4 6 9 10 14 15 21 22 25 26
K = 3:   8 12 18 20 27 28 30 42 44 45
K = 4:   16 24 36 40 54 56 60 81 84 88
K = 5:   32 48 72 80 108 112 120 162 168 176

ALGOL 68[edit]

Worth noticing is the n(...)(...) picture in the printf and the WHILE ... DO SKIP OD idiom which is quite common in ALgol 68.

BEGIN
   INT examples=10, classes=5;
   MODE SEMIPRIME = STRUCT ([examples]INT data, INT count);
   [classes]SEMIPRIME semi primes;
   PROC num facs = (INT n) INT :
COMMENT
   Return number of not necessarily distinct prime factors of n.
   Not very efficient for large n ...
COMMENT
   BEGIN
      INT tf := 2, residue := n, count := 1;
      WHILE tf < residue DO
	 INT remainder = residue MOD tf;
	 ( remainder = 0 | count +:= 1; residue %:= tf | tf +:= 1 )
      OD;
      count
   END;
   PROC update table = (REF []SEMIPRIME table, INT i) BOOL :
COMMENT
   Add i to the appropriate row of the table, if any, unless that row
   is already full. Return a BOOL which is TRUE when all of the table
   is full.
COMMENT
   BEGIN
      INT k := num facs(i);
      IF k <= classes
      THEN
	 INT c = 1 + count OF table[k];
	 ( c <= examples | (data OF table[k])[c] := i; count OF table[k] := c )
      FI;
      INT sum := 0;
      FOR i TO classes DO sum +:= count OF table[i] OD;
      sum < classes * examples
   END;
   FOR i TO classes DO count OF semi primes[i] := 0 OD;
   FOR i FROM 2 WHILE update table (semi primes, i) DO SKIP OD;
   FOR i TO classes
   DO
      printf (($"k = ", d, ":", n(examples)(xg(0))l$, i, data OF semi primes[i]))
   OD
END

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

AutoHotkey[edit]

Translation of the C Version

kprime(n,k) {
	p:=2, f:=0
	while( (f<k) && (p*p<=n) ) {
		while ( 0==mod(n,p) ) {
			n/=p
			f++
		}
		p++
	}
	return f + (n>1) == k
}
 
k:=1, results:=""
while( k<=5 ) {
	i:=2, c:=0, results:=results "k =" k ":"
	while( c<10 ) {
		if (kprime(i,k)) {
			results:=results " " i
			c++
		}
		i++
	}
	results:=results "`n"
	k++
}
 
MsgBox % results

Output (Msgbox):

k =1: 2 3 5 7 11 13 17 19 23 29
k =2: 4 6 9 10 14 15 21 22 25 26
k =3: 8 12 18 20 27 28 30 42 44 45
k =4: 16 24 36 40 54 56 60 81 84 88
k =5: 32 48 72 80 108 112 120 162 168 176

AWK[edit]

 
# syntax: GAWK -f ALMOST_PRIME.AWK
BEGIN {
    for (k=1; k<=5; k++) {
      printf("%d:",k)
      c = 0
      i = 1
      while (c < 10) {
        if (kprime(++i,k)) {
          printf(" %d",i)
          c++
        }
      }
      printf("\n")
    }
    exit(0)
}
function kprime(n,k,  f,p) {
    for (p=2; f<k && p*p<=n; p++) {
      while (n % p == 0) {
        n /= p
        f++
      }
    }
    return(f + (n > 1) == k)
}

Output:

1: 2 3 5 7 11 13 17 19 23 29
2: 4 6 9 10 14 15 21 22 25 26
3: 8 12 18 20 27 28 30 42 44 45
4: 16 24 36 40 54 56 60 81 84 88
5: 32 48 72 80 108 112 120 162 168 176

Befunge[edit]

Translation of: C

The extra spaces are to ensure it's readable on buggy interpreters that don't include a space after numeric output.

1>::48*"= k",,,,02p.":",01v
|^ v0!`\*:g40:<p402p300:+1<
K| >2g03g`*#v_ 1`03g+02g->|
[email protected]>/03g1+03p>vpv+1\.:,*48 <
P#|!\g40%g40:<4>:9`>#v_\1^|
|^>#!1#`+#50#:^#+1,+5>#5$<|

Output:

k = 1 : 2  3  5  7  11  13  17  19  23  29
k = 2 : 4  6  9  10  14  15  21  22  25  26
k = 3 : 8  12  18  20  27  28  30  42  44  45
k = 4 : 16  24  36  40  54  56  60  81  84  88
k = 5 : 32  48  72  80  108  112  120  162  168  176

C[edit]

#include <stdio.h>
 
int kprime(int n, int k)
{
	int p, f = 0;
	for (p = 2; f < k && p*p <= n; p++)
		while (0 == n % p)
			n /= p, f++;
 
	return f + (n > 1) == k;
}
 
int main(void)
{
	int i, c, k;
 
	for (k = 1; k <= 5; k++) {
		printf("k = %d:", k);
 
		for (i = 2, c = 0; c < 10; i++)
			if (kprime(i, k)) {
				printf(" %d", i);
				c++;
			}
 
		putchar('\n');
	}
 
	return 0;
}

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

C++[edit]

Translation of: Kotlin

#include <cstdlib>
#include <iostream>
#include <sstream>
#include <iomanip>
#include <list>
 
bool k_prime(unsigned n, unsigned k) {
    unsigned f = 0;
    for (unsigned p = 2; f < k && p * p <= n; p++)
        while (0 == n % p) { n /= p; f++; }
    return f + (n > 1 ? 1 : 0) == k;
}
 
std::list<unsigned> primes(unsigned k, unsigned n)  {
    std::list<unsigned> list;
    for (unsigned i = 2;list.size() < n;i++)
        if (k_prime(i, k)) list.push_back(i);
    return list;
}
 
int main(const int argc, const char* argv[]) {
    using namespace std;
    for (unsigned k = 1; k <= 5; k++) {
        ostringstream os("");
        const list<unsigned> l = primes(k, 10);
        for (list<unsigned>::const_iterator i = l.begin(); i != l.end(); i++)
            os << setw(4) << *i;
        cout << "k = " << k << ':' << os.str() << endl;
    }
 
	return EXIT_SUCCESS;
}

Output:

k = 1:   2   3   5   7  11  13  17  19  23  29
k = 2:   4   6   9  10  14  15  21  22  25  26
k = 3:   8  12  18  20  27  28  30  42  44  45
k = 4:  16  24  36  40  54  56  60  81  84  88
k = 5:  32  48  72  80 108 112 120 162 168 176

C#[edit]

using System;
using System.Collections.Generic;
using System.Linq;
 
namespace AlmostPrime
{
    class Program
    {
        static void Main(string[] args)
        {
            foreach (int k in Enumerable.Range(1, 5))
            {
                KPrime kprime = new KPrime() { K = k };
                Console.WriteLine("k = {0}: {1}",
                    k, string.Join<int>(" ", kprime.GetFirstN(10)));
            }
        }
    }
 
    class KPrime
    {
        public int K { get; set; }
 
        public bool IsKPrime(int number)
        {
            int primes = 0;
            for (int p = 2; p * p <= number && primes < K; ++p)
            {
                while (number % p == 0 && primes < K)
                {
                    number /= p;
                    ++primes;
                }
            }
            if (number > 1)
            {
                ++primes;
            }
            return primes == K;
        }
 
        public List<int> GetFirstN(int n)
        {
            List<int> result = new List<int>();
            for (int number = 2; result.Count < n; ++number)
            {
                if (IsKPrime(number))
                {
                    result.Add(number);
                }
            }
            return result;
        }
    }
}

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

Clojure[edit]

 
 
(ns clojure.examples.almostprime
	(:gen-class))
 
(defn divisors [n]
    " Finds divisors by looping through integers 2, 3,...i.. up to sqrt (n) [note: rather than compute sqrt(), test with i*i <=n] "
    (let [div (some #(if (= 0 (mod n %)) % nil) (take-while #(<= (* % %) n) (iterate inc 2)))]
        (if div                                                         ; div = nil (if no divisor found else its the divisor) 
            (into [] (concat (divisors div) (divisors (/ n div))))      ; Concat the two divisors of the two divisors
            [n])))                                                      ; Number is prime so only itself as a divisor
 
(defn divisors-k [k n]
    " Finds n numbers with k divisors.  Does this by looping through integers 2, 3, ... filtering (passing) ones with k divisors and 
      taking the first n "
    (->> (iterate inc 2)            ; infinite sequence of numbers starting at 2
         (map divisors)             ; compute divisor of each element of sequence
         (filter #(= (count %) k))  ; filter to take only elements with k divisors
         (take n)                   ; take n elements from filtered sequence
         (map #(apply * %))))       ; compute number by taking product of divisors
 
(println (for [k (range 1 6)]
          (println "k:" k (divisors-k k 10))))
 
}

Output:

(k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176)
nil

Common Lisp[edit]

(defun start ()
  (loop for k from 1 to 5
    do (format t "k = ~a: ~a~%" k (collect-k-almost-prime k))))
 
(defun collect-k-almost-prime (k &optional (d 2) (lst nil))
  (cond ((= (length lst) 10) (reverse lst))
        ((= (?-primality d) k) (collect-k-almost-prime k (+ d 1) (cons d lst)))
        (t (collect-k-almost-prime k (+ d 1) lst))))
 
(defun ?-primality (n &optional (d 2) (c 0))
  (cond ((> d (isqrt n)) (+ c 1))
        ((zerop (rem n d)) (?-primality (/ n d) d (+ c 1)))
        (t (?-primality n (+ d 1) c))))

Output:

k = 1: (2 3 5 7 11 13 17 19 23 29)
k = 2: (4 6 9 10 14 15 21 22 25 26)
k = 3: (8 12 18 20 27 28 30 42 44 45)
k = 4: (16 24 36 40 54 56 60 81 84 88)
k = 5: (32 48 72 80 108 112 120 162 168 176)
NIL

D[edit]

This contains a copy of the function decompose from the Prime decomposition task.

Translation of: Ada

import std.stdio, std.algorithm, std.traits;
 
Unqual!T[] decompose(T)(in T number) pure nothrow
in {
    assert(number > 1);
} body {
    typeof(return) result;
    Unqual!T n = number;
 
    for (Unqual!T i = 2; n % i == 0; n /= i)
        result ~= i;
    for (Unqual!T i = 3; n >= i * i; i += 2)
        for (; n % i == 0; n /= i)
            result ~= i;
 
    if (n != 1)
        result ~= n;
    return result;
}
 
void main() {
    enum outLength = 10; // 10 k-th almost primes.
 
    foreach (immutable k; 1 .. 6) {
        writef("K = %d: ", k);
        auto n = 2; // The "current number" to be checked.
        foreach (immutable i; 1 .. outLength + 1) {
            while (n.decompose.length != k)
                n++;
            // Now n is K-th almost prime.
            write(n, " ");
            n++;
        }
        writeln;
    }
}

Output:

K = 1: 2 3 5 7 11 13 17 19 23 29
K = 2: 4 6 9 10 14 15 21 22 25 26
K = 3: 8 12 18 20 27 28 30 42 44 45
K = 4: 16 24 36 40 54 56 60 81 84 88
K = 5: 32 48 72 80 108 112 120 162 168 176

EchoLisp[edit]

Small numbers : filter the sequence [ 2 .. n]

 
(define (almost-prime? p k)
	(= k (length (prime-factors p))))
 
(define (almost-primes k nmax)
	(take (filter (rcurry almost-prime? k) [2 ..]) nmax))
 
(define (task (kmax 6) (nmax 10))
	(for ((k [1 .. kmax]))
		(write 'k= k '|)
		(for-each write (almost-primes k nmax))
		(writeln)))

Output:

 
(task)
 
k= 1 | 2 3 5 7 11 13 17 19 23 29
k= 2 | 4 6 9 10 14 15 21 22 25 26
k= 3 | 8 12 18 20 27 28 30 42 44 45
k= 4 | 16 24 36 40 54 56 60 81 84 88
k= 5 | 32 48 72 80 108 112 120 162 168 176

Large numbers : generate - combinations with repetitions - k-almost-primes up to pmax.

 
(lib 'match)
(define-syntax-rule (: v i) (vector-ref v i))
(reader-infix ':) ;; abbrev (vector-ref v i) === [v : i]
 
 
(lib 'bigint)
(define cprimes (list->vector (primes 10000)))
 
;; generates next k-almost-prime < pmax
;; c = vector of k primes indices c[i] <= c[j]
;; p = vector of intermediate products prime[c[0]]*prime[c[1]]*..
;; p[k-1] is the generated k-almost-prime
;; increment one c[i] at each step
 
(define (almost-next pmax k c p)
    (define almost-prime #f)
    (define cp 0)
 
    (for ((i (in-range (1- k) -1 -1))) ;; look backwards for c[i] to increment
        (vector-set! c i (1+ [c : i])) ;; increment c[i]
        (set! cp [cprimes : [c : i]]) 
        (vector-set! p i (if (> i 0) (* [ p : (1- i)] cp) cp)) ;; update partial product
 
        (when (< [p : i) pmax)
	    (set! almost-prime
            (and  ;; set followers to c[i] value
	       (for ((j (in-range (1+ i) k)))
	       (vector-set! c j [c : i])
	       (vector-set! p j (*  [ p : (1- j)] cp))
	       #:break (>= [p : j] pmax) => #f )
	       [p  : (1- k)]
	  ) ;; // and
	  ) ;; set!
	  ) ;; when
    #:break almost-prime 
    ) ;; // for i
    almost-prime )
 
;; not sorted list of k-almost-primes < pmax
(define (almost-primes k nmax)
    (define base (expt 2 k)) ;; first one is 2^k
    (define pmax (* base nmax))
    (define c (make-vector k #0))
    (define p (build-vector k (lambda(i) (expt #2 (1+ i)))))
 
    (cons base
	(for/list 
	((almost-prime (in-producer almost-next pmax k c p )))
	 almost-prime)))

Output:

 
;; we want  500-almost-primes from the 10000-th.
(take (drop (list-sort < (almost-primes 500 10000)) 10000 ) 10)
 
(7241149198492252834202927258094752774597239286103014697435725917649659974371690699721153852986
440733637405206125678822081264723636566725108094369093648384 
etc ...
 
;; The first one is 2^497 * 3 * 17 * 347 , same result as Haskell.

Elixir[edit]

Translation of: Erlang

defmodule Factors do
  def factors(n), do: factors(n,2,[])
 
  defp factors(1,_,acc), do: acc
  defp factors(n,k,acc) when rem(n,k)==0, do: factors(div(n,k),k,[k|acc])
  defp factors(n,k,acc)                 , do: factors(n,k+1,acc)
 
  def kfactors(n,k), do: kfactors(n,k,1,1,[])
 
  defp kfactors(_tn,tk,_n,k,_acc) when k == tk+1, do: IO.puts "done! "
  defp kfactors(tn,tk,_n,k,acc) when length(acc) == tn do
    IO.puts "K: #{k} #{inspect acc}"
    kfactors(tn,tk,2,k+1,[])
  end
  defp kfactors(tn,tk,n,k,acc) do
    case length(factors(n)) do
      ^k -> kfactors(tn,tk,n+1,k,acc++[n])
      _  -> kfactors(tn,tk,n+1,k,acc)
    end
  end
end
 
Factors.kfactors(10,5)

Output:

K: 1 [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
K: 2 [4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
K: 3 [8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
K: 4 [16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
K: 5 [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
done!

Erlang[edit]

Using the factors function from Prime_decomposition#Erlang.

 
-module(factors).                                         
-export([factors/1,kfactors/0,kfactors/2]).               
 
factors(N) ->                                             
     factors(N,2,[]).                                     
 
factors(1,_,Acc) -> Acc;                                  
factors(N,K,Acc) when N rem K == 0 ->                     
    factors(N div K,K, [K|Acc]);                          
factors(N,K,Acc) ->                                       
    factors(N,K+1,Acc).                                   
 
kfactors() -> kfactors(10,5,1,1,[]).                      
kfactors(N,K) -> kfactors(N,K,1,1,[]).                    
kfactors(_Tn,Tk,_N,K,_Acc) when K == Tk+1 ->  io:fwrite("Done! ");            
kfactors(Tn,Tk,N,K,Acc) when length(Acc) == Tn  ->        
    io:format("K: ~w ~w ~n", [K, Acc]),                   
    kfactors(Tn,Tk,2,K+1,[]);                             
 
kfactors(Tn,Tk,N,K,Acc) ->                                
    case length(factors(N)) of K ->                       
     kfactors(Tn,Tk, N+1,K, Acc ++ [ N ] );               
      _ ->                                                
      kfactors(Tn,Tk, N+1,K, Acc) end.

Output:

9> factors:kfactors(10,5). 
K: 1 [2,3,5,7,11,13,17,19,23,29] 
K: 2 [4,6,9,10,14,15,21,22,25,26] 
K: 3 [8,12,18,20,27,28,30,42,44,45] 
K: 4 [16,24,36,40,54,56,60,81,84,88] 
K: 5 [32,48,72,80,108,112,120,162,168,176] 
Done! ok
10> factors:kfactors(15,10).
K: 1 [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47] 
K: 2 [4,6,9,10,14,15,21,22,25,26,33,34,35,38,39] 
K: 3 [8,12,18,20,27,28,30,42,44,45,50,52,63,66,68] 
K: 4 [16,24,36,40,54,56,60,81,84,88,90,100,104,126,132] 
K: 5 [32,48,72,80,108,112,120,162,168,176,180,200,208,243,252] 
K: 6 [64,96,144,160,216,224,240,324,336,352,360,400,416,486,504] 
K: 7 [128,192,288,320,432,448,480,648,672,704,720,800,832,972,1008] 
K: 8 [256,384,576,640,864,896,960,1296,1344,1408,1440,1600,1664,1944,2016] 
K: 9 [512,768,1152,1280,1728,1792,1920,2592,2688,2816,2880,3200,3328,3888,4032] 
K: 10 [1024,1536,2304,2560,3456,3584,3840,5184,5376,5632,5760,6400,6656,7776,8064] 
Done! ok

ERRE[edit]

 
PROGRAM ALMOST_PRIME
 
!
! for rosettacode.org
!
 
!$INTEGER
 
PROCEDURE KPRIME(N,K->KP)
  LOCAL P,F
  FOR P=2 TO 999 DO
      EXIT IF NOT((F<K) AND (P*P<=N))
      WHILE (N MOD P)=0 DO
         N/=P
         F+=1
      END WHILE
  END FOR
  KP=(F-(N>1)=K)
END PROCEDURE
 
BEGIN
  PRINT(CHR$(12);)  !CLS
  FOR K=1 TO 5 DO
     PRINT("k =";K;":";)
     C=0
     FOR I=2 TO 999 DO
        EXIT IF NOT(C<10)
        KPRIME(I,K->KP)
        IF KP THEN
            PRINT(I;)
            C+=1
        END IF
     END FOR
     PRINT
  END FOR
END PROGRAM

Output:

K = 1: 2  3  5  7  11  13  17  19  23  29
K = 2: 4  6  9  10  14  15  21  22  25  26
K = 3: 8  12  18  20  27  28  30  42  44  45
K = 4: 16  24  36  40  54  56  60  81  84  88
K = 5: 32  48  72  80  108  112  120  162  168  176

FreeBASIC[edit]

' FB 1.05.0 Win64
 
Function kPrime(n As Integer, k As Integer) As Boolean
   Dim f As Integer = 0
   For i As Integer = 2 To n
     While n Mod i = 0
       If f = k Then Return false 
       f += 1
       n \= i
     Wend
   Next
   Return f = k
End Function
 
Dim As Integer i, c, k
For k = 1 To 5
  Print "k = "; k; " : ";
  i = 2
  c = 0
  While c < 10  
    If kPrime(i, k) Then
      Print Using "### "; i;
      c += 1
    End If
    i += 1
  Wend
  Print
Next
 
Print
Print "Press any key to quit"
Sleep

Output:

k =  1 :   2   3   5   7  11  13  17  19  23  29
k =  2 :   4   6   9  10  14  15  21  22  25  26
k =  3 :   8  12  18  20  27  28  30  42  44  45
k =  4 :  16  24  36  40  54  56  60  81  84  88
k =  5 :  32  48  72  80 108 112 120 162 168 176

Frink[edit]

for k = 1 to 5
{
   n=2
   count = 0
   print["k=$k:"]
   do
   {
      if length[factorFlat[n]] == k
      {
         print[" $n"]
         count = count + 1
      }
      n = n + 1
   } while count < 10
 
   println[]
}

Output:

k=1: 2 3 5 7 11 13 17 19 23 29
k=2: 4 6 9 10 14 15 21 22 25 26
k=3: 8 12 18 20 27 28 30 42 44 45
k=4: 16 24 36 40 54 56 60 81 84 88
k=5: 32 48 72 80 108 112 120 162 168 176

Futhark[edit]

 
fun kprime(n: int, k: int): bool =
  let (p,f) = (2, 0)
  loop ((n, p, f)) = while f < k && p*p <= n do
    loop ((n,f)) = while 0 == n % p do
      (n/p, f+1)
    in (n, p+1, f)
  in f + (if n > 1 then 1 else 0) == k
 
fun main(m: int): [][]int =
  map (fn k: [10]int =>
         let ps = replicate 10 0
         loop ((i,c,ps) = (2,0,ps)) = while c < 10 do
           if kprime(i,k) then
             unsafe let ps[c] = i
                    in (i+1, c+1, ps)
           else (i+1, c, ps)
         in ps)
  (map (1+) (iota m))

F#[edit]

let rec genFactor (f, n) =
    if f > n then None
    elif n % f = 0 then Some (f, (f, n/f))
    else genFactor (f+1, n)
 
 
let factorsOf (num) =
    Seq.unfold (fun (f, n) -> genFactor (f, n)) (2, num)
 
let kFactors k = Seq.unfold (fun n ->
    let rec loop m =
        if Seq.length (factorsOf m) = k then m
        else loop (m+1)
    let next = loop n
    Some(next, next+1)) 2
 
[1 .. 5]
|> List.iter (fun k ->
        printfn "%A" (Seq.take 10 (kFactors k) |> Seq.toList))

Output:

[2; 3; 5; 7; 11; 13; 17; 19; 23; 29]
[4; 6; 9; 10; 14; 15; 21; 22; 25; 26]
[8; 12; 18; 20; 27; 28; 30; 42; 44; 45]
[16; 24; 36; 40; 54; 56; 60; 81; 84; 88]
[32; 48; 72; 80; 108; 112; 120; 162; 168; 176]

Go[edit]

package main
 
import "fmt"
 
func kPrime(n, k int) bool {
    nf := 0
    for i := 2; i <= n; i++ {
        for n%i == 0 {
            if nf == k {
                return false
            }
            nf++
            n /= i
        }
    }
    return nf == k
}
 
func gen(k, n int) []int {
    r := make([]int, n)
    n = 2
    for i := range r {
        for !kPrime(n, k) {
            n++
        }
        r[i] = n
        n++
    }
    return r
}
 
func main() {
    for k := 1; k <= 5; k++ {
        fmt.Println(k, gen(k, 10))
    }
}

Output:

1 [2 3 5 7 11 13 17 19 23 29]
2 [4 6 9 10 14 15 21 22 25 26]
3 [8 12 18 20 27 28 30 42 44 45]
4 [16 24 36 40 54 56 60 81 84 88]
5 [32 48 72 80 108 112 120 162 168 176]

Groovy[edit]

 
 public class almostprime
{
public static boolean kprime(int n,int k)
  {
    int i,div=0;
     for(i=2;(i*i <= n) && (div<k);i++)
      {
        while(n%i==0)
          {
            n = n/i;
            div++;
          }
      }
   return div + ((n > 1)?1:0) == k;
  }
  public static void main(String[] args)
    {
      int i,l,k;
       for(k=1;k<=5;k++)
        {
          println("k = " + k + ":");
           l = 0;
            for(i=2;l<10;i++)
              {
                if(kprime(i,k))
                {
                  print(i + " ");
                  l++;
                }
              }
          println();
        }
     }
}

Output:

k = 1:
2 3 5 7 11 13 17 19 23 29 
k = 2:
4 6 9 10 14 15 21 22 25 26 
k = 3:
8 12 18 20 27 28 30 42 44 45 
k = 4:
16 24 36 40 54 56 60 81 84 88 
k = 5:
32 48 72 80 108 112 120 162 168 176

Haskell[edit]

isPrime :: Integral a => a -> Bool
isPrime n = not $ any ((0 ==) . (mod n)) [2..(truncate $ sqrt $ fromIntegral n)]
 
primes :: [Integer]
primes = filter isPrime [2..]
 
isKPrime :: (Num a, Eq a) => a -> Integer -> Bool
isKPrime 1 n = isPrime n
isKPrime k n = any (isKPrime (k - 1)) sprimes
  where
    sprimes = map fst $ filter ((0 ==) . snd) $ map (divMod n) $ takeWhile (< n) primes
 
kPrimes :: (Num a, Eq a) => a -> [Integer]
kPrimes k = filter (isKPrime k) [2..]
 
main :: IO ()
main = flip mapM_ [1..5] $ \k ->
  putStrLn $ "k = " ++ show k ++ ": " ++ (unwords $ map show (take 10 $ kPrimes k))

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

Larger ks require more complicated methods:

primes = 2:3:[n | n <- [5,7..], foldr (\p r-> p*p > n || rem n p > 0 && r) 
	True (drop 1 primes)]
 
merge aa@(a:as) bb@(b:bs)
	| a < b = a:merge as bb
	| otherwise = b:merge aa bs
 
-- n-th item is all k-primes not divisible by any of the first n primes
notdivs k = f primes $ kprimes (k-1) where
	f (p:ps) s = map (p*) s : f ps (filter ((/=0).(`mod`p)) s)
 
kprimes k
	| k == 1 = primes
	| otherwise = f (head ndk) (tail ndk) (tail $ map (^k) primes) where
		ndk = notdivs k
		-- tt is the thresholds for merging in next sequence
		-- it is equal to "map head seqs", but don't do that
		f aa@(a:as) seqs tt@(t:ts)
			| a < t = a : f as seqs tt
			| otherwise = f (merge aa $ head seqs) (tail seqs) ts
 
main = do 
	-- next line is for task requirement:
	mapM_ (\x->print (x, take 10 $ kprimes x)) [1 .. 5]
 
	putStrLn "\n10000th to 10100th 500-amost primes:"
	mapM_ print $ take 100 $ drop 10000 $ kprimes 500

Output:

(1,[2,3,5,7,11,13,17,19,23,29])
(2,[4,6,9,10,14,15,21,22,25,26])
(3,[8,12,18,20,27,28,30,42,44,45])
(4,[16,24,36,40,54,56,60,81,84,88])
(5,[32,48,72,80,108,112,120,162,168,176])

10000th to 10100th 500-amost primes:
7241149198492252834202927258094752774597239286103014697435725917649659974371690699721153852986440733637405206125678822081264723636566725108094369093648384
        <...snipped 99 more equally unreadable numbers...>

Icon and Unicon[edit]

Works in both languages.

link "factors"
 
procedure main()
    every writes(k := 1 to 5,": ") do
        every writes(right(genKap(k),5)\10|"\n")
end
 
procedure genKap(k)
    suspend (k = *factors(n := seq(q)), n)
end

Output:

->ap
1:     2    3    5    7   11   13   17   19   23   29
2:     4    6    9   10   14   15   21   22   25   26
3:     8   12   18   20   27   28   30   42   44   45
4:    16   24   36   40   54   56   60   81   84   88
5:    32   48   72   80  108  112  120  162  168  176
->

J[edit]

   (10 {. [:~.[:/:~[:,*/~)^:(i.5)~p:i.10
 2  3  5  7  11  13  17  19  23  29
 4  6  9 10  14  15  21  22  25  26
 8 12 18 20  27  28  30  42  44  45
16 24 36 40  54  56  60  81  84  88
32 48 72 80 108 112 120 162 168 176

Explanation:

Generate 10 primes.
Multiply each of them by the first ten primes
Sort and find unique values, take the first ten of those
Multiply each of them by the first ten primes
Sort and find unique values, take the first ten of those

...

The results of the odd steps in this procedure are the desired result.

Java[edit]

public class AlmostPrime {
    public static void main(String[] args) {
        for (int k = 1; k <= 5; k++) {
            System.out.print("k = " + k + ":");
 
            for (int i = 2, c = 0; c < 10; i++) {
                if (kprime(i, k)) {
                    System.out.print(" " + i);
                    c++;
                }
            }
 
            System.out.println("");
        }
    }
 
    public static boolean kprime(int n, int k) {
        int f = 0;
        for (int p = 2; f < k && p * p <= n; p++) {
            while (n % p == 0) {
                n /= p;
                f++;
            }
        }
        return f + ((n > 1) ? 1 : 0) == k;
    }
}

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

JavaScript[edit]

function almostPrime (n, k) {
    var divisor = 2, count = 0
    while(count < k + 1 && n != 1) {
        if (n % divisor == 0) {
            n = n / divisor
            count = count + 1
        } else {
            divisor++
        }
    }
    return count == k
}
 
for (var k = 1; k <= 5; k++) {
    document.write("<br>k=", k, ": ")
    var count = 0, n = 0
    while (count <= 10) {
        n++
        if (almostPrime(n, k)) {
            document.write(n, " ")
            count++
        }
    }
}

Output:

k=1: 2 3 5 7 11 13 17 19 23 29 31
k=2: 4 6 9 10 14 15 21 22 25 26 33
k=3: 8 12 18 20 27 28 30 42 44 45 50
k=4: 16 24 36 40 54 56 60 81 84 88 90
k=5: 32 48 72 80 108 112 120 162 168 176 180

jq[edit]

Works with: jq version 1.4

Infrastructure:

# Recent versions of jq (version > 1.4) have the following definition of "until":
def until(cond; next):
  def _until:
    if cond then . else (next|_until) end;
  _until;
 
# relatively_prime(previous) tests whether the input integer is prime
# relative to the primes in the array "previous":
def relatively_prime(previous):
  . as $in
  | (previous|length) as $plen
  # state: [found, ix]
  |  [false, 0]
  | until( .[0] or .[1] >= $plen;
           [ ($in % previous[.[1]]) == 0, .[1] + 1] )
  | .[0] | not ;
 
# Emit a stream in increasing order of all primes (from 2 onwards)
# that are less than or equal to mx:
def primes(mx):
 
  # The helper function, next, has arity 0 for tail recursion optimization;
  # it expects its input to be the array of previously found primes:
  def next:
     . as $previous
     | ($previous | .[length-1]) as $last
     | if ($last >= mx) then empty
       else ((2 + $last)
       | until( relatively_prime($previous) ; . + 2)) as $nextp
       | if $nextp <= mx
         then $nextp, (( $previous + [$nextp] ) | next)
	 else empty
         end
       end;
  if mx <= 1 then empty
  elif mx == 2 then 2
  else (2, 3, ( [2,3] | next))
  end
;
 
# Return an array of the distinct prime factors of . in increasing order
def prime_factors:
 
  # Return an array of prime factors of . given that "primes"
  # is an array of relevant primes:
  def pf(primes):
    if . <= 1 then []
    else . as $in
    | if ($in | relatively_prime(primes)) then [$in]
      else reduce primes[] as $p
             ([];
              if ($in % $p) != 0 then .
 	      else . + [$p] +  (($in / $p) | pf(primes))
	      end)
      end
      | unique
    end;
 
  if . <= 1 then []
  else . as $in
  | pf( [ primes( (1+$in) | sqrt | floor)  ] )
  end;
 
# Return an array of prime factors of . repeated according to their multiplicities:
def prime_factors_with_multiplicities:
  # Emit p according to the multiplicity of p
  # in the input integer assuming p > 1
  def multiplicity(p):
    if   .  < p     then empty
    elif . == p     then p
    elif (. % p) == 0 then
       ((./p) | recurse( if (. % p) == 0 then (. / p) else empty end) | p)
    else empty
    end;
 
  if . <= 1 then []
  else . as $in
  | prime_factors as $primes
  | if ($in|relatively_prime($primes)) then [$in]
    else reduce $primes[]  as $p
           ([];
            if ($in % $p) == 0 then . + [$in|multiplicity($p)] else . end )
    end
  end;

isalmostprime

def isalmostprime(k): (prime_factors_with_multiplicities | length) == k;
 
# Emit a stream of the first N almost-k primes
def almostprimes(N; k):
  if N <= 0 then empty
  else
    # state [remaining, candidate, answer]
    [N, 1, null]
    | recurse( if .[0] <= 0 then empty
	       elif (.[1] | isalmostprime(k)) then [.[0]-1, .[1]+1, .[1]]
	       else [.[0], .[1]+1, null]
               end)
    | .[2] | select(. != null)
  end;

The task:

range(1;6) as $k | "k=\($k): \([almostprimes(10;$k)])"

Output:

$ jq -c -r -n -f Almost_prime.jq
k=1: [2,3,5,7,11,13,17,19,23,29]
k=2: [4,6,9,10,14,15,21,22,25,26]
k=3: [8,12,18,20,27,28,30,42,44,45]
k=4: [16,24,36,40,54,56,60,81,84,88]
k=5: [32,48,72,80,108,112,120,162,168,176]

Julia[edit]

isalmostprime(n, k) = sum(values(factor(n))) == k
function almostprimes(N, k) # return first N almost-k primes
    P = Array(Int, N)
    i = 0; n = 2
    while i < N
        if isalmostprime(n, k); P[i += 1] = n; end
        n += 1
    end
    return P
end

Output:

julia> [almostprimes(10, k) for k in 1:5]
5-element Array{Array{Int64,1},1}:
 [2,3,5,7,11,13,17,19,23,29]          
 [4,6,9,10,14,15,21,22,25,26]         
 [8,12,18,20,27,28,30,42,44,45]       
 [16,24,36,40,54,56,60,81,84,88]      
 [32,48,72,80,108,112,120,162,168,176]

Lua[edit]

-- Returns boolean indicating whether n is k-almost prime
function almostPrime (n, k)
    local divisor, count = 2, 0
    while count < k + 1 and n ~= 1 do
        if n % divisor == 0 then
            n = n / divisor
            count = count + 1
        else
            divisor = divisor + 1
        end
    end
    return count == k
end
 
-- Generates table containing first ten k-almost primes for given k
function kList (k)
    local n, kTab = 2^k, {}
    while #kTab < 10 do
        if almostPrime(n, k) then
            table.insert(kTab, n)
        end
        n = n + 1
    end
    return kTab
end
 
-- Main procedure, displays results from five calls to kList()
for k = 1, 5 do
    io.write("k=" .. k .. ": ")
    for _, v in pairs(kList(k)) do
        io.write(v .. ", ")
    end
    print("...")
end

Output:

k=1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
k=2: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...
k=3: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, ...
k=4: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, ...
k=5: 32, 48, 72, 80, 108, 112, 120, 162, 168, 176, ...

Kotlin[edit]

Translation of: Java

fun Int.k_prime(x: Int): Boolean {
    var n = x
    var f = 0
    var p = 2
    while (f < this && p * p <= n) {
        while (0 == n % p) { n /= p; f++ }
        p++
    }
    return f + (if (n > 1) 1 else 0) == this
}
 
fun Int.primes(n : Int) : List<Int> {
    var i = 2
    var list = listOf<Int>()
    while (list.size < n) {
        if (k_prime(i)) list += i
        i++
    }
    return list
}
 
fun main(args: Array<String>) {
    for (k in 1..5)
        println("k = $k: " + k.primes(10))
}

Output:

k = 1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
k = 2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
k = 3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
k = 4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
k = 5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

Mathematica / Wolfram Language[edit]

kprimes[k_,n_] :=
  (* generates a list of the n smallest k-almost-primes *)
  Module[{firstnprimes, runningkprimes = {}},
  firstnprimes = Prime[Range[n]];
  runningkprimes = firstnprimes;
  Do[
   runningkprimes = 
     Outer[Times, firstnprimes , runningkprimes ] // Flatten // Union  // Take[#, n] & ; 
   (* only keep lowest n numbers in our running list *)
   , {i, 1, k - 1}];
  runningkprimes
  ]
(* now to create table with n=10 and k ranging from 1 to 5 *)
Table[Flatten[{"k = " <> ToString[i] <> ": ", kprimes[i, 10]}], {i,1,5}] // TableForm

Output:

k = 1: 	2	3	5	7	11	13	17	19	23	29
k = 2: 	4	6	9	10	14	15	21	22	25	26
k = 3: 	8	12	18	20	27	28	30	42	44	45
k = 4: 	16	24	36	40	54	56	60	81	84	88
k = 5: 	32	48	72	80	108	112	120	162	168	176

Nim[edit]

proc prime(k: int, listLen: int): seq[int] =
 result = @[]
 var
  test: int = 2
  curseur: int = 0
 while curseur < listLen:
  var
   i: int = 2
   compte = 0
   n = test
  while i <= n:
   if (n mod i)==0:
    n = n div i
    compte += 1
   else:
    i += 1
  if compte == k:
   result.add(test)
   curseur += 1
  test += 1
 
for k in 1..5:
 echo "k = ",k," : ",prime(k,10)

Output:

k = 1 : @[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
k = 2 : @[4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
k = 3 : @[8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
k = 4 : @[16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
k = 5 : @[32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

Objeck[edit]

Translation of: C

class Kth_Prime {
  function : native : kPrime(n : Int, k : Int) ~ Bool {
    f := 0;
    for (p := 2; f < k & p*p <= n; p+=1;) {
      while (0 = n % p) {
        n /= p; f+=1;
      };
    };
 
    return f + ((n > 1) ? 1 : 0) = k;
  }
 
  function : Main(args : String[]) ~ Nil {
    for (k := 1; k <= 5; k+=1;) {
      "k = {$k}:"->Print();
 
      c := 0;
      for (i := 2; c < 10; i+=1;) {
        if (kPrime(i, k)) {
          " {$i}"->Print();
          c+=1;
        };
      };
      '\n'->Print();
    };
  }
}

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

Oforth[edit]

: kprime(n, k)
| i |
   0 2 n for: i [ while(n i /mod swap 0 &= ) [ ->n 1+ ] drop ] k == ;
 
: table(k)
| l |
   ListBuffer new ->l
   2 while (l size 10 <>) [ dup k kprime ifTrue: [ dup l add ] 1+ ]
   drop l ;

Output:

>#[ table .cr ] 5 seqEach
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
[4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
[8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
[16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
[32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

PARI/GP[edit]

almost(k)=my(n); for(i=1,10,while(bigomega(n++)!=k,); print1(n", "));
for(k=1,5,almost(k);print)

Output:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
4, 6, 9, 10, 14, 15, 21, 22, 25, 26,
8, 12, 18, 20, 27, 28, 30, 42, 44, 45,
16, 24, 36, 40, 54, 56, 60, 81, 84, 88,
32, 48, 72, 80, 108, 112, 120, 162, 168, 176,

Pascal[edit]

Library: primTrial

Works with: Free Pascal

program AlmostPrime;
{$IFDEF FPC}
  {$Mode Delphi}
{$ENDIF}
uses
  primtrial;
var
  i,K,cnt : longWord;
BEGIN
  K := 1;
  repeat
    cnt := 0;
    i := 2;
    write('K=',K:2,':');
    repeat
      if isAlmostPrime(i,K) then
      Begin
        write(i:6,' ');
        inc(cnt);
      end;
      inc(i);
    until cnt = 9;
    writeln;
    inc(k);
  until k > 10;
END.

output

K= 1 :    2     3     5     7    11    13    17    19    23    29
K= 2 :    4     6     9    10    14    15    21    22    25    26
K= 3 :    8    12    18    20    27    28    30    42    44    45
K= 4 :   16    24    36    40    54    56    60    81    84    88
K= 5 :   32    48    72    80   108   112   120   162   168   176
K= 6 :   64    96   144   160   216   224   240   324   336   352
K= 7 :  128   192   288   320   432   448   480   648   672   704
K= 8 :  256   384   576   640   864   896   960  1296  1344  1408
K= 9 :  512   768  1152  1280  1728  1792  1920  2592  2688  2816
K=10 : 1024  1536  2304  2560  3456  3584  3840  5184  5376  5632

Perl[edit]

Using a CPAN module, which is simple and fast:

Library: ntheory

use ntheory qw/factor/;
sub almost {
  my($k,$n) = @_;
  my $i = 1;
  map { $i++ while scalar factor($i) != $k; $i++ } 1..$n;
}
say "$_ : ", join(" ", almost($_,10)) for 1..5;

Output:

1 : 2 3 5 7 11 13 17 19 23 29
2 : 4 6 9 10 14 15 21 22 25 26
3 : 8 12 18 20 27 28 30 42 44 45
4 : 16 24 36 40 54 56 60 81 84 88
5 : 32 48 72 80 108 112 120 162 168 176

or writing everything by hand:

use strict;
use warnings;
 
sub k_almost_prime;
 
for my $k ( 1 .. 5 ) {
	my $almost = 0;
	print join(", ", map {
		1 until k_almost_prime ++$almost, $k;
		"$almost";
	} 1 .. 10), "\n";
}
 
sub nth_prime;
 
sub k_almost_prime {
	my ($n, $k) = @_;
	return if $n <= 1 or $k < 1;
	my $which_prime = 0;
	for my $count ( 1 .. $k ) {
		while( $n % nth_prime $which_prime ) {
			++$which_prime;
		}
		$n /= nth_prime $which_prime;
		return if $n == 1 and $count != $k;
	}
	($n == 1) ? 1 : ();
}
 
BEGIN {
	# This is loosely based on one of the python solutions
	# to the RC Sieve of Eratosthenes task.
	my @primes = (2, 3, 5, 7);
	my $p_iter = 1;
	my $p = $primes[$p_iter];
	my $q = $p*$p;
	my %sieve;
	my $candidate = $primes[-1] + 2;
	sub nth_prime {
		my $n = shift;
		return if $n < 0;
		OUTER: while( $#primes < $n ) {
			while( my $s = delete $sieve{$candidate} ) {
				my $next = $s + $candidate;
				$next += $s while exists $sieve{$next};
				$sieve{$next} = $s;
				$candidate += 2;
			}
			while( $candidate < $q ) {
				push @primes, $candidate;
				$candidate += 2;
				next OUTER if exists $sieve{$candidate};
			}
			my $twop = 2 * $p;
			my $next = $q + $twop;
			$next += $twop while exists $sieve{$next};
			$sieve{$next} = $twop;
			$p = $primes[++$p_iter];
			$q = $p * $p;	
			$candidate += 2;
		}
		return $primes[$n];
	}
}

Output:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29
4, 6, 9, 10, 14, 15, 21, 22, 25, 26
8, 12, 18, 20, 27, 28, 30, 42, 44, 45
16, 24, 36, 40, 54, 56, 60, 81, 84, 88
32, 48, 72, 80, 108, 112, 120, 162, 168, 176

Perl 6[edit]

Translation of: C

Works with: Rakudo version 2015.12

sub is-k-almost-prime($n is copy, $k) returns Bool {
    loop (my ($p, $f) = 2, 0; $f < $k && $p*$p <= $n; $p++) {
        $n /= $p, $f++ while $n %% $p;
    }
    $f + ($n > 1) == $k;
}
 
for 1 .. 5 -> $k {
    say ~.[^10]
        given grep { is-k-almost-prime($_, $k) }, 2 .. *
}

Output:

2 3 5 7 11 13 17 19 23 29
4 6 9 10 14 15 21 22 25 26
8 12 18 20 27 28 30 42 44 45
16 24 36 40 54 56 60 81 84 88
32 48 72 80 108 112 120 162 168 176

Here is a solution with identical output based on the factors routine from Count_in_factors#Perl_6 (to be included manually until we decide where in the distribution to put it).

constant @primes = 2, |(3, 5, 7 ... *).grep: *.is-prime;
 
multi sub factors(1) { 1 }
multi sub factors(Int $remainder is copy) {
    gather for @primes -> $factor {
        # if remainder < factor², we're done
        if $factor * $factor > $remainder {
            take $remainder if $remainder > 1;
            last;
        }
        # How many times can we divide by this prime?
        while $remainder %% $factor {
            take $factor;
            last if ($remainder div= $factor) === 1;
        }
    }
}
 
constant @factory = lazy 0..* Z=> flat (0, 0, map { +factors($_) }, 2..*);
 
sub almost($n) { map *.key, grep *.value == $n, @factory }
 
put almost($_)[^10] for 1..5;

Phix[edit]

 
-- Naieve stuff, mostly, but coded with enthuiasm!
-- Following the idea behind (but not the code from!) the J submission:
--  Generate 10 primes (kept in p10)                            -- (print K=1)
--  Multiply each of them by the first ten primes 
--  Sort and find unique values, take the first ten of those    -- (print K=2)
--  Multiply each of them by the first ten primes 
--  Sort and find unique values, take the first ten of those    -- (print K=3)
--  ...
-- However I just keep a "top 10", using a bubble insertion, and stop 
--  multiplying as soon as everything else for p10[i] will be too big.
 
-- (as calculated earlier from this routine,
--  or that "return 1" in pi() works just fine.)
--constant f17={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59}
constant f17={2,3,5,7,11,13,17}
 
function pi(integer n)
-- approximates the number of primes less than or equal to n
--  if n<=10 then return 4 end if
--  -- best estimate
--  return floor(n/(log(n)-1))
--  if n<=20 then return 1 end if -- (or use a table:)
    if n<17 then
        for i=1 to length(f17) do
            if n<=f17[i] then return i end if
        end for
    end if
--  -- upper bound for n>=17 (Rosser and Schoenfeld 1962):
--  return floor(1.25506*n/log(n))
    -- lower bound for n>=17 (Rosser and Schoenfeld 1962):
    return floor(n/log(n))
end function
 
function primes(integer n)
-- return the first n prime numbers (tested 0 to 20,000, which took ~86s)
sequence prime
integer count = 0
integer lowN, highN, midN
 
    -- First, iteratively estimate the sieve size required
    lowN = 2*n
    highN = n*n+1
    while lowN<highN do
        midN = floor((lowN+highN)/2)
        if pi(midN)>n then
            highN = midN
        else
            lowN = midN+1
        end if
    end while
    -- Then apply standard sieve and store primes as we find
    -- them towards the (no longer used) start of the sieve.
    prime = repeat(1,highN)
    for i=2 to highN do
        if prime[i] then
            count += 1
            prime[count] = i
            if count>=n then exit end if
            for k=i+i to highN by i do
                prime[k] = 0
            end for 
        end if
    end for
    return prime[1..n]
end function
 
procedure display(integer k, sequence kprimes)
    printf(1,"%d: ",k)
    for i=1 to length(kprimes) do
        printf(1,"%5d",kprimes[i])
    end for
    puts(1,"\n")
end procedure
 
function bubble(sequence next, integer v)
-- insert v into next (discarding next[$]), keeping next in ascending order
-- (relies on next[1] /always/ being smaller that anything that we insert.)
    for i=length(next)-1 to 1 by -1 do
        if v>next[i] then
            next[i+1] = v
            exit
        end if
        next[i+1] = next[i]
    end for
    return next
end function
 
procedure almost_prime()
sequence p10 = primes(10)
sequence apk = p10  -- (almostprime[k])
sequence next = repeat(0,length(p10))
integer high, test
    for k=1 to 5 do
        display(k,apk)
        if k=5 then exit end if
        next = apk
        for i=1 to length(p10) do
--          next[i] = apk[i]*p10[1]
            next[i] = apk[i]*2
        end for
        high = next[$]
        for i=2 to length(p10) do
            for j=1 to length(next) do
                test = apk[j]*p10[i]
                if not find(test,next) then
                    if test>high then exit end if
                    next = bubble(next,test)
                    high = next[$]
                end if
            end for
        end for
        apk = next
    end for
    if getc(0) then end if
end procedure
 
    almost_prime()

Output:

1:     2    3    5    7   11   13   17   19   23   29
2:     4    6    9   10   14   15   21   22   25   26
3:     8   12   18   20   27   28   30   42   44   45
4:    16   24   36   40   54   56   60   81   84   88
5:    32   48   72   80  108  112  120  162  168  176

and a translation of the C version, with improved variable names and some extra notes

 
 
function kprime(integer n, integer k)
--
-- returns true if n has exactly k factors
--
-- p is a "pseudo prime" in that 2,3,4,5,6,7,8,9,10,11 will behave 
--  exactly like 2,3,5,7,11, ie the remainder(n,4)=0 (etc) will never 
--  succeed because remainder(n,2) would have succeeded twice first.
--  Hence for larger n consider replacing p+=1 with p=next_prime(),
--  then again, on "" this performs an obscene number of divisions..
--
integer p = 2, 
        factors = 0
 
    while factors<k and p*p<=n do
        while remainder(n,p)=0 do
            n = n/p
            factors += 1
        end while
        p += 1
    end while 
    factors += (n>1)
    return factors==k
end function
 
procedure almost_primeC()
integer nextkprime, count
 
    for k=1 to 5 do
        printf(1,"k = %d: ", k);
        nextkprime = 2
        count = 0
        while count<10 do
            if kprime(nextkprime, k) then
                printf(1," %4d", nextkprime)
                count += 1
            end if 
            nextkprime += 1
        end while
        puts(1,"\n")
    end for
    if getc(0) then end if
end procedure
 
    almost_primeC()

Output:

k = 1:     2    3    5    7   11   13   17   19   23   29
k = 2:     4    6    9   10   14   15   21   22   25   26
k = 3:     8   12   18   20   27   28   30   42   44   45
k = 4:    16   24   36   40   54   56   60   81   84   88
k = 5:    32   48   72   80  108  112  120  162  168  176

PicoLisp[edit]

(de factor (N)
   (make
      (let
         (D 2
            L (1 2 2 . (4 2 4 2 4 6 2 6 .))
            M (sqrt N) )
         (while (>= M D)
            (if (=0 (% N D))
               (setq M 
                  (sqrt (setq N (/ N (link D)))) )
               (inc 'D (pop 'L)) ) )
         (link N) ) ) )
 
(de almost (N)
   (let (X 2  Y 0)
      (make
         (loop
            (when (and (nth (factor X) N) (not (cdr @)))
               (link X)
               (inc 'Y) )
            (T (= 10 Y) 'done)
            (inc 'X) ) ) ) )
 
(for I 5
   (println I '-> (almost I) ) )
 
(bye)

Potion[edit]

# Converted from C
kprime = (n, k):
  p = 2, f = 0
  while (f < k && p*p <= n):
    while (0 == n % p):
      n /= p
      f++.
    p++.
  n = if (n > 1): 1.
      else: 0.
  f + n == k.
 
1 to 5 (k):
  "k = " print, k print, ":" print
  i = 2, c = 0
  while (c < 10):
    if (kprime(i, k)): " " print, i print, c++.
    i++
  .
  "" say.

C and Potion take 0.006s, Perl5 0.028s

Prolog[edit]

% almostPrime(K, +Take, List) succeeds if List can be unified with the
% first Take K-almost-primes.
% Notice that K need not be specified.
% To avoid having to cache or recompute the first Take primes, we define
% almostPrime/3 in terms of almostPrime/4 as follows:
%
almostPrime(K, Take, List) :-
  % Compute the list of the first Take primes:
  nPrimes(Take, Primes),   
  almostPrime(K, Take, Primes, List).
 
almostPrime(1, Take, Primes, Primes).
 
almostPrime(K, Take, Primes, List) :- 
  generate(2, K),  % generate K >= 2
  K1 is K - 1,
  almostPrime(K1, Take, Primes, L),
  multiplylist( Primes, L, Long),
  sort(Long, Sorted), % uniquifies
  take(Take, Sorted, List).

That's it. The rest is machinery. For portability, a compatibility section is included below.

nPrimes( M, Primes) :- nPrimes( [2], M, Primes).
 
nPrimes( Accumulator, I, Primes) :-
	next_prime(Accumulator, Prime),
	append(Accumulator, [Prime], Next),
	length(Next, N),
	( N = I -> Primes = Next; nPrimes( Next, I, Primes)).
 
% next_prime(+Primes, NextPrime) succeeds if NextPrime is the next
% prime after a list, Primes, of consecutive primes starting at 2.
next_prime([2], 3).
next_prime([2|Primes], P) :-
	last(Primes, PP),
	P2 is PP + 2,
	generate(P2, N),
	1 is N mod 2,		        % odd
	Max is floor(sqrt(N+1)),	% round-off paranoia 
	forall( (member(Prime, [2|Primes]),
		 (Prime =< Max -> true
		 ; (!, fail))), N mod Prime > 0 ),
	!,
        P = N.
 
% multiply( +A, +List, Answer )
multiply( A, [], [] ).
multiply( A, [X|Xs], [AX|As] ) :-
  AX is A * X, 
  multiply(A, Xs, As).
 
% multiplylist( L1, L2, List ) succeeds if List is the concatenation of X * L2
% for successive elements X of L1.
multiplylist( [], B, [] ).
multiplylist( [A|As], B, List ) :-
   multiply(A, B, L1),
   multiplylist(As, B, L2),
   append(L1, L2, List).
 
take(N, List, Head) :- 
  length(Head, N), 
  append(Head,X,List).

%%%%% compatibility section %%%%%
 
:- if(current_prolog_flag(dialect, yap)).
generate(Min, I) :- between(Min, inf, I).
 
append([],L,L).
append([X|Xs], L, [X|Ls]) :- append(Xs,L,Ls).
 
:- endif.
 
:- if(current_prolog_flag(dialect, swi)).
generate(Min, I) :- between(Min, inf, I).
:- endif.
 
:- if(current_prolog_flag(dialect, yap)).
append([],L,L).
append([X|Xs], L, [X|Ls]) :- append(Xs,L,Ls).
 
last([X], X).
last([_|Xs],X) :- last(Xs,X).
 
:- endif.
 
:- if(current_prolog_flag(dialect, gprolog)).
generate(Min, I) :- 
  current_prolog_flag(max_integer, Max),
  between(Min, Max, I).
:- endif.

Example using SWI-Prolog:

?- between(1,5,I),
   (almostPrime(I, 10, L) -> writeln(L)), fail.

[2,3,5,7,11,13,17,19,23,29]
[4,6,9,10,14,15,21,22,25,26]
[8,12,18,20,27,28,30,42,44,45]
[16,24,36,40,54,56,60,81,84,88]
[32,48,72,80,108,112,120,162,168,176]

?- time( (almostPrime(5, 10, L), writeln(L))).
[32,48,72,80,108,112,120,162,168,176]
% 1,906 inferences, 0.001 CPU in 0.001 seconds (84% CPU, 2388471 Lips)

Python[edit]

This imports Prime decomposition#Python

from prime_decomposition import decompose
from itertools import islice, count
try: 
    from functools import reduce
except: 
    pass
 
 
def almostprime(n, k=2):
    d = decompose(n)
    try:
        terms = [next(d) for i in range(k)]
        return reduce(int.__mul__, terms, 1) == n
    except:
        return False
 
if __name__ == '__main__':
    for k in range(1,6):
        print('%i: %r' % (k, list(islice((n for n in count() if almostprime(n, k)), 10))))

Output:

1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

R[edit]

This uses the function from Prime decomposition#R

#===============================================================
# Find k-Almost-primes
# R implementation
#===============================================================
#---------------------------------------------------------------
# Function for prime factorization from Rosetta Code
#---------------------------------------------------------------
 
findfactors <- function(n) {
  d <- c()
  div <- 2; nxt <- 3; rest <- n
  while( rest != 1 ) {
    while( rest%%div == 0 ) {
      d <- c(d, div)
      rest <- floor(rest / div)
    }
    div <- nxt
    nxt <- nxt + 2
  }
  d
}
 
#---------------------------------------------------------------
# Find k-Almost-primes
#---------------------------------------------------------------
 
almost_primes <- function(n = 10, k = 5) {
 
  # Set up matrix for storing of the results
 
  res <- matrix(NA, nrow = k, ncol = n)
  rownames(res) <- paste("k = ", 1:k, sep = "")
  colnames(res) <- rep("", n)
 
  # Loop over k
 
  for (i in 1:k) {
 
    tmp <- 1 
 
    while (any(is.na(res[i, ]))) { # Keep looping if there are still missing entries in the result-matrix
      if (length(findfactors(tmp)) == i) { # Check number of factors
        res[i, which.max(is.na(res[i, ]))] <- tmp
      }
      tmp <- tmp + 1
    }
  }
  print(res)
}

Output:

k = 1  2  3  5  7  11  13  17  19  23  29
k = 2  4  6  9 10  14  15  21  22  25  26
k = 3  8 12 18 20  27  28  30  42  44  45
k = 4 16 24 36 40  54  56  60  81  84  88
k = 5 32 48 72 80 108 112 120 162 168 176

Racket[edit]

#lang racket
(require (only-in math/number-theory factorize))
 
(define ((k-almost-prime? k) n)
  (= k (for/sum ((f (factorize n))) (cadr f))))
 
(define KAP-table-values
  (for/list ((k (in-range 1 (add1 5))))
    (define kap? (k-almost-prime? k))
    (for/list ((j (in-range 10)) (i (sequence-filter kap? (in-naturals 1))))
      i)))
 
(define (format-table t)
  (define longest-number-length
    (add1 (order-of-magnitude (argmax order-of-magnitude (cons (length t) (apply append t))))))
  (define (fmt-val v) (~a v #:width longest-number-length #:align 'right))
  (string-join
   (for/list ((r t) (k (in-naturals 1)))
     (string-append
      (format "║ k = ~a║ " (fmt-val k))
      (string-join (for/list ((c r)) (fmt-val c)) "| ")
      "║"))
   "\n"))
 
(displayln (format-table KAP-table-values))

Output:

║ k =   1║   2|   3|   5|   7|  11|  13|  17|  19|  23|  29║
║ k =   2║   4|   6|   9|  10|  14|  15|  21|  22|  25|  26║
║ k =   3║   8|  12|  18|  20|  27|  28|  30|  42|  44|  45║
║ k =   4║  16|  24|  36|  40|  54|  56|  60|  81|  84|  88║
║ k =   5║  32|  48|  72|  80| 108| 112| 120| 162| 168| 176║

REXX[edit]

naive version[edit]

The method used is to count the number of factors in the number to determine the K-primality.

The first three k-almost primes for each K group are computed directly (rather than found).

/*REXX program  computes and displays  the  first  N  K─almost  primes  from   1 ──► K. */
parse arg N K .                                  /*get optional arguments from the C.L. */
if N=='' | N==","  then N=10                     /*N  not specified?   Then use default.*/
if K=='' | K==","  then K= 5                     /*K   "      "          "   "     "    */
                                                 /*W: is the width of K, used for output*/
    do m=1  for  K;     $=2**m;  fir=$           /*generate & assign 1st K─almost prime.*/
    #=1;                if #==N  then leave      /*#: K─almost primes; Enough are found?*/
    #=2;                $=$  3*(2**(m-1))        /*generate & append 2nd K─almost prime.*/
    if #==N  then leave                          /*#: K─almost primes; Enough are found?*/
    if m==1  then _=fir + fir                    /* [↓]  gen & append 3rd K─almost prime*/
             else do;  _=9 * (2**(m-2));    #=3;    $=$  _;    end
        do j=_ + m - 1   until #==N              /*process an  K─almost prime  N  times.*/
        if factr()\==m  then iterate             /*not the correct  K─almost  prime?    */
        #=# + 1;         $=$ j                   /*bump K─almost counter; append it to $*/
        end   /*j*/                              /* [↑]   generate  N  K─almost  primes.*/
    say right(m, length(K))"─almost ("N') primes:'     $
    end       /*m*/                              /* [↑]  display a line for each K─prime*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
factr: z=j;                    do f=0  while z// 2==0;  z=z% 2;  end  /*divisible by  2.*/
                               do f=f  while z// 3==0;  z=z% 3;  end  /*divisible  "  3.*/
                               do f=f  while z// 5==0;  z=z% 5;  end  /*divisible  "  5.*/
                               do f=f  while z// 7==0;  z=z% 7;  end  /*divisible  "  7.*/
                               do f=f  while z//11==0;  z=z%11;  end  /*divisible  " 11.*/
                               do f=f  while z//13==0;  z=z%13;  end  /*divisible  " 13.*/
         do p=17  by 6  while  p<=z              /*insure  P  isn't divisible by three. */
         parse var  p   ''  -1  _                /*obtain the right─most decimal digit. */
                                                 /* [↓]  fast check for divisible by 5. */
         if _\==5  then do; do f=f+1  while z//p==0; z=z%p; end;  f=f-1; end  /*÷ by P? */
         if _ ==3  then iterate                  /*fast check for  X  divisible by five.*/
         x=p+2;             do f=f+1  while z//x==0; z=z%x; end;  f=f-1       /*÷ by X? */
         end   /*i*/                             /* [↑]  find all the factors in  Z.    */
 
       if f==0  then return 1                    /*if  prime (f==0),  then return unity.*/
                     return f                    /*return to invoker the number of divs.*/

output when using the default input:

1─almost (10) primes: 2 3 5 7 11 13 17 19 23 29
2─almost (10) primes: 4 6 9 10 14 15 21 22 25 26
3─almost (10) primes: 8 12 18 20 27 28 30 42 44 45
4─almost (10) primes: 16 24 36 40 54 56 60 81 84 88
5─almost (10) primes: 32 48 72 80 108 112 120 162 168 176

output when using the input of: 20 12

 1─almost (20) primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
 2─almost (20) primes: 4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57
 3─almost (20) primes: 8 12 18 20 27 28 30 42 44 45 50 52 63 66 68 70 75 76 78 92
 4─almost (20) primes: 16 24 36 40 54 56 60 81 84 88 90 100 104 126 132 135 136 140 150 152
 5─almost (20) primes: 32 48 72 80 108 112 120 162 168 176 180 200 208 243 252 264 270 272 280 300
 6─almost (20) primes: 64 96 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600
 7─almost (20) primes: 128 192 288 320 432 448 480 648 672 704 720 800 832 972 1008 1056 1080 1088 1120 1200
 8─almost (20) primes: 256 384 576 640 864 896 960 1296 1344 1408 1440 1600 1664 1944 2016 2112 2160 2176 2240 2400
 9─almost (20) primes: 512 768 1152 1280 1728 1792 1920 2592 2688 2816 2880 3200 3328 3888 4032 4224 4320 4352 4480 4800
10─almost (20) primes: 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632 5760 6400 6656 7776 8064 8448 8640 8704 8960 9600
11─almost (20) primes: 2048 3072 4608 5120 6912 7168 7680 10368 10752 11264 11520 12800 13312 15552 16128 16896 17280 17408 17920 19200
12─almost (20) primes: 4096 6144 9216 10240 13824 14336 15360 20736 21504 22528 23040 25600 26624 31104 32256 33792 34560 34816 35840 38400

optimized version[edit]

This optimized REXX version can be over a hundred times faster than the naive version.

Some of the optimizations are:

calculating the first 2^(K-1) K─almost primes for each K group
generating the primes (up to the limit) instead of dividing by (most) divisors.
extending the up-front prime divisors in the factr function.

The 1^st optimization (bullet) allows the direct computation (instead of searching) of all K─almost primes up to the first odd prime in the list.

Once the required primes are generated, the finding of the K─almost primes is almost instantaneous.

/*REXX program  computes and displays  the first    N    K─almost primes from  1 ──► K. */
parse arg N K .                                  /*obtain optional arguments from the CL*/
if N=='' | N==','  then N=10                     /*N  not specified?   Then use default.*/
if K=='' | K==','  then K= 5                     /*K   "      "          "   "     "    */
nn=N;  N=abs(N);   w=length(K)                   /*N positive? Then show K─almost primes*/
limit= (2**K) * N / 2                            /*this is the limit for most K-primes. */
if N==1  then limit=limit * 2                    /*  "   "  "    "    "  a    N    of 1.*/
if K==1  then limit=limit * 4                    /*  "   "  "    "    "  a K─prime  " 2.*/
if K==2  then limit=limit * 2                    /*  "   "  "    "    "  "    "     " 4.*/
if K==3  then limit=limit * 3 % 2                /*  "   "  "    "    "  "    "     " 8.*/
call genPrimes  limit + 1                        /*generate primes up to the  LIMIT + 1.*/
say 'The highest prime computed: '        @.#        " (under the limit of " limit').'
say                                              /* [↓]  define where 1st K─prime is odd*/
d.=0;  d.2=  2;  d.3 =  4;  d.4 =  7;  d.5 = 13;  d.6 = 22;  d.7 =  38;   d.8=63
       d.9=102;  d.10=168;  d.11=268;  d.12=426;  d.13=673;  d.14=1064
d!=0
    do m=1  for  K;    d!=max(d!,d.m)            /*generate & assign 1st K─almost prime.*/
    mr=right(m,w);     mm=m-1
 
    $=;           do #=1  to min(N, d!)          /*assign some doubled K─almost primes. */
                  $=$  d.mm.# * 2
                  end   /*#*/
    #=#-1
    if m==1  then from=2
             else from=1 + word($, words($) )
 
        do j=from   until  #==N                  /*process an  K─almost prime  N  times.*/
        if factr()\==m  then iterate             /*not the correct  K─almost  prime?    */
        #=#+1;   $=$ j                           /*bump K─almost counter; append it to $*/
        end   /*j*/                              /* [↑]   generate  N  K─almost  primes.*/
 
    if nn>0  then say mr"─almost ("N') primes:'     $
             else say '    the last'  mr  "K─almost prime: "   word($, words($))
                                               /* [↓]  assign K─almost primes.*/
          do q=1  for #;     d.m.q=word($,q)             ;   end  /*q*/
          do q=1  for #;  if d.m.q\==d.mm.q*2  then leave;   end  /*q*/
                                               /* [↑]  count doubly-duplicates*/
/*──── say copies('─',40)  'for '   m", "   q-1   'numbers were doubly─duplicated.' ────*/
/*──── say                                                                          ────*/
    end       /*m*/                              /* [↑]  display a line for each K─prime*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
factr: if #.j\==.  then return #.j
       z=j;                                do f=0 while z// 2==0; z=z% 2; end   /*÷ by 2*/
                                           do f=f while z// 3==0; z=z% 3; end   /*÷ "  3*/
                                           do f=f while z// 5==0; z=z% 5; end   /*÷ "  5*/
                                           do f=f while z// 7==0; z=z% 7; end   /*÷ "  7*/
                                           do f=f while z//11==0; z=z%11; end   /*÷ " 11*/
                                           do f=f while z//13==0; z=z%13; end   /*÷ " 13*/
                                           do f=f while z//17==0; z=z%17; end   /*÷ " 17*/
                                           do f=f while z//19==0; z=z%19; end   /*÷ " 19*/
 
         do i=9    while  @.i<=z;       [email protected].i    /*divide by some higher primes.        */
           do f=f  while z//d==0;   z=z%d;  end  /*is  Z  divisible by the  prime  D ?  */
         end   /*i*/                             /* [↑]  find all factors in  Z.        */
 
       if f==0  then f=1;   #.j=f;   return f    /*Is prime (f≡0)?   Then return unity. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
genPrimes: arg x;             @.=;      @.1=2;     @.2=3;    #.=.;     #=2;     s.#[email protected].#**2
             do [email protected].# +2  by 2  to x             /*only find odd primes from here on.   */
                do p=2  while s.p<=j             /*divide by some known low odd primes. */
                if j//@.p==0  then iterate j     /*Is  J  divisible by X?  Then ¬ prime.*/
                end   /*p*/                      /* [↓]  a prime  (J)  has been found.  */
             #=#+1;    @.#=j;   #.j=1;   s.#=j*j /*bump prime count, and also assign ···*/
             end      /*j*/                      /* ··· the # of factors, prime, prime².*/
           return                                /* [↑]  not an optimal prime generator.*/

output when using the input of: 20 16

The highest prime computed:  655357  (under the limit of  655360).

 1─almost (20) primes:  2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
 2─almost (20) primes:  4 6 9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57
 3─almost (20) primes:  8 12 18 20 27 28 30 42 44 45 50 52 63 66 68 70 75 76 78 92
 4─almost (20) primes:  16 24 36 40 54 56 60 81 84 88 90 100 104 126 132 135 136 140 150 152
 5─almost (20) primes:  32 48 72 80 108 112 120 162 168 176 180 200 208 243 252 264 270 272 280 300
 6─almost (20) primes:  64 96 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600
 7─almost (20) primes:  128 192 288 320 432 448 480 648 672 704 720 800 832 972 1008 1056 1080 1088 1120 1200
 8─almost (20) primes:  256 384 576 640 864 896 960 1296 1344 1408 1440 1600 1664 1944 2016 2112 2160 2176 2240 2400
 9─almost (20) primes:  512 768 1152 1280 1728 1792 1920 2592 2688 2816 2880 3200 3328 3888 4032 4224 4320 4352 4480 4800
10─almost (20) primes:  1024 1536 2304 2560 3456 3584 3840 5184 5376 5632 5760 6400 6656 7776 8064 8448 8640 8704 8960 9600
11─almost (20) primes:  2048 3072 4608 5120 6912 7168 7680 10368 10752 11264 11520 12800 13312 15552 16128 16896 17280 17408 17920 19200
12─almost (20) primes:  4096 6144 9216 10240 13824 14336 15360 20736 21504 22528 23040 25600 26624 31104 32256 33792 34560 34816 35840 38400
13─almost (20) primes:  8192 12288 18432 20480 27648 28672 30720 41472 43008 45056 46080 51200 53248 62208 64512 67584 69120 69632 71680 76800
14─almost (20) primes:  16384 24576 36864 40960 55296 57344 61440 82944 86016 90112 92160 102400 106496 124416 129024 135168 138240 139264 143360 153600
15─almost (20) primes:  32768 49152 73728 81920 110592 114688 122880 165888 172032 180224 184320 204800 212992 248832 258048 270336 276480 278528 286720 307200
16─almost (20) primes:  65536 98304 147456 163840 221184 229376 245760 331776 344064 360448 368640 409600 425984 497664 516096 540672 552960 557056 573440 614400

Ring[edit]

 
for ap = 1 to 5
    see "k = " + ap + ":" 
    aList = []
    for n = 1 to 200
        num = 0
        for nr = 1 to n
            if n%nr=0 and isPrime(nr)=1
               num = num + 1 
               pr = nr
               while true
                     pr = pr * nr
                     if n%pr = 0
                        num = num + 1
                     else exit ok
               end ok
        next  
        if (ap = 1 and isPrime(n) = 1) or (ap > 1 and num = ap)
           add(aList, n)
           if len(aList)=10 exit ok ok
     next
     for m = 1 to len(aList)
           see " " + aList[m]
     next 
     see nl
next
 
func isPrime num
     if (num <= 1) return 0 ok
     if (num % 2 = 0 and num != 2) return 0 ok
     for i = 3 to floor(num / 2) -1 step 2
         if (num % i = 0) return 0 ok
     next
     return 1

Output:

k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176

Ruby[edit]

require 'prime'
 
def almost_primes(k=2)
  return to_enum(:almost_primes, k) unless block_given?
  n = 0
  loop do 
    n += 1
    yield n if n.prime_division.map( &:last ).inject( &:+ ) == k
  end
end
 
(1..5).each{|k| puts almost_primes(k).take(10).join(", ")}

Output:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29
4, 6, 9, 10, 14, 15, 21, 22, 25, 26
8, 12, 18, 20, 27, 28, 30, 42, 44, 45
16, 24, 36, 40, 54, 56, 60, 81, 84, 88
32, 48, 72, 80, 108, 112, 120, 162, 168, 176

Translation of: J

require 'prime'
 
p ar = pr = Prime.take(10)
4.times{p ar = ar.product(pr).map{|(a,b)| a*b}.uniq.sort.take(10)}

Output:

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
[4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
[8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
[16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
[32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

Rust[edit]

fn is_kprime(n: u32, k: u32) -> bool {
    let mut primes = 0;
    let mut f = 2;
    let mut rem = n;
    while primes < k && rem > 1{
        while (rem % f) == 0 && rem > 1{
            rem /= f;
            primes += 1;
        }
        f += 1;
    }
    rem == 1 && primes == k
}
 
struct KPrimeGen {
    k: u32,
    n: u32,
}
 
impl Iterator for KPrimeGen {
    type Item = u32;
    fn next(&mut self) -> Option<u32> {
        self.n += 1;
        while !is_kprime(self.n, self.k) {
            self.n += 1;
        }
        Some(self.n)
    }
}
 
fn kprime_generator(k: u32) -> KPrimeGen {
    KPrimeGen {k: k, n: 1}
}
 
fn main() {
    for k in 1..6 {
        println!("{}: {:?}", k, kprime_generator(k).take(10).collect::<Vec<_>>());
    }
}

Output:

1: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
2: [4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
3: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
4: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
5: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

Scala[edit]

def isKPrime(n: Int, k: Int, d: Int = 2): Boolean = (n, k, d) match {
    case (n, k, _) if n == 1 => k == 0
    case (n, _, d) if n % d == 0 => isKPrime(n / d, k - 1, d)
    case (_, _, _) => isKPrime(n, k, d + 1)
}
 
def kPrimeStream(k: Int): Stream[Int] = {
    def loop(n: Int): Stream[Int] =
        if (isKPrime(n, k)) n #:: loop(n+ 1)
        else loop(n + 1)
    loop(2)
}
 
for (k <- 1 to 5) {
    println( s"$k: [${ kPrimeStream(k).take(10) mkString " " }]" )
}

Output:

1: [2 3 5 7 11 13 17 19 23 29]
2: [4 6 9 10 14 15 21 22 25 26]
3: [8 12 18 20 27 28 30 42 44 45]
4: [16 24 36 40 54 56 60 81 84 88]
5: [32 48 72 80 108 112 120 162 168 176]

SequenceL[edit]

import <Utilities/Conversion.sl>;
import <Utilities/Sequence.sl>;
 
main(args(2)) :=
	let
		result := firstNKPrimes(1 ... 5, 10);
 
		output[i] := "k = " ++ intToString(i) ++ ": " ++ delimit(intToString(result[i]), ' ');
	in
		delimit(output, '\n');
 
firstNKPrimes(k, N) := firstNKPrimesHelper(k, N, 2, []);
 
firstNKPrimesHelper(k, N, current, result(1)) :=
	let
		newResult := result when not isKPrime(k, current) else result ++ [current]; 
	in
		result when size(result) = N
	else
		firstNKPrimesHelper(k, N, current + 1, newResult);
 
isKPrime(k, n) := size(primeFactorization(n)) = k;

Using Prime Decomposition Solution [1]

Output:

main.exe
"k = 1: 2 3 5 7 11 13 17 19 23 29
k = 2: 4 6 9 10 14 15 21 22 25 26
k = 3: 8 12 18 20 27 28 30 42 44 45
k = 4: 16 24 36 40 54 56 60 81 84 88
k = 5: 32 48 72 80 108 112 120 162 168 176"

Sidef[edit]

Translation of: Perl 6

func is_k_almost_prime(n, k) {
    for (var (p, f) = (2, 0); (f < k) && (p*p <= n); ++p) {
        (n /= p; ++f) while (p `divides` n)
    }
    n > 1 ? (f.inc == k) : (f == k)
}
 
{ |k|
    var x = 10
    say gather {
        { |i|
            if (is_k_almost_prime(i, k)) {
                take(i)
                --x == 0 && break
            }
        } << 1..Inf
    }
} << 1..5

Output:

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
[4, 6, 9, 10, 14, 15, 21, 22, 25, 26]
[8, 12, 18, 20, 27, 28, 30, 42, 44, 45]
[16, 24, 36, 40, 54, 56, 60, 81, 84, 88]
[32, 48, 72, 80, 108, 112, 120, 162, 168, 176]

Tcl[edit]

Works with: Tcl version 8.6

Library: Tcllib (Package: math::numtheory)

package require Tcl 8.6
package require math::numtheory
 
proc firstNprimes n {
    for {set result {};set i 2} {[llength $result] < $n} {incr i} {
	if {[::math::numtheory::isprime $i]} {
	    lappend result $i
	}
    }
    return $result
}
 
proc firstN_KalmostPrimes {n k} {
    set p [firstNprimes $n]
    set i [lrepeat $k 0]
    set c {}
 
    while true {
	dict set c [::tcl::mathop::* {*}[lmap j $i {lindex $p $j}]] ""
	for {set x 0} {$x < $k} {incr x} {
	    lset i $x [set xx [expr {([lindex $i $x] + 1) % $n}]]
	    if {$xx} break
	}
	if {$x == $k} break
    }
    return [lrange [lsort -integer [dict keys $c]] 0 [expr {$n - 1}]]
}
 
for {set K 1} {$K <= 5} {incr K} {
    puts "$K => [firstN_KalmostPrimes 10 $K]"
}

Output:

1 => 2 3 5 7 11 13 17 19 23 29
2 => 4 6 9 10 14 15 21 22 25 26
3 => 8 12 18 20 27 28 30 42 44 45
4 => 16 24 36 40 54 56 60 81 84 88
5 => 32 48 72 80 108 112 120 162 168 176

uBasic/4tH[edit]

Translation of: C

Local(3)
 
For [email protected] = 1 To 5
  Print "k = ";[email protected];": ";
 
  [email protected]=0
 
  For [email protected] = 2 Step 1 While [email protected] < 10
    If FUNC(_kprime ([email protected],[email protected])) Then
       [email protected] = [email protected] + 1
       Print " ";[email protected];
    EndIf
  Next
 
  Print
Next
 
End
 
_kprime Param(2)
  Local(2)
 
  [email protected] = 0
  For [email protected] = 2 Step 1 While ([email protected] < [email protected]) * (([email protected] * [email protected]) < ([email protected] + 1))
    Do While ([email protected] % [email protected]) = 0
      [email protected] = [email protected] / [email protected]
      [email protected] = [email protected] + 1
    Loop
  Next
Return ([email protected] = ([email protected] + ([email protected] > 1)))

Output:

k = 1:  2 3 5 7 11 13 17 19 23 29
k = 2:  4 6 9 10 14 15 21 22 25 26
k = 3:  8 12 18 20 27 28 30 42 44 45
k = 4:  16 24 36 40 54 56 60 81 84 88
k = 5:  32 48 72 80 108 112 120 162 168 176

0 OK, 0:200

VBScript[edit]

Repurposed the VBScript code for the Prime Decomposition task.

 
For k = 1 To 5
	count = 0
	increment = 1
	WScript.StdOut.Write "K" & k & ": "
	Do Until count = 10
		If PrimeFactors(increment) = k Then
			WScript.StdOut.Write increment & " "
			count = count + 1
		End If
		increment = increment + 1
	Loop
	WScript.StdOut.WriteLine
Next
 
Function PrimeFactors(n)
	PrimeFactors = 0
	arrP = Split(ListPrimes(n)," ")
	divnum = n
	Do Until divnum = 1
		For i = 0 To UBound(arrP)-1
			If divnum = 1 Then
				Exit For
			ElseIf divnum Mod arrP(i) = 0 Then
				divnum = divnum/arrP(i) 
				PrimeFactors = PrimeFactors + 1
			End If
		Next
	Loop
End Function
 
Function IsPrime(n)
	If n = 2 Then
		IsPrime = True
	ElseIf n <= 1 Or n Mod 2 = 0 Then
		IsPrime = False
	Else
		IsPrime = True
		For i = 3 To Int(Sqr(n)) Step 2
			If n Mod i = 0 Then
				IsPrime = False
				Exit For
			End If
		Next
	End If
End Function
 
Function ListPrimes(n)
	ListPrimes = ""
	For i = 1 To n
		If IsPrime(i) Then
			ListPrimes = ListPrimes & i & " "
		End If
	Next
End Function

Output:

K1: 2 3 5 7 11 13 17 19 23 29 
K2: 4 6 9 10 14 15 21 22 25 26 
K3: 8 12 18 20 27 28 30 42 44 45 
K4: 16 24 36 40 54 56 60 81 84 88 
K5: 32 48 72 80 108 112 120 162 168 176

zkl[edit]

Translation of: Ruby

Translation of: J

Using the prime generator from task Extensible prime generator#zkl.

Can't say I entirely understand this algorithm. Uses list comprehension to calculate the outer/tensor product (p10 ⊗ ar).

primes:=Utils.Generator(Import("sieve").postponed_sieve);
(p10:=ar:=primes.walk(10)).println();
do(4){
   (ar=([[(x,y);ar;p10;'*]] : Utils.Helpers.listUnique(_).sort()[0,10])).println();
}

Output:

L(2,3,5,7,11,13,17,19,23,29)
L(4,6,9,10,14,15,21,22,25,26)
L(8,12,18,20,27,28,30,42,44,45)
L(16,24,36,40,54,56,60,81,84,88)
L(32,48,72,80,108,112,120,162,168,176)

ZX Spectrum Basic[edit]

Translation of: AWK

10 FOR k=1 TO 5
20 PRINT k;":";
30 LET c=0: LET i=1
40 IF c=10 THEN GO TO 100
50 LET i=i+1
60 GO SUB 1000
70 IF r THEN PRINT " ";i;: LET c=c+1
90 GO TO 40
100 PRINT 
110 NEXT k
120 STOP 
1000 REM kprime
1010 LET p=2: LET n=i: LET f=0
1020 IF f=k OR (p*p)>n THEN GO TO 1100
1030 IF n/p=INT (n/p) THEN LET n=n/p: LET f=f+1: GO TO 1030
1040 LET p=p+1: GO TO 1020
1100 LET r=(f+(n>1)=k)
1110 RETURN

Output:

1: 2 3 5 7 11 13 17 19 23 29
2: 4 6 9 10 14 15 21 22 25 26
3: 8 12 18 20 27 28 30 42 44 45
4: 16 24 36 40 54 56 60 81 84 88
5: 32 48 72 80 108 112 120 162 168 176

Almost prime

Contents

Ada[edit]

ALGOL 68[edit]

AutoHotkey[edit]

AWK[edit]

Befunge[edit]

C[edit]

C++[edit]

C#[edit]

Clojure[edit]

Common Lisp[edit]

D[edit]

EchoLisp[edit]

Elixir[edit]

Erlang[edit]

ERRE[edit]

FreeBASIC[edit]

Frink[edit]

Futhark[edit]

F#[edit]

Go[edit]

Groovy[edit]

Haskell[edit]

Icon and Unicon[edit]

J[edit]

Java[edit]

JavaScript[edit]

jq[edit]

Julia[edit]

Lua[edit]

Kotlin[edit]

Mathematica / Wolfram Language[edit]

Nim[edit]

Objeck[edit]

Oforth[edit]

PARI/GP[edit]

Pascal[edit]

Perl[edit]

Perl 6[edit]

Phix[edit]

PicoLisp[edit]

Potion[edit]

Prolog[edit]

Python[edit]

R[edit]

Racket[edit]

REXX[edit]

naive version[edit]

optimized version[edit]

Ring[edit]

Ruby[edit]

Rust[edit]

Scala[edit]

SequenceL[edit]

Sidef[edit]

Tcl[edit]

uBasic/4tH[edit]

VBScript[edit]

zkl[edit]

ZX Spectrum Basic[edit]