Almost prime
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. So, for example, 1-almost-primes, where , are the prime numbers themselves; 2-almost-primes are the semiprimes.
The task is to 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 .
- Cf.
Ada
This imports the package Prime_Numbers from Prime decomposition#Ada.
<lang 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;</lang>
- 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
Worth noticing is the n(...)(...) picture in the printf and the WHILE ... DO SKIP OD idiom which is quite common in ALgol 68. <lang algol68>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</lang>
- 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
Translation of the C Version <lang AutoHotkey>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</lang>
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
Befunge
The extra spaces are to ensure it's readable on buggy interpreters that don't include a space after numeric output.
<lang befunge>1>::48*"= k",,,,02p.":",01v |^ v0!`\*:g40:<p402p300:+1< K| >2g03g`*#v_ 1`03g+02g->| F@>/03g1+03p>vpv+1\.:,*48 < P#|!\g40%g40:<4>:9`>#v_\1^| |^>#!1#`+#50#:^#+1,+5>#5$<|</lang>
- 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
<lang c>#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; }</lang>
- 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#
<lang csharp>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;
}
}
}</lang>
- 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
Common Lisp
<lang lisp>(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))))</lang>
- 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
This contains a copy of the function decompose from the Prime decomposition task.
<lang d>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;
}
}</lang>
- 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
Small numbers : filter the sequence [ 2 .. n] <lang scheme> (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))) </lang>
- Output:
<lang scheme> (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 </lang> Large numbers : generate - combinations with repetitions - k-almost-primes up to pmax. <lang scheme> (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)))
</lang>
- Output:
<lang scheme>
- 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.
</lang>
Elixir
<lang elixir>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)</lang>
- 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
Using the factors function from Prime_decomposition#Erlang.
<lang 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.
</lang>
- 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
<lang ERRE> 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$(12);) !CLS
FOR K=1 TO 5 DO
PRINT("k =";K;":";)
C=0
FOR I=2 TO 999 DO
EXIT IF NOT(C<10)
KPRIME(I,K->KP)
IF KP THEN
PRINT(I;)
C+=1
END IF
END FOR
PRINT
END FOR
END PROGRAM </lang>
- Output:
K = 1: 2 3 5 7 11 13 17 19 23 29 K = 2: 4 6 9 10 14 15 21 22 25 26 K = 3: 8 12 18 20 27 28 30 42 44 45 K = 4: 16 24 36 40 54 56 60 81 84 88 K = 5: 32 48 72 80 108 112 120 162 168 176
Frink
<lang frink>for k = 1 to 5 {
n=2
count = 0
print["k=$k:"]
do
{
if length[factorFlat[n]] == k
{
print[" $n"]
count = count + 1
}
n = n + 1
} while count < 10
println[]
}</lang>
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
F#
<lang fsharp>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))</lang>
- 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
<lang go>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))
}
}</lang>
- 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]
Haskell
<lang Haskell>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))</lang>
- 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: <lang haskell>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</lang>
- 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...>
Objeck
<lang objeck>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();
};
}
}</lang>
- 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
Icon and Unicon
Works in both languages. <lang unicon>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</lang>
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
<lang J> (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</lang> 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
<lang java>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 (0 == n % p){
n /= p;
f++;
}
return f + ((n > 1)?1:0) == k;
}
}</lang>
- 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
jq
Infrastructure: <lang jq># 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;</lang>
isalmostprime <lang jq>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;</lang> The task:
<lang jq>range(1;6) as $k | "k=\($k): \([almostprimes(10;$k)])"</lang>
- Output:
<lang sh>$ 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]</lang>
Julia
<lang julia>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</lang>
- 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]
Mathematica / Wolfram Language
<lang Mathematica>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</lang>
- 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
<lang Oforth>func: kprime(n, k) { | i |
0 2 n for: i [ while(n i /mod swap 0 &==) [ ->n 1 + ] drop ] k ==
}
func: table(k) { | l |
ListBuffer new ->l 2 while (l size 10 <>) [ k over kprime ifTrue: [ dup l add ] 1 + ] drop l
}</lang>
- Output:
>5 seq apply(#[ table println ]) [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
<lang parigp>almost(k)=my(n); for(i=1,10,while(bigomega(n++)!=k,); print1(n", ")); for(k=1,5,almost(k);print)</lang>
- 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
<lang Pascal>program AlmostPrime; {$IFDEF FPC}
{$Mode Delphi}
{$ENDIF} uses
primtrial;
var
i,K,cnt : longWord;
BEGIN
K := 1;
repeat
cnt := 0;
i := 2;
write('K=',K:2,':');
repeat
if isAlmostPrime(i,K) then
Begin
write(i:6,' ');
inc(cnt);
end;
inc(i);
until cnt = 9;
writeln;
inc(k);
until k > 10;
END.</lang>
- output
K= 1 : 2 3 5 7 11 13 17 19 23 29 K= 2 : 4 6 9 10 14 15 21 22 25 26 K= 3 : 8 12 18 20 27 28 30 42 44 45 K= 4 : 16 24 36 40 54 56 60 81 84 88 K= 5 : 32 48 72 80 108 112 120 162 168 176 K= 6 : 64 96 144 160 216 224 240 324 336 352 K= 7 : 128 192 288 320 432 448 480 648 672 704 K= 8 : 256 384 576 640 864 896 960 1296 1344 1408 K= 9 : 512 768 1152 1280 1728 1792 1920 2592 2688 2816 K=10 : 1024 1536 2304 2560 3456 3584 3840 5184 5376 5632
Perl
Using a CPAN module, which is simple and fast:
<lang perl>use ntheory qw/factor/; sub almost {
my($k,$n) = @_;
my $i = 1;
map { $i++ while scalar factor($i) != $k; $i++ } 1..$n;
} say "$_ : ", join(" ", almost($_,10)) for 1..5;</lang>
- 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: <lang perl>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]; } }</lang>
- 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
<lang perl6>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 -> $k {
say .[^10]
given grep { is-k-almost-prime($_, $k) }, 2 .. *
}</lang>
- Output:
2 3 5 7 11 13 17 19 23 29 4 6 9 10 14 15 21 22 25 26 8 12 18 20 27 28 30 42 44 45 16 24 36 40 54 56 60 81 84 88 32 48 72 80 108 112 120 162 168 176
Here is a solution with identical output based on the factors routine from Count_in_factors#Perl_6 (to be included manually until we decide where in the distribution to put it). <lang perl6>constant factory = 0..* Z=> (0, 0, map { +factors($_) }, 2..*);
sub almost($n) { map *.key, grep *.value == $n, factory }
say almost($_)[^10] for 1..5;</lang>
Phix
<lang Phix> -- 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()
</lang>
- 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 <lang Phix>
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()
</lang>
- 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
<lang PicoLisp>(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)</lang>
Potion
<lang potion># 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.</lang>
C and Potion take 0.006s, Perl5 0.028s
Prolog
<lang prolog>% 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).
</lang>That's it. The rest is machinery. For portability, a compatibility section is included below. <lang Prolog>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).
</lang> <lang Prolog>%%%%% 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.
</lang>
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
This imports Prime decomposition#Python <lang 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))))</lang>
- 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]
Racket
<lang racket>#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))</lang>
- 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
The method used is to count the number of factors in the number to determine the K-primality.
The first two k-almost primes for each K group are computed directly (rather than found). <lang rexx>/*REXX program computes & displays N numbers of the first K k─almost primes.*/ 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== | J==',' 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.*/
do j=fir+fir+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 /*÷ 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 i=11 by 6 while i<=z /*insure I isn't divisible by three. */
parse var i -1 _ /*obtain the right─most decimal digit. */
/* [↓] fast check for divisible by 5. */
if _\==5 then do; do f=f+1 while z//i==0; z=z%i; end; f=f-1; end /*÷ by I*/
if _==3 then iterate /*fast check for X divisible by five.*/
x=i+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. */
return max(f,1) /*if prime (f==0), then return unity.*/</lang> 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
Ruby
<lang ruby>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(", ")}</lang>
- 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
<lang ruby>require 'prime'
p ar = pr = Prime.take(10) 4.times{p ar = ar.product(pr).map{|(a,b)| a*b}.uniq.sort.take(10)}</lang>
- 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
<lang rust>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<_>>());
}
}</lang>
- 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
<lang Scala>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 " " }]" )
}</lang>
- 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]
Sidef
<lang ruby>func is_k_almost_prime(n, k) {
for (var (p, f) = (2, 0); (f < k) && (p*p <= n); ++p) {
(n /= p; ++f) while (n %% p);
}
f + (n > 1) == k
}
5.times { |k|
var x = 10
say gather {
Math.inf.times { |i|
if (is_k_almost_prime(i, k)) {
take(i); (--x).is_zero && break;
}
}
}
}</lang>
- 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
<lang tcl>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]"
}</lang>
- 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
<lang>Local(3)
For c@ = 1 To 5
Print "k = ";c@;": ";
b@=0
For a@ = 2 Step 1 While b@ < 10
If FUNC(_kprime (a@,c@)) Then
b@ = b@ + 1
Print " ";a@;
EndIf
Next
Next
End
_kprime Param(2)
Local(2)
d@ = 0
For c@ = 2 Step 1 While (d@ < b@) * ((c@ * c@) < (a@ + 1))
Do While (a@ % c@) = 0
a@ = a@ / c@
d@ = d@ + 1
Loop
Next
Return (b@ = (d@ + (a@ > 1)))</lang>
- 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
Repurposed the VBScript code for the Prime Decomposition task. <lang vb> 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 </lang>
- 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
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). <lang zkl>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();
}</lang>
- 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)
- Programming Tasks
- Prime Numbers
- Ada
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