Almost prime
From Rosetta Code
You are encouraged to solve this task according to the task description, using any language you may know.
A k-Almost-prime is a natural number that is the product of (possibly identical) primes.
- Example
1-almost-primes, where , are the prime numbers themselves.
2-almost-primes, where , 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 .
- Related tasks
Contents
- 1 11l
- 2 Ada
- 3 ALGOL 68
- 4 AutoHotkey
- 5 AWK
- 6 Befunge
- 7 C
- 8 C++
- 9 C#
- 10 Clojure
- 11 Common Lisp
- 12 D
- 13 EchoLisp
- 14 Elixir
- 15 Erlang
- 16 ERRE
- 17 Factor
- 18 FreeBASIC
- 19 Frink
- 20 Futhark
- 21 F#
- 22 Go
- 23 Groovy
- 24 GW-BASIC
- 25 Haskell
- 26 Icon and Unicon
- 27 J
- 28 Java
- 29 JavaScript
- 30 jq
- 31 Julia
- 32 Lua
- 33 Kotlin
- 34 Maple
- 35 Mathematica / Wolfram Language
- 36 Modula-2
- 37 Nim
- 38 Objeck
- 39 Oforth
- 40 PARI/GP
- 41 Pascal
- 42 Perl
- 43 Perl 6
- 44 Phix
- 45 PicoLisp
- 46 Potion
- 47 Prolog
- 48 PureBasic
- 49 Python
- 50 R
- 51 Racket
- 52 REXX
- 53 Ring
- 54 Ruby
- 55 Rust
- 56 Scala
- 57 SequenceL
- 58 Sidef
- 59 Tcl
- 60 uBasic/4tH
- 61 VBA
- 62 VBScript
- 63 Visual Basic .NET
- 64 Yabasic
- 65 zkl
- 66 ZX Spectrum Basic
11l[edit]
F k_prime(k, =n)
V f = 0
V p = 2
L f < k & p * p <= n
L n % p == 0
n /= p
f++
p++
R f + (I n > 1 {1} E 0) == k
F primes(k, n)
V i = 2
Array[Int] list
L list.len < n
I k_prime(k, i)
list.append(i)
i++
R list
L(k) 1..5
print(‘k = ’k‘: ’, end' ‘’)
print(primes(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]
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]
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]
#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.
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]
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$(12);) !CLS
FOR K=1 TO 5 DO
PRINT("k =";K;":";)
C=0
FOR I=2 TO 999 DO
EXIT IF NOT(C<10)
KPRIME(I,K->KP)
IF KP THEN
PRINT(I;)
C+=1
END IF
END FOR
END FOR
END PROGRAM
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
Factor[edit]
USING: formatting fry kernel lists lists.lazy locals
math.combinatorics math.primes.factors math.ranges sequences ;
IN: rosetta-code.almost-prime
: k-almost-prime? ( n k -- ? )
'[ factors _ <combinations> [ product ] map ]
[ [ = ] curry ] bi any? ;
:: first10 ( k -- seq )
10 0 lfrom [ k k-almost-prime? ] lfilter ltake list>array ;
5 [1,b] [ dup first10 "K = %d: %[%3d, %]\n" printf ] each
- Output:
K = 1: { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 }
K = 2: { 4, 6, 9, 10, 14, 15, 21, 22, 25, 26 }
K = 3: { 8, 12, 18, 20, 27, 28, 30, 42, 44, 45 }
K = 4: { 16, 24, 36, 40, 54, 56, 60, 81, 84, 88 }
K = 5: { 32, 48, 72, 80, 108, 112, 120, 162, 168, 176 }
FreeBASIC[edit]
' FB 1.05.0 Win64
Function kPrime(n As Integer, k As Integer) As Boolean
Dim f As Integer = 0
For i As Integer = 2 To n
While n Mod i = 0
If f = k Then Return false
f += 1
n \= i
Wend
Next
Return f = k
End Function
Dim As Integer i, c, k
For k = 1 To 5
Print "k = "; k; " : ";
i = 2
c = 0
While c < 10
If kPrime(i, k) Then
Print Using "### "; i;
c += 1
End If
i += 1
Wend
Next
Print "Press any key to quit"
Sleep
- Output:
k = 1 : 2 3 5 7 11 13 17 19 23 29 k = 2 : 4 6 9 10 14 15 21 22 25 26 k = 3 : 8 12 18 20 27 28 30 42 44 45 k = 4 : 16 24 36 40 54 56 60 81 84 88 k = 5 : 32 48 72 80 108 112 120 162 168 176
Frink[edit]
for k = 1 to 5
{
n=2
count = 0
print["k=$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
GW-BASIC[edit]
10 'Almost prime
20 FOR K% = 1 TO 5
30 PRINT "k = "; K%; ": ";
40 LET I% = 2
50 LET C% = 0
60 WHILE C% < 10
70 LET AN% = I%: LET AK% = K%: GOSUB 1000
80 IF ISKPRIME <> 0 THEN PRINT USING "### "; I%;: LET C% = C% + 1
90 LET I% = I% + 1
100 WEND
110 PRINT
120 NEXT K%
130 END
995 ' Check if n (AN%) is a k (AK%) prime
1000 LET F% = 0
1010 FOR II% = 2 TO AN%
1020 WHILE AN% MOD II% = 0
1030 IF F% = AK% THEN LET ISKPRIME = 0: RETURN
1040 LET F% = F% + 1
1050 LET AN% = AN% \ II%
1060 WEND
1070 NEXT II%
1080 LET ISKPRIME = (F% = AK%)
1090 RETURN
- 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]
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]
using Primes
isalmostprime(n::Integer, k::Integer) = sum(values(factor(n))) == k
function almostprimes(N::Integer, k::Integer) # return first N almost-k primes
P = Vector{typeof(k)}(N)
i = 0; n = 2
while i < N
if isalmostprime(n, k) P[i += 1] = n end
n += 1
end
return P
end
for k in 1:5
println("$k-Almost-primes: ", join(almostprimes(10, k), ", "), "...")
end
- Output:
1-Almost-primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... 2-Almost-primes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26... 3-Almost-primes: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45... 4-Almost-primes: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88... 5-Almost-primes: 32, 48, 72, 80, 108, 112, 120, 162, 168, 176...
using Primes
isalmostprime(n::Integer, k::Integer) = sum(values(factor(n))) == k
function almostprimes(N::Integer, k::Integer) # return first N almost-k primes
P = Vector{typeof(k)}(undef,N)
i = 0; n = 2
while i < N
if isalmostprime(n, k) P[i += 1] = n end
n += 1
end
return P
end
for k in 1:5
println("$k-Almost-primes: ", join(almostprimes(10, k), ", "), "...")
end
- Output:
1-Almost-primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... 2-Almost-primes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26... 3-Almost-primes: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45... 4-Almost-primes: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88... 5-Almost-primes: 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]
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 = mutableListOf<Int>()
while (list.size < n) {
if (k_prime(i)) list.add(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]
Maple[edit]
AlmostPrimes:=proc(k, numvalues::posint:=10)
local aprimes, i, intfactors;
aprimes := Array([]);
i := 0;
do
i := i + 1;
intfactors := ifactors(i)[2];
intfactors := [seq(seq(intfactors[i][1], j=1..intfactors[i][2]),i = 1..numelems(intfactors))];
if numelems(intfactors) = k then
ArrayTools:-Append(aprimes,i);
end if;
until numelems(aprimes) = 10:
aprimes;
end proc:
<seq( AlmostPrimes(i), i = 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]]
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
Modula-2[edit]
MODULE AlmostPrime;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE KPrime(n,k : INTEGER) : BOOLEAN;
VAR p,f : INTEGER;
BEGIN
f := 0;
p := 2;
WHILE (f<k) AND (p*p<=n) DO
WHILE n MOD p = 0 DO
n := n DIV p;
INC(f)
END;
INC(p)
END;
IF n>1 THEN
RETURN f+1 = k
END;
RETURN f = k
END KPrime;
VAR
buf : ARRAY[0..63] OF CHAR;
i,c,k : INTEGER;
BEGIN
FOR k:=1 TO 5 DO
FormatString("k = %i:", buf, k);
WriteString(buf);
i:=2;
c:=0;
WHILE c<10 DO
IF KPrime(i,k) THEN
FormatString(" %i", buf, i);
WriteString(buf);
INC(c)
END;
INC(i)
END;
WriteLn;
END;
ReadChar;
END AlmostPrime.
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]
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)) {
" {$i}"->Print();
c+=1;
};
};
'\n'->Print();
};
}
}
- Output:
k = 1: 2 3 5 7 11 13 17 19 23 29 k = 2: 4 6 9 10 14 15 21 22 25 26 k = 3: 8 12 18 20 27 28 30 42 44 45 k = 4: 16 24 36 40 54 56 60 81 84 88 k = 5: 32 48 72 80 108 112 120 162 168 176
Oforth[edit]
: kprime?( n k -- b )
| i |
0 2 n for: i [
while( n i /mod swap 0 = ) [ ->n 1+ ] drop
]
k ==
;
: table( k -- [] )
| l |
Array new dup ->l
2 while (l size 10 <>) [ dup k kprime? if dup l add then 1+ ]
drop
;
- Output:
>#[ table .cr ] 5 each [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] [8, 12, 18, 20, 27, 28, 30, 42, 44, 45] [16, 24, 36, 40, 54, 56, 60, 81, 84, 88] [32, 48, 72, 80, 108, 112, 120, 162, 168, 176]
PARI/GP[edit]
almost(k)=my(n); for(i=1,10,while(bigomega(n++)!=k,); print1(n", "));
for(k=1,5,almost(k);print)
- Output:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 32, 48, 72, 80, 108, 112, 120, 162, 168, 176,
Pascal[edit]
program AlmostPrime;
{$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:
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]
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 theThat's it. The rest is machinery. For portability, a compatibility section is included below.
% 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).
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 %%%%%Example using SWI-Prolog:
:- 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.
?- 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)
PureBasic[edit]
EnableExplicit
Procedure.b kprime(n.i, k.i)
Define p.i = 2,
f.i = 0
While f < k And p*p <= n
While n % p = 0
n / p
f + 1
Wend
p + 1
Wend
ProcedureReturn Bool(f + Bool(n > 1) = k)
EndProcedure
;___main____
If Not OpenConsole("Almost prime")
End -1
EndIf
Define i.i,
c.i,
k.i
For k = 1 To 5
Print("k = " + Str(k) + ":")
i = 2
c = 0
While c < 10
If kprime(i, k)
Print(RSet(Str(i),4))
c + 1
EndIf
i + 1
Wend
PrintN("")
Next
Input()
- 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
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]
An updated version with no import dependencies.[edit]
# k-Almost-primes
# Python 3.6.3
# no imports
def prime_factors(m=2):
for i in range(2, m):
r, q = divmod(m, i)
if not q:
return [i] + prime_factors(r)
return [m]
def k_almost_primes(n, k=2):
multiples = set()
lists = list()
for x in range(k+1):
lists.append([])
for i in range(2, n+1):
if i not in multiples:
if len(lists[1]) < 10:
lists[1].append(i)
multiples.update(range(i*i, n+1, i))
print("k=1: {}".format(lists[1]))
for j in range(2, k+1):
for m in multiples:
l = prime_factors(m)
ll = len(l)
if ll == j and len(lists[j]) < 10:
lists[j].append(m)
print("k={}: {}".format(j, lists[j]))
k_almost_primes(200, 5)
# try:
#k_almost_primes(6000, 10)
- Output:
>>> %Run k_almost_primes.py 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]
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 1st 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?
1.step {|n| yield n if n.prime_division.sum( &:last ) == k }
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
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]
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]
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]
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
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
VBA[edit]
Private Function kprime(ByVal n As Integer, k As Integer) As Boolean
Dim p As Integer, factors As Integer
p = 2
factors = 0
Do While factors < k And p * p <= n
Do While n Mod p = 0
n = n / p
factors = factors + 1
Loop
p = p + 1
Loop
factors = factors - (n > 1) 'true=-1
kprime = factors = k
End Function
Private Sub almost_primeC()
Dim nextkprime As Integer, count As Integer
Dim k As Integer
For k = 1 To 5
Debug.Print "k ="; k; ":";
nextkprime = 2
count = 0
Do While count < 10
If kprime(nextkprime, k) Then
Debug.Print " "; Format(CStr(nextkprime), "@@@@@");
count = count + 1
End If
nextkprime = nextkprime + 1
Loop
Debug.Print
Next k
End Sub
- 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
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
Visual Basic .NET[edit]
Module Module1
Class KPrime
Public K As Integer
Public Function IsKPrime(number As Integer) As Boolean
Dim primes = 0
Dim p = 2
While p * p <= number AndAlso primes < K
While number Mod p = 0 AndAlso primes < K
number = number / p
primes = primes + 1
End While
p = p + 1
End While
If number > 1 Then
primes = primes + 1
End If
Return primes = K
End Function
Public Function GetFirstN(n As Integer) As List(Of Integer)
Dim result As New List(Of Integer)
Dim number = 2
While result.Count < n
If IsKPrime(number) Then
result.Add(number)
End If
number = number + 1
End While
Return result
End Function
End Class
Sub Main()
For Each k In Enumerable.Range(1, 5)
Dim kprime = New KPrime With {
.K = k
}
Console.WriteLine("k = {0}: {1}", k, String.Join(" ", kprime.GetFirstN(10)))
Next
End Sub
End Module
- 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
Yabasic[edit]
// Returns boolean indicating whether n is k-almost prime
sub almostPrime(n, k)
local divisor, count
divisor = 2
while(count < (k + 1) and n <> 1)
if not mod(n, divisor) then
n = n / divisor
count = count + 1
else
divisor = divisor + 1
end if
wend
return count = k
end sub
// Generates table containing first ten k-almost primes for given k
sub kList(k, kTab())
local n, i
n = 2^k : i = 1
while(i < 11)
if almostPrime(n, k) then
kTab(i) = n
i = i + 1
end if
n = n + 1
wend
end sub
// Main procedure, displays results from five calls to kList()
dim kTab(10)
for k = 1 to 5
print "k = ", k, " : ";
kList(k, kTab())
for n = 1 to 10
print kTab(n), ", ";
next
print "..."
next
zkl[edit]
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]
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
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