# Multiplicatively perfect numbers

(Redirected from Special numbers)
Multiplicatively perfect numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

If the product of the divisors of an integer n (including n itself) is equal to n^2, then n is a multiplicatively perfect number. Alternatively: the product of the proper divisors of n (i.e. excluding n) is equal to n.

Note that the integer '1' qualifies under the first definition (1 = 1 x 1), but not under the second since '1' has no proper divisors. Given this ambiguity, it is optional whether '1' is included or not in solutions to this task.

Find and show on this page the multiplicatively perfect numbers below 500.

Stretch

Find and show the number of multiplicatively perfect numbers under 500, 5,000, 50,000 and 500,000 and for each of these limits deduce (avoid counting separately) and show the number of semi-primes (numbers which are the product of exactly two primes) under that limit.

## ALGOL 68

Uses sieves, counts 1 as a multiplicatively perfect number.
As with the Phix and probably other samples, uses the fact that the multiplicatively perfect numbers (other than 1) must have three proper divisors - see OIES A007422.

```BEGIN # find multiplicatively perfect numbers - numbers whose proper         #
# divisor product is the number itself ( includes 1 )                  #
PR read "primes.incl.a68" PR          # include prime utilties           #
INT max number = 500 000;             # largest number we wlil consider  #
[]BOOL prime = PRIMESIEVE max number; # sieve the primes to max number   #
[ 1 : max number ]INT pdc;            # table of proper divisor counts   #
[ 1 : max number ]INT pfc;            # table of prime factor counts     #
# count the proper divisors and prime factors                            #
FOR n TO UPB pdc DO pdc[ n ] := 1; pfc[ n ] := 0 OD;
FOR i FROM 2 TO UPB pdc DO
BOOL is prime = prime[ i ];
INT m := 1;                       # j will be the mth multiple of i  #
FOR j FROM i + i BY i TO UPB pdc DO
pdc[ j ] +:= 1;
IF prime[ m +:= 1 ] THEN      # j is a prime multiple of i       #
pfc[ j ] +:= 1;
IF i = m THEN              # j is i squared                   #
pfc[ j ] +:= 1
FI
ELSE                          # j is not a prime multiple of i   #
pfc[ j ] +:= pfc[ m ]      # add the prime factor count of m  #
FI
OD
OD;
# find the mutiplicatively perfect numbers                               #
INT mp count     :=   0;
INT sp count     :=   0;
INT next to show := 500;
FOR n TO UPB pdc DO
IF n = 1 OR pdc[ n ] = 3 THEN
# n is 1 or has 3 proper divisors so is multiplicatively perfect #
# number - see OEIS A007422                                      #
mp count +:= 1;
IF n < 500 THEN
print( ( " ", whole( n, -3 ) ) );
IF mp count MOD 10 = 0 THEN print( ( newline ) ) FI
FI
FI;
IF pfc[ n ] = 2 THEN
# 2 prime factors, n is semi-prime                               #
sp count +:= 1
FI;
IF n = next to show THEN
print( ( "Up to ", whole( next to show, -8 )
, " there are ", whole( mp count, -6 )
, " multiplicatively perfect numbers and ", whole( sp count, -6 )
, " semi-primes", newline
)
);
next to show *:= 10
FI
OD
END```
Output:
```   1   6   8  10  14  15  21  22  26  27
33  34  35  38  39  46  51  55  57  58
62  65  69  74  77  82  85  86  87  91
93  94  95 106 111 115 118 119 122 123
125 129 133 134 141 142 143 145 146 155
158 159 161 166 177 178 183 185 187 194
201 202 203 205 206 209 213 214 215 217
218 219 221 226 235 237 247 249 253 254
259 262 265 267 274 278 287 291 295 298
299 301 302 303 305 309 314 319 321 323
326 327 329 334 335 339 341 343 346 355
358 362 365 371 377 381 382 386 391 393
394 395 398 403 407 411 413 415 417 422
427 437 445 446 447 451 453 454 458 466
469 471 473 478 481 482 485 489 493 497
Up to      500 there are    150 multiplicatively perfect numbers and    153 semi-primes
Up to     5000 there are   1354 multiplicatively perfect numbers and   1365 semi-primes
Up to    50000 there are  12074 multiplicatively perfect numbers and  12110 semi-primes
Up to   500000 there are 108223 multiplicatively perfect numbers and 108326 semi-primes
```

## BASIC

### BASIC256

Translation of: FreeBASIC
```arraybase 1
limit = 500
dim Divisors(1)
c = 0

print "Special numbers under "; limit; ":"

for n = 1 to limit
pro = 1
for m = 2 to ceil(n / 2)
if n mod m = 0 then
pro *= m
c += 1
redim Divisors(c)
Divisors[c] = m
end if
next m
ub = Divisors[?]
if n = pro and ub > 2 then
print rjust(string(n), 3); " = "; rjust(string(Divisors[ub-1]), 2); " x "; rjust(string(Divisors[ub]), 3)
end if
next n```
Output:
`Same as FreeBASIC entry.`

### True BASIC

Translation of: FreeBASIC
```LET limit = 50
DIM divisors(0)
LET c = 1

PRINT "Special numbers under"; limit; ":"

FOR n = 1 TO limit
LET pro = 1
FOR m = 2 TO ceil(n/2)
IF remainder(n, m) = 0 THEN
LET pro = pro*m
MAT REDIM divisors(c)
LET divisors(c) = m
LET c = c+1
END IF
NEXT m
LET ub = UBOUND(divisors)
IF n = pro AND ub > 1 THEN PRINT  USING "### = ## x ###": n, divisors(ub-1), divisors(ub)
NEXT n
END
```
Output:
`Same as FreeBASIC entry.`

### PureBasic

Translation of: FreeBASIC
```Macro Ceil (_x_)
Round(_x_, #PB_Round_Up)
EndMacro

OpenConsole()
Define.i limit = 500
Dim Divisors.i(1)
Define.i n, pro, m, c = 0, ub

PrintN("Special numbers under" + Str(limit) + ":")

For n = 1 To limit
pro = 1
For m = 2 To Ceil(n / 2)
If Mod(n, m) = 0:
pro * m
ReDim Divisors(c) : Divisors(c) = m
c + 1
EndIf
Next m
ub = ArraySize(Divisors())
If n = pro And ub > 1:
PrintN(RSet(Str(n),3) + " = " + RSet(Str(Divisors(ub-1)),2) + " x " + RSet(Str(Divisors(ub)),3))
EndIf
Next n
Input()
CloseConsole()```
Output:
`Same as FreeBASIC entry.`

### Yabasic

Translation of: FreeBASIC
```limit = 500
dim Divisors(1)
c = 1

print "Special numbers under", limit, ":"

for n = 1 to limit
pro = 1
for m = 2 to ceil(n / 2)
if mod(n, m) = 0 then
pro = pro * m
dim Divisors(c) : Divisors(c) = m
c = c + 1
fi
next m
ub = arraysize(Divisors(), 1)
if n = pro and ub > 1 then
print n using "###", " = ", Divisors(ub-1) using "##", " x ", Divisors(ub) using "###"
fi
next n```
Output:
`Same as FreeBASIC entry.`

## C

This includes '1' as an MPN. Run time around 2.3 seconds.

```#include <stdio.h>
#include <stdbool.h>
#include <locale.h>

bool isPrime(int n) {
if (n < 2) return false;
if (n%2 == 0) return n == 2;
if (n%3 == 0) return n == 3;
int d = 5;
while (d*d <= n) {
if (n%d == 0) return false;
d += 2;
if (n%d == 0) return false;
d += 4;
}
return true;
}

void divisors(int n, int *divs, int *length) {
int i, j, k = 1, c = 0;
if (n%2) k = 2;
for (i = 1; i*i <= n; i += k) {
if (i == 1) continue; // exclude 1 and n
if (!(n%i)) {
divs[c++] = i;
if (c > 2) break; // not eligible if has > 2 divisors
j = n / i;
if (j != i) divs[c++] = j;
}
}
*length = c;
}

int main() {
int i, d, j, k, t, length, prod;
int divs[4], count = 0, limit = 500, s = 3, c = 3, squares = 1, cubes = 1;
printf("Multiplicatively perfect numbers under %d:\n", limit);
setlocale(LC_NUMERIC, "");
for (i = 1; ; ++i) {
if (i != 1) {
divisors(i, divs, &length);
} else {
divs[1] = divs[0] = 1;
length = 2;
}
if (length == 2 && divs[0] * divs[1] == i) {
++count;
if (i < 500) {
printf("%3d  ", i);
if (!(count%10)) printf("\n");
}
}
if (i == 499) printf("\n");
if (i >= limit - 1) {
for (j = s; j * j < limit; j += 2) if (isPrime(j)) ++squares;
for (k = c; k * k * k < limit; k +=2 ) if (isPrime(k)) ++cubes;
t = count + squares - cubes - 1;
printf("Counts under %'9d: MPNs = %'7d  Semi-primes = %'7d\n", limit, count, t);
if (limit == 5000000) break;
s = j;
c = k;
limit *= 10;
}
}
return 0;
}
```
Output:
```Multiplicatively perfect numbers under 500:
1    6    8   10   14   15   21   22   26   27
33   34   35   38   39   46   51   55   57   58
62   65   69   74   77   82   85   86   87   91
93   94   95  106  111  115  118  119  122  123
125  129  133  134  141  142  143  145  146  155
158  159  161  166  177  178  183  185  187  194
201  202  203  205  206  209  213  214  215  217
218  219  221  226  235  237  247  249  253  254
259  262  265  267  274  278  287  291  295  298
299  301  302  303  305  309  314  319  321  323
326  327  329  334  335  339  341  343  346  355
358  362  365  371  377  381  382  386  391  393
394  395  398  403  407  411  413  415  417  422
427  437  445  446  447  451  453  454  458  466
469  471  473  478  481  482  485  489  493  497

Counts under       500: MPNs =     150  Semi-primes =     153
Counts under     5,000: MPNs =   1,354  Semi-primes =   1,365
Counts under    50,000: MPNs =  12,074  Semi-primes =  12,110
Counts under   500,000: MPNs = 108,223  Semi-primes = 108,326
Counts under 5,000,000: MPNs = 978,983  Semi-primes = 979,274
```

## C++

```#include <iostream>
#include <vector>
#include <numeric>

std::vector<int> divisors( int n ) {
std::vector<int> divisors ;
for ( int i = 1 ; i < n + 1 ; i++ ) {
if ( n % i == 0 )
divisors.push_back( i ) ;
}
return divisors ;
}

int main( ) {
std::vector<int> multi_perfect ;
for ( int i = 1 ; i < 501 ; i++ ) {
std::vector<int> divis { divisors( i ) } ;
if ( std::accumulate( divis.begin( ) , divis.end( ) , 1 ,
std::multiplies<int>() ) == (i * i ) )
multi_perfect.push_back( i ) ;
}
std::cout << '(' ;
int count = 1 ;
for ( int i : multi_perfect ) {
std::cout << i << ' ' ;
if ( count % 15 == 0 )
std::cout << '\n' ;
count++ ;
}
std::cout << ")\n" ;
return 0 ;
}
```
Output:
```(1 6 8 10 14 15 21 22 26 27 33 34 35 38 39
46 51 55 57 58 62 65 69 74 77 82 85 86 87 91
93 94 95 106 111 115 118 119 122 123 125 129 133 134 141
142 143 145 146 155 158 159 161 166 177 178 183 185 187 194
201 202 203 205 206 209 213 214 215 217 218 219 221 226 235
237 247 249 253 254 259 262 265 267 274 278 287 291 295 298
299 301 302 303 305 309 314 319 321 323 326 327 329 334 335
339 341 343 346 355 358 362 365 371 377 381 382 386 391 393
394 395 398 403 407 411 413 415 417 422 427 437 445 446 447
451 453 454 458 466 469 471 473 478 481 482 485 489 493 497
)
```

## FreeBASIC

```#define ceil(x) (-((-x*2.0-0.5) Shr 1))
Dim As Integer limit = 500
Dim As Integer n, pro, Divisors(), m, c = 0, ub
Print "Special numbers under"; limit; ":"

For n = 1 To limit
pro = 1
For m = 2 To ceil(n / 2)
If n Mod m = 0 Then
pro *= m
Redim Preserve Divisors(c) : Divisors(c) = m
c += 1
End If
Next m
ub = Ubound(Divisors)
If n = pro And ub > 1 Then
Print Using "### = ## x ###"; n; Divisors(ub-1); Divisors(ub)
End If
Next n

Sleep```
Output:
`Similar to Ring entry.`

## Go

Translation of: C
Library: Go-rcu

Run time around 9.2 seconds.

```package main

import (
"fmt"
"rcu"
)

// library method customized for this task
func Divisors(n int) []int {
var divisors []int
i := 1
k := 1
if n%2 == 1 {
k = 2
}
for ; i*i <= n; i += k {
if i > 1 && n%i == 0 { // exclude 1 and n
divisors = append(divisors, i)
if len(divisors) > 2 { // not eligible if has > 2 divisors
break
}
j := n / i
if j != i {
divisors = append(divisors, j)
}
}
}
return divisors
}

func main() {
count := 0
limit := 500
s := 3
c := 3
squares := 1
cubes := 1
fmt.Printf("Multiplicatively perfect numbers under %d:\n", limit)
var divs []int
for i := 0; ; i++ {
if i != 1 {
divs = Divisors(i)
} else {
divs = []int{1, 1}
}
if len(divs) == 2 && divs[0]*divs[1] == i {
count++
if i < 500 {
fmt.Printf("%3d  ", i)
if count%10 == 0 {
fmt.Println()
}
}
}
if i == 499 {
fmt.Println()
}
if i >= limit-1 {
var j, k int
for j = s; j*j < limit; j += 2 {
if rcu.IsPrime(j) {
squares++
}
}
for k = c; k*k*k < limit; k += 2 {
if rcu.IsPrime(k) {
cubes++
}
}
t := count + squares - cubes - 1
slimit := rcu.Commatize(limit)
scount := rcu.Commatize(count)
st := rcu.Commatize(t)
fmt.Printf("Counts under %9s: MPNs = %7s  Semi-primes = %7s\n", slimit, scount, st)
if limit == 5000000 {
break
}
s, c = j, k
limit *= 10
}
}
}
```
Output:
```Same as C example.
```

```divisors :: Int -> [Int]
divisors n = [d | d <- [1..n] , mod n d == 0 ]

isMultiplicativelyPerfect :: Int -> Bool
isMultiplicativelyPerfect n = (product \$ divisors n) == n ^ 2

solution :: [Int]
solution = filter isMultiplicativelyPerfect [1..500]
```
Output:
```[1,6,8,10,14,15,21,22,26,27,33,34,35,38,39,46,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94,95,106,111,115,118,119,122,123,125,129,133,134,141,142,143,145,146,155,158,159,161,166,177,178,183,185,187,194,201,202,203,205,206,209,213,214,215,217,218,219,221,226,235,237,247,249,253,254,259,262,265,267,274,278,287,291,295,298,299,301,302,303,305,309,314,319,321,323,326,327,329,334,335,339,341,343,346,355,358,362,365,371,377,381,382,386,391,393,394,395,398,403,407,411,413,415,417,422,427,437,445,446,447,451,453,454,458,466,469,471,473,478,481,482,485,489,493,497]
```

## J

Implementation:

```factors=: [: /:~@, */&>@{@((^ i.@>:)&.>/)@q:~&__
isMPerfect=: *: = */@(factors ::_:)"0
```

```   #(#~ isMPerfect) i.500
150
10 15\$(#~ isMPerfect) i.500
1   6   8  10  14  15  21  22  26  27  33  34  35  38  39
46  51  55  57  58  62  65  69  74  77  82  85  86  87  91
93  94  95 106 111 115 118 119 122 123 125 129 133 134 141
142 143 145 146 155 158 159 161 166 177 178 183 185 187 194
201 202 203 205 206 209 213 214 215 217 218 219 221 226 235
237 247 249 253 254 259 262 265 267 274 278 287 291 295 298
299 301 302 303 305 309 314 319 321 323 326 327 329 334 335
339 341 343 346 355 358 362 365 371 377 381 382 386 391 393
394 395 398 403 407 411 413 415 417 422 427 437 445 446 447
451 453 454 458 466 469 471 473 478 481 482 485 489 493 497
```

For the stretch goal, we need to determine the number of semi-primes, given the number of multiplicatively perfect numbers less than N:

```adjSemiPrime=: + _1 + %: -&(p:inv) 3&%:
```

Thus (first number in following results is count of multiplicatively perfect numbers, second is count of semiprimes):

```   {{ (, adjSemiPrime&y) +/isMPerfect i.y}} 500
150 153
{{ (, adjSemiPrime&y) +/isMPerfect i.y}} 5000
1354 1365
{{ (, adjSemiPrime&y) +/isMPerfect i.y}} 50000
12074 12110
{{ (, adjSemiPrime&y) +/isMPerfect i.y}} 500000
108223 108326
```

## Julia

```using Printf
using Primes

""" Find and count multiplicatively perfect integers up to thresholds """
function testmultiplicativelyperfects(thresholds = [500, 5000, 50_000, 500_000])
mpcount, scount = 0, 0
println("Multiplicatively perfect numbers under \$(thresholds[begin]):")
for n in 1:thresholds[end]
f = factor(n).pe
flen = length(f)
if flen == 2 && f[1][2] == 1 == f[2][2] || flen == 1 && f[1][2] == 3
mpcount += 1
if n < thresholds[begin]
@printf("%3d * %3d = %3d   ", f[1][1], n ÷ f[1][1], n)
mpcount % 5 == 0 && println()
end
end
if n in thresholds
scount = mpcount - cbsum + sqsum
@printf("\nCounts under %d: MPNs = %7d  Semi-primes = %7d\n", n, mpcount, scount)
end
end
end

testmultiplicativelyperfects()
```
Output:
```
Multiplicatively perfect numbers under 500:
2 *   3 =   6     2 *   4 =   8     2 *   5 =  10     2 *   7 =  14     3 *   5 =  15
3 *   7 =  21     2 *  11 =  22     2 *  13 =  26     3 *   9 =  27     3 *  11 =  33
2 *  17 =  34     5 *   7 =  35     2 *  19 =  38     3 *  13 =  39     2 *  23 =  46
3 *  17 =  51     5 *  11 =  55     3 *  19 =  57     2 *  29 =  58     2 *  31 =  62
5 *  13 =  65     3 *  23 =  69     2 *  37 =  74     7 *  11 =  77     2 *  41 =  82
5 *  17 =  85     2 *  43 =  86     3 *  29 =  87     7 *  13 =  91     3 *  31 =  93
2 *  47 =  94     5 *  19 =  95     2 *  53 = 106     3 *  37 = 111     5 *  23 = 115
2 *  59 = 118     7 *  17 = 119     2 *  61 = 122     3 *  41 = 123     5 *  25 = 125
3 *  43 = 129     7 *  19 = 133     2 *  67 = 134     3 *  47 = 141     2 *  71 = 142
11 *  13 = 143     5 *  29 = 145     2 *  73 = 146     5 *  31 = 155     2 *  79 = 158
3 *  53 = 159     7 *  23 = 161     2 *  83 = 166     3 *  59 = 177     2 *  89 = 178
3 *  61 = 183     5 *  37 = 185    11 *  17 = 187     2 *  97 = 194     3 *  67 = 201
2 * 101 = 202     7 *  29 = 203     5 *  41 = 205     2 * 103 = 206    11 *  19 = 209
3 *  71 = 213     2 * 107 = 214     5 *  43 = 215     7 *  31 = 217     2 * 109 = 218
3 *  73 = 219    13 *  17 = 221     2 * 113 = 226     5 *  47 = 235     3 *  79 = 237
13 *  19 = 247     3 *  83 = 249    11 *  23 = 253     2 * 127 = 254     7 *  37 = 259
2 * 131 = 262     5 *  53 = 265     3 *  89 = 267     2 * 137 = 274     2 * 139 = 278
7 *  41 = 287     3 *  97 = 291     5 *  59 = 295     2 * 149 = 298    13 *  23 = 299
7 *  43 = 301     2 * 151 = 302     3 * 101 = 303     5 *  61 = 305     3 * 103 = 309
2 * 157 = 314    11 *  29 = 319     3 * 107 = 321    17 *  19 = 323     2 * 163 = 326
3 * 109 = 327     7 *  47 = 329     2 * 167 = 334     5 *  67 = 335     3 * 113 = 339
11 *  31 = 341     7 *  49 = 343     2 * 173 = 346     5 *  71 = 355     2 * 179 = 358
2 * 181 = 362     5 *  73 = 365     7 *  53 = 371    13 *  29 = 377     3 * 127 = 381
2 * 191 = 382     2 * 193 = 386    17 *  23 = 391     3 * 131 = 393     2 * 197 = 394
5 *  79 = 395     2 * 199 = 398    13 *  31 = 403    11 *  37 = 407     3 * 137 = 411
7 *  59 = 413     5 *  83 = 415     3 * 139 = 417     2 * 211 = 422     7 *  61 = 427
19 *  23 = 437     5 *  89 = 445     2 * 223 = 446     3 * 149 = 447    11 *  41 = 451
3 * 151 = 453     2 * 227 = 454     2 * 229 = 458     2 * 233 = 466     7 *  67 = 469
3 * 157 = 471    11 *  43 = 473     2 * 239 = 478    13 *  37 = 481     2 * 241 = 482
5 *  97 = 485     3 * 163 = 489    17 *  29 = 493     7 *  71 = 497
Counts under 500: MPNs =     149  Semi-primes =     153

Counts under 5000: MPNs =    1353  Semi-primes =    1365

Counts under 50000: MPNs =   12073  Semi-primes =   12110

Counts under 500000: MPNs =  108222  Semi-primes =  108326
```

## Nim

In our solution, 1 is not considered to be a multiplicatively perfect number.

Using 32 bits integers rather than 64 bits integers improves considerably the performance. With a limit set to 5_000_000 and 64 bits integers, the program runs in around 10 seconds. This time drops to 3 seconds if 32 bits integers are used.

```import std/strformat

func isMPN(n: int32): bool =
## Return true if "n" is a multiplicatively perfect number.
## We consider than 1 is not an MPN.
var first, second = 0i32   # First and second proper divisors.
let delta = 1 + (n and 1)
var d = delta + 1
while d * d <= n:
if n mod d == 0:
if second != 0: return false  # More than two proper divisors.
first = d
let q = n div d
if q != d: second = q
inc d, delta
result = first * second == n

var count = 0
for n in 1i32..499i32:
if n.isMPN:
inc count
stdout.write &"{n:3}"
stdout.write if count mod 10 == 0: '\n' else: ' '
echo '\n'

func isPrime(n: int32): bool =
## Return true if "n" is prime.
if n < 2: return false
if (n and 1) == 0: return n == 2
if n mod 3 == 0: return n == 3
var k = 5
var delta = 2
while k * k <= n:
if n mod k == 0: return false
inc k, delta
delta = 6 - delta
result = true

var mpnCount = 0
var limit = 500i32
var ns, nc = 3i32
var squares, cubes = 1i32
var n = 1i32
while true:
inc n
if n == limit:
while ns * ns < limit:
if ns.isPrime: inc squares
inc ns, 2
while nc * nc * nc < limit:
if nc.isPrime: inc cubes
inc nc, 2
echo &"Under {limit} there are {mpnCount} MPNs and {mpnCount - cubes + squares} semi-primes."
if limit == 500_000: break
limit *= 10
if n.isMPN: inc mpnCount
```

{{out]]

```  6   8  10  14  15  21  22  26  27  33
34  35  38  39  46  51  55  57  58  62
65  69  74  77  82  85  86  87  91  93
94  95 106 111 115 118 119 122 123 125
129 133 134 141 142 143 145 146 155 158
159 161 166 177 178 183 185 187 194 201
202 203 205 206 209 213 214 215 217 218
219 221 226 235 237 247 249 253 254 259
262 265 267 274 278 287 291 295 298 299
301 302 303 305 309 314 319 321 323 326
327 329 334 335 339 341 343 346 355 358
362 365 371 377 381 382 386 391 393 394
395 398 403 407 411 413 415 417 422 427
437 445 446 447 451 453 454 458 466 469
471 473 478 481 482 485 489 493 497

Under 500 there are 149 MPNs and 153 semi-primes.
Under 5000 there are 1353 MPNs and 1365 semi-primes.
Under 50000 there are 12073 MPNs and 12110 semi-primes.
Under 500000 there are 108222 MPNs and 108326 semi-primes.
```

## Perl

Translation of: Raku
Library: ntheory
```use v5.36;
use enum    <false true>;
use ntheory <is_prime nth_prime is_semiprime gcd>;

sub comma      { reverse ((reverse shift) =~ s/.{3}\K/,/gr) =~ s/^,//r }
sub table (@V) { my \$t = 10 * (my \$w = 5); ( sprintf( ('%'.\$w.'d')x@V, @V) ) =~ s/.{1,\$t}\K/\n/gr }

sub find_factor (\$n, \$constant = 1) {   # NB: required that n > 1
my(\$x, \$rho, \$factor) = (2, 1, 1);
while (\$factor == 1) {
\$rho *= 2;
my \$fixed = \$x;
for (0..\$rho) {
\$x = ( \$x * \$x + \$constant ) % \$n;
\$factor = gcd((\$x-\$fixed), \$n);
last if 1 < \$factor;
}
}
\$factor = find_factor(\$n, \$constant+1) if \$n == \$factor;
\$factor
}

# Call with range 1..\$limit
sub is_mpn(\$n) {
state \$cube = 1; \$cube = 1 if \$n == 1; # set and reset
\$n == 1 ? return true : is_prime(\$n) ? return false : ();
++\$cube, return true if \$n == nth_prime(\$cube)**3;
my \$factor = find_factor(\$n);
my \$div    = int \$n/\$factor;
return true if is_prime \$factor and is_prime \$div and \$div != \$factor;
false
}

say "Multiplicatively perfect numbers less than 500:\n" . table grep is_mpn(\$_), 1..499;

say 'There are:';
for my \$limit (5e2, 5e3, 5e4, 5e5, 5e6) {
my(\$m,\$s) = (0,0);
is_mpn       \$_ and \$m++ for 1..\$limit-1;
is_semiprime \$_ and \$s++ for 1..\$limit-1;
printf "%8s MPNs less than %8s, %8s semiprimes\n", comma(\$m), \$limit, comma \$s
}
```
Output:
```Multiplicatively perfect numbers less than 500:
1    6    8   10   14   15   21   22   26   27
33   34   35   38   39   46   51   55   57   58
62   65   69   74   77   82   85   86   87   91
93   94   95  106  111  115  118  119  122  123
125  129  133  134  141  142  143  145  146  155
158  159  161  166  177  178  183  185  187  194
201  202  203  205  206  209  213  214  215  217
218  219  221  226  235  237  247  249  253  254
259  262  265  267  274  278  287  291  295  298
299  301  302  303  305  309  314  319  321  323
326  327  329  334  335  339  341  343  346  355
358  362  365  371  377  381  382  386  391  393
394  395  398  403  407  411  413  415  417  422
427  437  445  446  447  451  453  454  458  466
469  471  473  478  481  482  485  489  493  497

There are:
150 MPNs less than      500,      153 semiprimes
1,354 MPNs less than     5000,    1,365 semiprimes
12,074 MPNs less than    50000,   12,110 semiprimes
108,223 MPNs less than   500000,  108,326 semiprimes
978,983 MPNs less than  5000000,  979,274 semi primes
```

## Phix

Includes 1, to exclude just start with multiplicatively_perfect_numbers = 0 and r = {}.

```with javascript_semantics
integer multiplicatively_perfect_numbers = 1,
semiprime_numbers = 0,
five_e_n = 5e2
sequence r = {1}
for n=1 to 5e6 do
sequence pn = vslice(prime_powers(n),2)
multiplicatively_perfect_numbers += 1
if n<=500 then r &= n end if
end if
if sum(pn)=2 then
semiprime_numbers += 1
end if
if n=five_e_n then
if n=5e2 then
printf(1,"%d multiplicatively perfect numbers under 500: %s\n",
{length(r),join(shorten(r,"",5,"%d"),",")})
end if
printf(1,"Counts under %,9d: MPNs = %,7d  Semi-primes = %,7d\n",
{five_e_n,multiplicatively_perfect_numbers,semiprime_numbers})
five_e_n *= 10
end if
end for
```
Output:
```150 multiplicatively perfect numbers under 500: 1,6,8,10,14,...,482,485,489,493,497
Counts under       500: MPNs =     150  Semi-primes =     153
Counts under     5,000: MPNs =   1,354  Semi-primes =   1,365
Counts under    50,000: MPNs =  12,074  Semi-primes =  12,110
Counts under   500,000: MPNs = 108,223  Semi-primes = 108,326
Counts under 5,000,000: MPNs = 978,983  Semi-primes = 979,274
```

## PL/M

Works with: 8080 PL/M Compiler

... under CP/M (or an emulator)

```100H: /* FIND MUKTIPLICATIVELY PERFECT NUMBERS - NUMBERS WHOSE PROPER        */
/* DIVISOR PRODUCT IS THE NUMBER ITSELF                                */
/* NOTE THIS IS EQUIVALENT TO FINDING NUMBERS WITH 3 PROPER DIVISORS   */
/* SEE OEIS A007422                                                    */

/* CP/M BDOS SYSTEM CALL                                                  */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
/* I/O ROUTINES                                                           */
PR\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C );  END;
PR\$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S );  END;
PR\$NL:     PROCEDURE;   CALL PR\$CHAR( 0DH ); CALL PR\$CHAR( 0AH ); END;
PR\$NUMBER: PROCEDURE( N ); /* PRINTS A NUMBER IN THE MINIMUN FIELD WIDTH  */
DECLARE V ADDRESS, N\$STR ( 6 )BYTE, W BYTE;
V = N;
W = LAST( N\$STR );
N\$STR( W ) = '\$';
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL PR\$STRING( .N\$STR( W ) );
END PR\$NUMBER;

/* FIND THE NUMBERS UP TO 500                                             */

DECLARE PDC      ( 501 )ADDRESS; /* TABLE OF PROPER DIVISOR COUNTS        */
DECLARE ( I, J, COUNT ) ADDRESS;

/* COUNT THE PROPER DIVISORS OF NUMBERS TO 500                            */
DO I = 0 TO LAST( PDC ); PDC( I ) = 1; END;
DO I = 2 TO LAST( PDC );
DO J = I + I TO LAST( PDC ) BY I;
PDC( J ) = PDC( J ) + 1;
END;
END;
PDC( 1 ) = 3;       /* PRETEND 1 HAS 3 PROPER DIVISORS SO IT IS INCLUDED  */

/* SHOW YHE MULTIPLICATIVELY PERFECT NUMBERS                              */
COUNT = 0;
DO I = 1 TO LAST( PDC );
IF PDC( I ) = 3 THEN DO;
CALL PR\$CHAR( ' ' );
IF I < 100 THEN DO;
IF I < 10 THEN CALL PR\$CHAR( ' ' );
CALL PR\$CHAR( ' ' );
END;
CALL PR\$NUMBER( I );
IF ( COUNT := COUNT + 1 ) MOD 10 = 0 THEN CALL PR\$NL;
END;
END;

EOF```
Output:
```   1   6   8  10  14  15  21  22  26  27
33  34  35  38  39  46  51  55  57  58
62  65  69  74  77  82  85  86  87  91
93  94  95 106 111 115 118 119 122 123
125 129 133 134 141 142 143 145 146 155
158 159 161 166 177 178 183 185 187 194
201 202 203 205 206 209 213 214 215 217
218 219 221 226 235 237 247 249 253 254
259 262 265 267 274 278 287 291 295 298
299 301 302 303 305 309 314 319 321 323
326 327 329 334 335 339 341 343 346 355
358 362 365 371 377 381 382 386 391 393
394 395 398 403 407 411 413 415 417 422
427 437 445 446 447 451 453 454 458 466
469 471 473 478 481 482 485 489 493 497
```

## Raku

```use List::Divvy;
use Lingua::EN::Numbers;

constant @primes = (^∞).grep: &is-prime;
constant @cubes  = @primes.map: *³;

state \$cube = 0;
sub is-mpn(Int \$n ) {
return False if \$n.is-prime;
if \$n == @cubes[\$cube] {
++\$cube;
return True
}
my \$factor = find-factor(\$n);
return True if (\$factor.is-prime && ( my \$div = \$n div \$factor ).is-prime && (\$div != \$factor));
False;
}

sub find-factor ( Int \$n, \$constant = 1 ) {
my \$x      = 2;
my \$rho    = 1;
my \$factor = 1;
while \$factor == 1 {
\$rho *= 2;
my \$fixed = \$x;
for ^\$rho {
\$x = ( \$x * \$x + \$constant ) % \$n;
\$factor = ( \$x - \$fixed ) gcd \$n;
last if 1 < \$factor;
}
}
\$factor = find-factor( \$n, \$constant + 1 ) if \$n == \$factor;
\$factor;
}

constant @mpn = lazy 1, |(2..*).grep: &is-mpn;

say 'Multiplicatively perfect numbers less than 500:';
put @mpn.&upto(500).batch(10)».fmt("%3d").join: "\n";

put "\nThere are:";
for 5e2, 5e3, 5e4, 5e5, 5e6 {
printf  "%8s MPNs less than %9s, %7s semiprimes.\n",
comma(my \$count = +@mpn.&upto(\$_)), .Int.&comma,
comma \$count + @primes.map(*²).&upto(\$_) - @cubes.&upto(\$_) - 1;
}
```
Output:
```Multiplicatively perfect numbers less than 500:
1   6   8  10  14  15  21  22  26  27
33  34  35  38  39  46  51  55  57  58
62  65  69  74  77  82  85  86  87  91
93  94  95 106 111 115 118 119 122 123
125 129 133 134 141 142 143 145 146 155
158 159 161 166 177 178 183 185 187 194
201 202 203 205 206 209 213 214 215 217
218 219 221 226 235 237 247 249 253 254
259 262 265 267 274 278 287 291 295 298
299 301 302 303 305 309 314 319 321 323
326 327 329 334 335 339 341 343 346 355
358 362 365 371 377 381 382 386 391 393
394 395 398 403 407 411 413 415 417 422
427 437 445 446 447 451 453 454 458 466
469 471 473 478 481 482 485 489 493 497

There are:
150 MPNs less than       500,     153 semiprimes.
1,354 MPNs less than     5,000,   1,365 semiprimes.
12,074 MPNs less than    50,000,  12,110 semiprimes.
108,223 MPNs less than   500,000, 108,326 semiprimes.
978,983 MPNs less than 5,000,000, 979,274 semiprimes.```

## Ring

```see "working..." + nl
see "Special numbers under 500:" + nl
limit = 500
Divisors = []
for n = 1 to limit
pro = 1
Divisors = []
for m = 2 to ceil(n/2)
if n % m = 0
pro = pro * m
ok
next
str = ""
if n = pro and len(Divisors) > 1
for m = 1 to len(Divisors)
str = str + Divisors[m] + " * "
if m = len(Divisors)
str = left(str,len(str)-2)
ok
next
see "" + n + " = " + str + nl
ok
next
see "done..." + nl```
Output:
```working...
Special numbers under 500:
6  =   2 x   3
8  =   2 x   4
10  =   2 x   5
14  =   2 x   7
15  =   3 x   5
21  =   3 x   7
22  =   2 x  11
26  =   2 x  13
27  =   3 x   9
33  =   3 x  11
34  =   2 x  17
35  =   5 x   7
38  =   2 x  19
39  =   3 x  13
46  =   2 x  23
51  =   3 x  17
55  =   5 x  11
57  =   3 x  19
58  =   2 x  29
62  =   2 x  31
65  =   5 x  13
69  =   3 x  23
74  =   2 x  37
77  =   7 x  11
82  =   2 x  41
85  =   5 x  17
86  =   2 x  43
87  =   3 x  29
91  =   7 x  13
93  =   3 x  31
94  =   2 x  47
95  =   5 x  19
106  =   2 x  53
111  =   3 x  37
115  =   5 x  23
118  =   2 x  59
119  =   7 x  17
122  =   2 x  61
123  =   3 x  41
125  =   5 x  25
129  =   3 x  43
133  =   7 x  19
134  =   2 x  67
141  =   3 x  47
142  =   2 x  71
143  =  11 x  13
145  =   5 x  29
146  =   2 x  73
155  =   5 x  31
158  =   2 x  79
159  =   3 x  53
161  =   7 x  23
166  =   2 x  83
177  =   3 x  59
178  =   2 x  89
183  =   3 x  61
185  =   5 x  37
187  =  11 x  17
194  =   2 x  97
201  =   3 x  67
202  =   2 x 101
203  =   7 x  29
205  =   5 x  41
206  =   2 x 103
209  =  11 x  19
213  =   3 x  71
214  =   2 x 107
215  =   5 x  43
217  =   7 x  31
218  =   2 x 109
219  =   3 x  73
221  =  13 x  17
226  =   2 x 113
235  =   5 x  47
237  =   3 x  79
247  =  13 x  19
249  =   3 x  83
253  =  11 x  23
254  =   2 x 127
259  =   7 x  37
262  =   2 x 131
265  =   5 x  53
267  =   3 x  89
274  =   2 x 137
278  =   2 x 139
287  =   7 x  41
291  =   3 x  97
295  =   5 x  59
298  =   2 x 149
299  =  13 x  23
301  =   7 x  43
302  =   2 x 151
303  =   3 x 101
305  =   5 x  61
309  =   3 x 103
314  =   2 x 157
319  =  11 x  29
321  =   3 x 107
323  =  17 x  19
326  =   2 x 163
327  =   3 x 109
329  =   7 x  47
334  =   2 x 167
335  =   5 x  67
339  =   3 x 113
341  =  11 x  31
343  =   7 x  49
346  =   2 x 173
355  =   5 x  71
358  =   2 x 179
362  =   2 x 181
365  =   5 x  73
371  =   7 x  53
377  =  13 x  29
381  =   3 x 127
382  =   2 x 191
386  =   2 x 193
391  =  17 x  23
393  =   3 x 131
394  =   2 x 197
395  =   5 x  79
398  =   2 x 199
403  =  13 x  31
407  =  11 x  37
411  =   3 x 137
413  =   7 x  59
415  =   5 x  83
417  =   3 x 139
422  =   2 x 211
427  =   7 x  61
437  =  19 x  23
445  =   5 x  89
446  =   2 x 223
447  =   3 x 149
451  =  11 x  41
453  =   3 x 151
454  =   2 x 227
458  =   2 x 229
466  =   2 x 233
469  =   7 x  67
471  =   3 x 157
473  =  11 x  43
478  =   2 x 239
481  =  13 x  37
482  =   2 x 241
485  =   5 x  97
489  =   3 x 163
493  =  17 x  29
497  =   7 x  71
done...
```

## RPL

Works with: HP version 49
```≪ { 1 }
2 ROT FOR j
IF j SQ j DIVIS ΠLIST == THEN j + END
NEXT

≪ (1,0)
2 ROT FOR j
j DIVIS
IF CASE
DUP SIZE DUP 4 > OVER " < THEN NOT END
3 == THEN 1 END
DUP 2 GET OVER 3 GET GCD 1 == END
THEN SWAP (0,1) + SWAP END
IF ΠLIST j SQ == THEN 1 + END
NEXT
≫ 'STRETCH' STO
```
```500 TASK
500 STRETCH
```
Output:
```2: { 1 6 8 10 14 15 21 22 26 27 33 34 35 38 39 46 51 55 57 58 62 65 69 74 77 82 85 86 87 91 93 94 95 106 111 115 118 119 122 123 125 129 133 134 141 142 143 145 146 155 158 159 161 166 177 178 183 185 187 194 201 202 203 205 206 209 213 214 215 217 218 219 221 226 235 237 247 249 253 254 259 262 265 267 274 278 287 291 295 298 299 301 302 303 305 309 314 319 321 323 326 327 329 334 335 339 341 343 346 355 358 362 365 371 377 381 382 386 391 393 394 395 398 403 407 411 413 415 417 422 427 437 445 446 447 451 453 454 458 466 469 471 473 478 481 482 485 489 493 497 }
1: (150,153)
```

## Rust

```fn divisors( num : u128 ) -> Vec<u128> {
(1..= num).filter( | &d | num % d == 0 ).collect( )
}

fn main() {
println!("{:?}" , (1..= 500).filter( | &d | {
let divis : Vec<u128> = divisors( d ) ;
let prod : u128 = divis.iter( ).product( ) ;
prod == d.checked_pow( 2 ).unwrap( )
}).collect::<Vec<u128>>( ) ) ;
}
```
Output:
```[1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 221, 226, 235, 237, 247, 249, 253, 254, 259, 262, 265, 267, 274, 278, 287, 291, 295, 298, 299, 301, 302, 303, 305, 309, 314, 319, 321, 323, 326, 327, 329, 334, 335, 339, 341, 343, 346, 355, 358, 362, 365, 371, 377, 381, 382, 386, 391, 393, 394, 395, 398, 403, 407, 411, 413, 415, 417, 422, 427, 437, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 473, 478, 481, 482, 485, 489, 493, 497]
```

## Sidef

```func multiplicatively_perfect_numbers(N) {
[1] + semiprimes(N).grep{|n| isqrt(n**tau(n)) == n.sqr } + N.icbrt.primes.map { .cube } -> sort
}

say ("Terms <= 500: ", multiplicatively_perfect_numbers(500).join(' '), "\n")

for n in (500, 5_000, 50_000, 500_000) {
var M = multiplicatively_perfect_numbers(n)
say "There are #{M.len} MPNs and #{semiprime_count(n)} semiprimes <= #{n.commify}."
}
```
Output:
```Terms <= 500: 1 6 8 10 14 15 21 22 26 27 33 34 35 38 39 46 51 55 57 58 62 65 69 74 77 82 85 86 87 91 93 94 95 106 111 115 118 119 122 123 125 129 133 134 141 142 143 145 146 155 158 159 161 166 177 178 183 185 187 194 201 202 203 205 206 209 213 214 215 217 218 219 221 226 235 237 247 249 253 254 259 262 265 267 274 278 287 291 295 298 299 301 302 303 305 309 314 319 321 323 326 327 329 334 335 339 341 343 346 355 358 362 365 371 377 381 382 386 391 393 394 395 398 403 407 411 413 415 417 422 427 437 445 446 447 451 453 454 458 466 469 471 473 478 481 482 485 489 493 497

There are 150 MPNs and 153 semiprimes <= 500.
There are 1354 MPNs and 1365 semiprimes <= 5,000.
There are 12074 MPNs and 12110 semiprimes <= 50,000.
There are 108223 MPNs and 108326 semiprimes <= 500,000.
```

## Wren

Library: Wren-math
Library: Wren-fmt

This includes '1' as an MPN. Not very quick at around 112 seconds but reasonable for Wren.

```import "./math" for Int, Nums
import "./fmt" for Fmt

// library method customized for this task
var divisors = Fn.new { |n|
var divisors = []
var i = 1
var k = (n%2 == 0) ? 1 : 2
while (i <= n.sqrt) {
if (i > 1 && n%i == 0) {  // exclude 1 and n
if (divisors.count > 2) break // not eligible if has > 2 divisors
var j = (n/i).floor
}
i = i + k
}
return divisors
}

var limit = 500
var count = 0
var i = 1
System.print("Multiplicatively perfect numbers under %(limit):")
while (true) {
var pd = (i != 1) ? divisors.call(i) : [1, 1]
if (pd.count == 2 && pd[0] * pd[1] == i) {
count = count + 1
if (i < 500) {
var pds = Fmt.swrite("\$3d x \$3d", pd[0], pd[1])
Fmt.write("\$3d  = \$s   ", i, pds)
if (count % 5 == 0) System.print()
}
}
if (i == 499) System.print()
if (i >= limit - 1) {
var squares = Int.primeCount((limit - 1).sqrt.floor)
var cubes   = Int.primeCount((limit - 1).cbrt.floor)
var count2 = count + squares - cubes - 1
Fmt.print("Counts under \$,9d: MPNs = \$,7d  Semi-primes = \$,7d", limit, count, count2)
if (limit == 5000000) return
limit = limit * 10
}
i = i + 1
}
```
Output:
```Multiplicatively perfect numbers under 500:
1  =   1 x   1     6  =   2 x   3     8  =   2 x   4    10  =   2 x   5    14  =   2 x   7
15  =   3 x   5    21  =   3 x   7    22  =   2 x  11    26  =   2 x  13    27  =   3 x   9
33  =   3 x  11    34  =   2 x  17    35  =   5 x   7    38  =   2 x  19    39  =   3 x  13
46  =   2 x  23    51  =   3 x  17    55  =   5 x  11    57  =   3 x  19    58  =   2 x  29
62  =   2 x  31    65  =   5 x  13    69  =   3 x  23    74  =   2 x  37    77  =   7 x  11
82  =   2 x  41    85  =   5 x  17    86  =   2 x  43    87  =   3 x  29    91  =   7 x  13
93  =   3 x  31    94  =   2 x  47    95  =   5 x  19   106  =   2 x  53   111  =   3 x  37
115  =   5 x  23   118  =   2 x  59   119  =   7 x  17   122  =   2 x  61   123  =   3 x  41
125  =   5 x  25   129  =   3 x  43   133  =   7 x  19   134  =   2 x  67   141  =   3 x  47
142  =   2 x  71   143  =  11 x  13   145  =   5 x  29   146  =   2 x  73   155  =   5 x  31
158  =   2 x  79   159  =   3 x  53   161  =   7 x  23   166  =   2 x  83   177  =   3 x  59
178  =   2 x  89   183  =   3 x  61   185  =   5 x  37   187  =  11 x  17   194  =   2 x  97
201  =   3 x  67   202  =   2 x 101   203  =   7 x  29   205  =   5 x  41   206  =   2 x 103
209  =  11 x  19   213  =   3 x  71   214  =   2 x 107   215  =   5 x  43   217  =   7 x  31
218  =   2 x 109   219  =   3 x  73   221  =  13 x  17   226  =   2 x 113   235  =   5 x  47
237  =   3 x  79   247  =  13 x  19   249  =   3 x  83   253  =  11 x  23   254  =   2 x 127
259  =   7 x  37   262  =   2 x 131   265  =   5 x  53   267  =   3 x  89   274  =   2 x 137
278  =   2 x 139   287  =   7 x  41   291  =   3 x  97   295  =   5 x  59   298  =   2 x 149
299  =  13 x  23   301  =   7 x  43   302  =   2 x 151   303  =   3 x 101   305  =   5 x  61
309  =   3 x 103   314  =   2 x 157   319  =  11 x  29   321  =   3 x 107   323  =  17 x  19
326  =   2 x 163   327  =   3 x 109   329  =   7 x  47   334  =   2 x 167   335  =   5 x  67
339  =   3 x 113   341  =  11 x  31   343  =   7 x  49   346  =   2 x 173   355  =   5 x  71
358  =   2 x 179   362  =   2 x 181   365  =   5 x  73   371  =   7 x  53   377  =  13 x  29
381  =   3 x 127   382  =   2 x 191   386  =   2 x 193   391  =  17 x  23   393  =   3 x 131
394  =   2 x 197   395  =   5 x  79   398  =   2 x 199   403  =  13 x  31   407  =  11 x  37
411  =   3 x 137   413  =   7 x  59   415  =   5 x  83   417  =   3 x 139   422  =   2 x 211
427  =   7 x  61   437  =  19 x  23   445  =   5 x  89   446  =   2 x 223   447  =   3 x 149
451  =  11 x  41   453  =   3 x 151   454  =   2 x 227   458  =   2 x 229   466  =   2 x 233
469  =   7 x  67   471  =   3 x 157   473  =  11 x  43   478  =   2 x 239   481  =  13 x  37
482  =   2 x 241   485  =   5 x  97   489  =   3 x 163   493  =  17 x  29   497  =   7 x  71

Counts under       500: MPNs =     150  Semi-primes =     153
Counts under     5,000: MPNs =   1,354  Semi-primes =   1,365
Counts under    50,000: MPNs =  12,074  Semi-primes =  12,110
Counts under   500,000: MPNs = 108,223  Semi-primes = 108,326
Counts under 5,000,000: MPNs = 978,983  Semi-primes = 979,274
```

## XPL0

```func Special(N);
int  N, D, P;
[D:= 2;  P:= 1;
while D < N do
[if rem(N/D) = 0 then P:= P*D;
D:= D+1;
];
return P = N;
];

int N, C;
[C:= 0;
Format(4, 0);
for N:= 2 to 500-1 do
if Special(N) then
[RlOut(0, float(N));
C:= C+1;
if rem(C/20) = 0 then CrLf(0);
];
]```
Output:
```   6   8  10  14  15  21  22  26  27  33  34  35  38  39  46  51  55  57  58  62
65  69  74  77  82  85  86  87  91  93  94  95 106 111 115 118 119 122 123 125
129 133 134 141 142 143 145 146 155 158 159 161 166 177 178 183 185 187 194 201
202 203 205 206 209 213 214 215 217 218 219 221 226 235 237 247 249 253 254 259
262 265 267 274 278 287 291 295 298 299 301 302 303 305 309 314 319 321 323 326
327 329 334 335 339 341 343 346 355 358 362 365 371 377 381 382 386 391 393 394
395 398 403 407 411 413 415 417 422 427 437 445 446 447 451 453 454 458 466 469
471 473 478 481 482 485 489 493 497```