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# Practical numbers

Practical 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.

A Practical number P has some selection of its proper divisors, (other than itself), that can be selected to sum to every integer less than itself.

Compute all the proper divisors/factors of an input number X, then, using all selections from the factors compute all possible sums of factors and see if all numbers from 1 to X-1 can be created from it.

Write a function that given X returns a boolean value of whether X is a Practical number, (using the above method).

• Show how many Practical numbers there are in the range 1..333, inclusive.
• Show that the Practical numbers in the above range start and end in:
1, 2, 4, 6, 8, 12, 16, 18, 20, 24 ... 288, 294, 300, 304, 306, 308, 312, 320, 324, 330
Stretch Goal
• Show if 666 is a Practical number

## APL

Works with: Dyalog APL
`pract ← ∧/⍳∊(+⌿⊢×[1](2/⍨≢)⊤(⍳2*≢))∘(⍸0=⍳|⊢)`
Output:
`      ⍸pract¨⍳333  ⍝ Which numbers from 1 to 333 are practical?1 2 4 6 8 12 16 18 20 24 28 30 32 36 40 42 48 54 56 60 64 66 72 78 80 84 88 90 96 100 104 108 112 120 126 128 132 140 144      150 156 160 162 168 176 180 192 196 198 200 204 208 210 216 220 224 228 234 240 252 256 260 264 270 272 276 280 288      294 300 304 306 308 312 320 324 330      pract 666    ⍝ Is 666 practical?1`

## C#

`using System.Collections.Generic; using System.Linq; using static System.Console; class Program {     static bool soas(int n, IEnumerable<int> f) {        if (n <= 0) return false; if (f.Contains(n)) return true;        switch(n.CompareTo(f.Sum())) { case 1: return false; case 0: return true;            case -1: var rf = f.Reverse().ToList(); var d = n - rf[0]; rf.RemoveAt(0);                return soas(d, rf) || soas(n, rf); } return true; }     static bool ip(int n) { var f = Enumerable.Range(1, n >> 1).Where(d => n % d == 0).ToList();        return Enumerable.Range(1, n - 1).ToList().TrueForAll(i => soas(i, f));  }     static void Main() {        int c = 0, m = 333; for (int i = 1; i <= m; i += i == 1 ? 1 : 2)            if (ip(i) || i == 1) Write("{0,3} {1}", i, ++c % 10 == 0 ? "\n" : "");         Write("\nFound {0} practical numbers between 1 and {1} inclusive.\n", c, m);        do Write("\n{0,5} is a{1}practical number.",            m = m < 500 ? m << 1 : m * 10 + 6, ip(m) ? " " : "n im"); while (m < 1e4); } }`
Output:
```  1   2   4   6   8  12  16  18  20  24
28  30  32  36  40  42  48  54  56  60
64  66  72  78  80  84  88  90  96 100
104 108 112 120 126 128 132 140 144 150
156 160 162 168 176 180 192 196 198 200
204 208 210 216 220 224 228 234 240 252
256 260 264 270 272 276 280 288 294 300
304 306 308 312 320 324 330
Found 77 practical numbers between 1 and 333 inclusive.

666 is a practical number.
6666 is a practical number.
66666 is an impractical number.```

## C++

Translation of: Phix
`#include <algorithm>#include <iostream>#include <numeric>#include <sstream>#include <vector> // Returns true if any subset of [begin, end) sums to n.template <typename iterator>bool sum_of_any_subset(int n, iterator begin, iterator end) {    if (begin == end)        return false;    if (std::find(begin, end, n) != end)        return true;    int total = std::accumulate(begin, end, 0);    if (n == total)        return true;    if (n > total)        return false;    --end;    int d = n - *end;    return (d > 0 && sum_of_any_subset(d, begin, end)) ||           sum_of_any_subset(n, begin, end);} // Returns the proper divisors of n.std::vector<int> factors(int n) {    std::vector<int> f{1};    for (int i = 2; i * i <= n; ++i) {        if (n % i == 0) {            f.push_back(i);            if (i * i != n)                f.push_back(n / i);        }    }    std::sort(f.begin(), f.end());    return f;} bool is_practical(int n) {    std::vector<int> f = factors(n);    for (int i = 1; i < n; ++i) {        if (!sum_of_any_subset(i, f.begin(), f.end()))            return false;    }    return true;} std::string shorten(const std::vector<int>& v, size_t n) {    std::ostringstream out;    size_t size = v.size(), i = 0;    if (n > 0 && size > 0)        out << v[i++];    for (; i < n && i < size; ++i)        out << ", " << v[i];    if (size > i + n) {        out << ", ...";        i = size - n;    }    for (; i < size; ++i)        out << ", " << v[i];    return out.str();} int main() {    std::vector<int> practical;    for (int n = 1; n <= 333; ++n) {        if (is_practical(n))            practical.push_back(n);    }    std::cout << "Found " << practical.size() << " practical numbers:\n"              << shorten(practical, 10) << '\n';    for (int n : {666, 6666, 66666, 672, 720, 222222})        std::cout << n << " is " << (is_practical(n) ? "" : "not ")                  << "a practical number.\n";    return 0;}`
Output:
```Found 77 practical numbers:
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, ..., 288, 294, 300, 304, 306, 308, 312, 320, 324, 330
666 is a practical number.
6666 is a practical number.
66666 is not a practical number.
672 is a practical number.
720 is a practical number.
222222 is a practical number.
```

See Pascal.

## FreeBASIC

`sub make_divisors( n as uinteger, div() as uinteger )     'produces a list of an integer's proper divisors     for i as uinteger = n/2 to 1 step -1         if n mod i = 0 then             redim preserve div(1 to 1 + ubound(div))             div(ubound(div)) = i         end if     next iend sub function sum_divisors( n as uinteger, div() as uinteger ) as uinteger    'takes a list of divisors and an integer which, when interpreted    'as binary, selects which terms to sum    dim as uinteger sum = 0, term = 1    while n        if n mod 2 = 1 then sum += div(term)        term += 1        n\=2    wend    return sumend function function is_practical( n as uinteger ) as boolean    'determines if an integer is practical    if n = 1 then return true    if n mod 2 = 1 then return false    'there can be no odd practicals other than 1    if n < 5 then return true           '2 and 4 are practical, but small enough to be handled specially    dim as uinteger hits(1 to n-1), nt, i, sd    redim as uinteger div(0 to 0)    make_divisors( n, div() )    nt = ubound(div)    for i = 1 to 2^nt-1        sd = sum_divisors(i, div())        if sd<n then hits(sd)+=1    next i    for i = 1 to n-1        if hits(i) = 0 then return false    next i    return trueend function print 1;" ";  'treat 1 as a special case for n as uinteger = 2 to 666    if is_practical(n) then print n;" ";next n:print`

All practical numbers up to and including the stretch goal of DCLXVI.

Output:
```
1 2 4 6 8 12 16 18 20 24 28 30 32 36 40 42 48 54 56 60 64 66 72 78 80 84 88 90 96 100 104 108 112 120 126 128 132 140 144 150 156 160 162 168 176 180 192 196 198 200 204 208 210 216 220 224 228 234 240 252 256 260 264 270 272 276 280 288 294 300 304 306 308 312 320 324 330 336 340 342 348 352 360 364 368 378 380 384 390 392 396 400 408 414 416 420 432 440 448 450 456 460 462 464 468 476 480 486 496 500 504 510 512 520 522 528 532 540 544 546 552 558 560 570 576 580 588 594 600 608 612 616 620 624 630 640 644 648 660 666```

## Go

Translation of: Wren
Library: Go-rcu
`package main import (    "fmt"    "rcu") func powerset(set []int) [][]int {    if len(set) == 0 {        return [][]int{{}}    }    head := set[0]    tail := set[1:]    p1 := powerset(tail)    var p2 [][]int    for _, s := range powerset(tail) {        h := []int{head}        h = append(h, s...)        p2 = append(p2, h)    }    return append(p1, p2...)} func isPractical(n int) bool {    if n == 1 {        return true    }    divs := rcu.ProperDivisors(n)    subsets := powerset(divs)    found := make([]bool, n) // all false by default    count := 0    for _, subset := range subsets {        sum := rcu.SumInts(subset)        if sum > 0 && sum < n && !found[sum] {            found[sum] = true            count++            if count == n-1 {                return true            }        }    }    return false} func main() {    fmt.Println("In the range 1..333, there are:")    var practical []int    for i := 1; i <= 333; i++ {        if isPractical(i) {            practical = append(practical, i)        }    }    fmt.Println(" ", len(practical), "practical numbers")    fmt.Println("  The first ten are", practical[0:10])    fmt.Println("  The final ten are", practical[len(practical)-10:])    fmt.Println("\n666 is practical:", isPractical(666))}`
Output:
```In the range 1..333, there are:
77 practical numbers
The first ten are [1 2 4 6 8 12 16 18 20 24]
The final ten are [288 294 300 304 306 308 312 320 324 330]

666 is practical: true
```

## J

Borrowed from the Proper divisors#J page:

`factors=: [: /:[email protected], */&>@{@((^ [email protected]>:)&.>/)@q:~&__properDivisors=: factors -. ]`

Borrowed from the Power set#J page:

`ps=:  #~ 2 #:@[email protected]^ #`

Implementation:

`isPrac=: ('' -:&# i. -. 0,+/"1@(ps ::empty)@properDivisors)"0`

`   +/ isPrac 1+i.333    NB. count practical numbers77   (#~ isPrac) 1+i.333  NB. list them1 2 4 6 8 12 16 18 20 24 28 30 32 36 40 42 48 54 56 60 64 66 72 78 80 84 88 90 96 100 104 108 112 120 126 128 132 140 144 150 156 160 162 168 176 180 192 196 198 200 204 208 210 216 220 224 228 234 240 252 256 260 264 270 272 276 280 288 294 300 304 306 30...   isPrac 666           NB. test1`

## Java

Translation of: Phix
`import java.util.*; public class PracticalNumbers {    public static void main(String[] args) {        final int from = 1;        final int to = 333;        List<Integer> practical = new ArrayList<>();        for (int i = from; i <= to; ++i) {            if (isPractical(i))                practical.add(i);        }        System.out.printf("Found %d practical numbers between %d and %d:\n%s\n",                practical.size(), from, to, shorten(practical, 10));         printPractical(666);        printPractical(6666);        printPractical(66666);        printPractical(672);        printPractical(720);        printPractical(222222);    }     private static void printPractical(int n) {        if (isPractical(n))            System.out.printf("%d is a practical number.\n", n);        else            System.out.printf("%d is not a practical number.\n", n);    }     private static boolean isPractical(int n) {        int[] divisors = properDivisors(n);        for (int i = 1; i < n; ++i) {            if (!sumOfAnySubset(i, divisors, divisors.length))                return false;        }        return true;    }     private static boolean sumOfAnySubset(int n, int[] f, int len) {        if (len == 0)            return false;        int total = 0;        for (int i = 0; i < len; ++i) {            if (n == f[i])                return true;            total += f[i];        }        if (n == total)            return true;        if (n > total)            return false;        --len;        int d = n - f[len];        return (d > 0 && sumOfAnySubset(d, f, len)) || sumOfAnySubset(n, f, len);    }     private static int[] properDivisors(int n) {        List<Integer> divisors = new ArrayList<>();        divisors.add(1);        for (int i = 2;; ++i) {            int i2 = i * i;            if (i2 > n)                break;            if (n % i == 0) {                divisors.add(i);                if (i2 != n)                    divisors.add(n / i);            }        }        int[] result = new int[divisors.size()];        for (int i = 0; i < result.length; ++i)            result[i] = divisors.get(i);        Arrays.sort(result);        return result;    }     private static String shorten(List<Integer> list, int n) {        StringBuilder str = new StringBuilder();        int len = list.size(), i = 0;        if (n > 0 && len > 0)            str.append(list.get(i++));        for (; i < n && i < len; ++i) {            str.append(", ");            str.append(list.get(i));        }        if (len > i + n) {            if (n > 0)                str.append(", ...");            i = len - n;        }        for (; i < len; ++i) {            str.append(", ");            str.append(list.get(i));        }        return str.toString();    }}`
Output:
```Found 77 practical numbers between 1 and 333:
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, ..., 288, 294, 300, 304, 306, 308, 312, 320, 324, 330
666 is a practical number.
6666 is a practical number.
66666 is not a practical number.
672 is a practical number.
720 is a practical number.
222222 is a practical number.
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

This implementation does not create an in-memory representation of the powerset; this saves some time and of course potentially a great deal of memory.

The version of `proper_divisors` at Proper_divisors#jq may be used and therefore its definition is not repeated here.

`# A reminder to include the definition of `proper_divisors`# include "proper_divisors" { search: "." }; # Input: an array, each of whose elements is treated as distinct.# Output: a stream of arrays.# If the items in the input array are distinct, then the items in the# stream represent the items in the powerset of the input array.  If# the items in the input array are sorted, then the items in each of# the output arrays will also be sorted.  The lengths of the output# arrays are non-decreasing.def powersetStream:  if length == 0 then []  else .[0] as \$first    | (.[1:] | powersetStream)     | ., ([\$first] + . )  end; def isPractical:  . as \$n  | if . == 1 then true    elif . % 2 == 1 then false    else [proper_divisors] as \$divs    | first(        foreach (\$divs|powersetStream) as \$subset (           {found: [],           count:  0 };           (\$subset|add) as \$sum           | if \$sum > 0 and \$sum < \$n and (.found[\$sum] | not)             then .found[\$sum] = true             | .count += 1             | if (.count == \$n - 1) then .emit = true               else .               end  	   else .	   end;           select(.emit).emit) )      // false   end; # Input: the upper bound of range(1,_) to consider (e.g. infinite)# Output: a stream of the practical numbers, in orderdef practical:  range(1;.)  | select(isPractical); def task(\$n):  (\$n + 1 | [practical]) as \$p  | ("In the range 1 .. \(\$n) inclusive, there are \(\$p|length) practical numbers.",     "The first ten are:", \$p[0:10],     "The last ten are:", \$p[-10:] ); task(333),(666,6666,66666 | "\nIs \(.) practical? \(if isPractical then "Yes." else "No." end)" )`
Output:
```In the range 1 .. 333 inclusive, there are 77 practical numbers.
The first ten are:
[1,2,4,6,8,12,16,18,20,24]
The last ten are:
[288,294,300,304,306,308,312,320,324,330]

Is 666 practical? Yes.

Is 6666 practical? Yes.

Is 66666 practical? No.
```

## Julia

Translation of: Python
`using Primes """ proper divisors of n """function proper_divisors(n)    f = [one(n)]    for (p,e) in factor(n)        f = reduce(vcat, [f*p^j for j in 1:e], init=f)    end    pop!(f)    return fend """ return true if any subset of f sums to n. """function sumofanysubset(n, f)    n in f && return true    total = sum(f)    n == total && return true    n > total && return false    rf = reverse(f)    d = n - popfirst!(rf)    return (d in rf) || (d > 0 && sumofanysubset(d, rf)) || sumofanysubset(n, rf)end function ispractical(n)    n == 1 && return true    isodd(n) && return false    f = proper_divisors(n)    return all(i -> sumofanysubset(i, f), 1:n-1)end const prac333 = filter(ispractical, 1:333)println("There are ", length(prac333), " practical numbers between 1 to 333 inclusive.")println("Start and end: ", filter(x -> x < 25, prac333), " ... ", filter(x -> x > 287, prac333), "\n")for n in [666, 6666, 66666, 222222]    println("\$n is ", ispractical(n) ? "" : "not ", "a practical number.")end `
Output:
```There are 77 practical numbers between 1 to 333 inclusive.
Start and end: [1, 2, 4, 6, 8, 12, 16, 18, 20, 24] ... [288, 294, 300, 304, 306, 308, 312, 320, 324, 330]

666 is a practical number.
6666 is a practical number.
66666 is not a practical number.
222222 is a practical number.
```

## Nim

`import intsets, math, sequtils, strutils func properDivisors(n: int): seq[int] =  result = @[1]  for i in 2..sqrt(n.toFloat).int:    if n mod i == 0:      let j = n div i      result.add i      if i != j: result.add j func allSums(n: Positive): IntSet =  let divs = n.properDivisors()  var currSet: IntSet  for d in divs:    currSet.assign(result)  # Make a copy of the set.    for sum in currSet:      result.incl sum + d   # Add a new sum to the set.    result.incl d           # Add the single value. func isPractical(n: Positive): bool =  toSeq(1..<n).toIntSet <= allSums(n) var count = 0for n in 1..333:  if n.isPractical:    inc count    stdout.write (\$n).align(3), if count mod 11 == 0: '\n' else: ' 'echo "Found ", count, " practical numbers between 1 and 333."echo()echo "666 is ", if 666.isPractical: "" else: "not ", "a practical number."`
Output:
```  1   2   4   6   8  12  16  18  20  24  28
30  32  36  40  42  48  54  56  60  64  66
72  78  80  84  88  90  96 100 104 108 112
120 126 128 132 140 144 150 156 160 162 168
176 180 192 196 198 200 204 208 210 216 220
224 228 234 240 252 256 260 264 270 272 276
280 288 294 300 304 306 308 312 320 324 330
Found 77 practical numbers between 1 and 333.

666 is a practical number.```

## Pascal

simple brute force.Marking sum of divs by shifting the former sum by the the next divisor.
SumAllSetsForPractical tries to break as soon as possible.Should try to check versus https://en.wikipedia.org/wiki/Practical_number#Characterization_of_practical_numbers

```...σ denotes the sum of the divisors of x. For example, 2 × 3^2 × 29 × 823 = 429606 is practical,
because the inequality above holds for each of its prime factors:
3 ≤ σ(2) + 1 = 4, 29 ≤ σ(2 × 3^2) + 1 = 40, and 823 ≤ σ(2 × 3^2 × 29) + 1 = 1171. ```
`program practicalnumbers;{\$IFDEF FPC}  {\$MODE DELPHI}{\$OPTIMIZATION ON,ALL}{\$ELSE}  {\$APPTYPE CONSOLE} {\$ENDIF} uses  sysutils{\$IFNDEF FPC}    ,Windows{\$ENDIF}  ; const  LOW_DIVS = 0;  HIGH_DIVS = 2048 - 1; type  tdivs = record    DivsVal: array[LOW_DIVS..HIGH_DIVS] of Uint32;    DivsMaxIdx, DivsNum, DivsSumProp: NativeUInt;  end; var  Divs: tDivs;  HasSum: array of byte; procedure GetDivisors(var Divs: tdivs; n: Uint32);//calc all divisors,keep sortedvar  i, quot, ug, og: UInt32;  sum: UInt64;begin  with Divs do  begin    DivsNum := n;    sum := 0;    ug := 0;    og := HIGH_DIVS;    i := 1;     while i * i < n do    begin      quot := n div i;      if n - quot * i = 0 then      begin        DivsVal[og] := quot;        Divs.DivsVal[ug] := i;        inc(sum, quot + i);        dec(og);        inc(ug);      end;      inc(i);    end;    if i * i = n then    begin      DivsVal[og] := i;      inc(sum, i);      dec(og);    end;  //move higher divisors down    while og < high_DIVS do    begin      inc(og);      DivsVal[ug] := DivsVal[og];      inc(ug);    end;    DivsMaxIdx := ug - 2;    DivsSumProp := sum - n;  end; //withend; function SumAllSetsForPractical(Limit: Uint32): boolean;//mark sum and than shift by next divisor == add//for practical numbers every sum must be markedvar  hs0, hs1: pByte;  idx, j, loLimit, maxlimit, delta: NativeUint;begin  Limit := trunc(Limit * (Limit / Divs.DivsSumProp));  loLimit := 0;  maxlimit := 0;  hs0 := @HasSum[0];  hs0[0] := 1; //empty set  for idx := 0 to Divs.DivsMaxIdx do  begin    delta := Divs.DivsVal[idx];    hs1 := @hs0[delta];    for j := maxlimit downto 0 do      hs1[j] := hs1[j] or hs0[j];    maxlimit := maxlimit + delta;    while hs0[loLimit] <> 0 do      inc(loLimit);    //IF there is a 0 < delta, it will never be set    //IF there are more than the Limit set,    //it will be copied by the following Delta's    if (loLimit < delta) or (loLimit > Limit) then      Break;  end;  result := (loLimit > Limit);end; function isPractical(n: Uint32): boolean;var  i: NativeInt;  sum: NativeUInt;begin  if n = 1 then    EXIT(True);  if ODD(n) then    EXIT(false);  if (n > 2) and not ((n mod 4 = 0) or (n mod 6 = 0)) then    EXIT(false);   GetDivisors(Divs, n);  i := n - 1;  sum := Divs.DivsSumProp;  if sum < i then    result := false  else  begin    if length(HasSum) > sum + 1 + 1 then      FillChar(HasSum[0], sum + 1, #0)    else    begin      setlength(HasSum, 0);      setlength(HasSum, sum + 8 + 1);    end;    result := SumAllSetsForPractical(i);  end;end; procedure OutIsPractical(n: nativeInt);begin  if isPractical(n) then    writeln(n, ' is practical')  else    writeln(n, ' is not practical');end; const  ColCnt = 10;  MAX = 333; var  T0: Int64;  n, col, count: NativeInt; begin  col := ColCnt;  count := 0;  for n := 1 to MAX do    if isPractical(n) then    begin      write(n: 5);      inc(count);      dec(col);      if col = 0 then      begin        writeln;        col := ColCnt;      end;    end;  writeln;  writeln('There are ', count, ' pratical numbers from 1 to ', MAX);  writeln;   T0 := GetTickCount64;  OutIsPractical(666);  OutIsPractical(6666);  OutIsPractical(66666);  OutIsPractical(954432);  OutIsPractical(720);  OutIsPractical(5384);  OutIsPractical(1441440);  writeln(Divs.DivsNum, ' has ', Divs.DivsMaxIdx + 1, ' proper divisors');  writeln((GetTickCount64 - T0) / 1000: 10: 3, ' s');  T0 := GetTickCount64;  OutIsPractical(99998640);  writeln(Divs.DivsNum, ' has ', Divs.DivsMaxIdx + 1, ' proper divisors ');  writeln((GetTickCount64 - T0) / 1000: 10: 3, ' s');  T0 := GetTickCount64;  OutIsPractical(99998640);  writeln(Divs.DivsNum, ' has ', Divs.DivsMaxIdx + 1, ' proper divisors ');  writeln((GetTickCount64 - T0) / 1000: 10: 3, ' s');  setlength(HasSum, 0);  {\$IFNDEF UNIX}  readln; {\$ENDIF}end. `
Output:
``` TIO.RUN.

1    2    4    6    8   12   16   18   20   24
28   30   32   36   40   42   48   54   56   60
64   66   72   78   80   84   88   90   96  100
104  108  112  120  126  128  132  140  144  150
156  160  162  168  176  180  192  196  198  200
204  208  210  216  220  224  228  234  240  252
256  260  264  270  272  276  280  288  294  300
304  306  308  312  320  324  330
There are 77 pratical numbers from 1 to 333

666 is practical
6666 is practical
66666 is not practical
954432 is not practical
720 is practical
5384 is not practical
1441440 is practical
1441440 has 287 proper divisors
0.017 s
99998640 is not practical
99998640 has 119 proper divisors
0.200 s // with reserving memory
99998640 is not practical
99998640 has 119 proper divisors
0.081 s // already reserved memory

Real time: 0.506 s CPU share: 87.94 %```

### alternative

Now without generating sum of allset.

`program practicalnumbers2; {\$IFDEF FPC}  {\$MODE DELPHI}{\$OPTIMIZATION ON,ALL}{\$ELSE}  {\$APPTYPE CONSOLE}{\$ENDIF}uses  SysUtils; type  tdivs = record    DivsVal: array[0..1024 - 1] of Uint32;  end; var  Divs: tDivs;   function CheckIsPractical(var Divs: tdivs; n: Uint32): boolean;    //calc all divisors,calc sum of divisors  var    sum: UInt64;    i :NativeInt;    quot,ug,sq,de: UInt32;   begin    with Divs do    begin      sum := 1;      ug := Low(tdivs.DivsVal);      i  := 2;      sq := 4;      de := 5;      while sq < n do      begin        quot := n div i;        if n - quot * i = 0 then        begin          if sum + 1 < i then            EXIT(false);          Inc(sum, i);          DivsVal[ug] := quot;          Inc(ug);        end;        Inc(i);        sq += de;        de := de+2;      end;      if sq = n then      begin        if sum + 1 < i then          EXIT(false);        DivsVal[ug] := i;        Inc(sum, i);        Inc(ug);      end;      //check higher      while ug > 0 do      begin        Dec(ug);        i := DivsVal[ug];        if sum + 1 < i then          EXIT(false);        Inc(sum, i);        if sum >= n - 1 then          break;      end;    end;//with    result := true;  end;   function isPractical(n: Uint32): boolean;  begin    if n in [1,2] then      EXIT(True);    if ODD(n) then      EXIT(False);    Result := CheckIsPractical(Divs, n);  end;   procedure OutIsPractical(n: nativeInt);  begin    if isPractical(n) then      writeln(n, ' is practical')    else      writeln(n, ' is not practical');  end; const  ColCnt = 10;  MAX = 333;var  T0 : int64;  n, col, Count: NativeInt;begin  col := ColCnt;  Count := 0;  for n := 1 to MAX do    if isPractical(n) then    begin      Write(n: 5);      Inc(Count);      Dec(col);      if col = 0 then      begin        writeln;        col := ColCnt;      end;    end;  writeln;  writeln('There are ', Count, ' pratical numbers from 1 to ', MAX);  writeln;    OutIsPractical(666);  OutIsPractical(6666);  OutIsPractical(66666);  OutIsPractical(954432);  OutIsPractical(720);  OutIsPractical(5184);  OutIsPractical(1441440);  OutIsPractical(99998640);   T0 := GetTickCOunt64;  count := 0;  For n := 1 to 1000*1000 do    inc(count,Ord(isPractical(n)));  writeln('Count of practical numbers til 1,000,000 ',count,(GetTickCount64-t0)/1000:8:4,' s');  {\$IFDEF WINDOWS}  readln;  {\$ENDIF}end. `
Output:
``` TIO.RUN
1    2    4    6    8   12   16   18   20   24
28   30   32   36   40   42   48   54   56   60
64   66   72   78   80   84   88   90   96  100
104  108  112  120  126  128  132  140  144  150
156  160  162  168  176  180  192  196  198  200
204  208  210  216  220  224  228  234  240  252
256  260  264  270  272  276  280  288  294  300
304  306  308  312  320  324  330
There are 77 pratical numbers from 1 to 333

666 is practical
6666 is practical
66666 is not practical
954432 is not practical
720 is practical
5184 is practical
1441440 is practical
99998640 is not practical
Count of practical numbers til 1,000,000 97385  2.1380 s

Real time: 2.277 s CPU share: 99.55 %```

## Perl

Library: ntheory
`use strict;use warnings;use feature 'say';use enum <False True>;use ntheory <divisors vecextract>;use List::AllUtils <sum uniq firstidx>; sub proper_divisors {  return 1 if 0 == (my \$n = shift);  my @d = divisors(\$n);  pop @d;  @d} sub powerset_sums { uniq map { sum vecextract(\@_,\$_) } 1..2**@_-1 } sub is_practical {    my(\$n) = @_;    return True  if \$n == 1;    return False if 0 != \$n % 2;    (\$n-2 == firstidx { \$_ == \$n-1 } powerset_sums(proper_divisors(\$n)) ) ? True : False;} my @pn;is_practical(\$_) and push @pn, \$_ for 1..333;say @pn . " matching numbers:\n" . (sprintf "@{['%4d' x @pn]}", @pn) =~ s/(.{40})/\$1\n/gr;say '';printf "%6d is practical? %s\n", \$_, is_practical(\$_) ? 'True' : 'False' for 666, 6666, 66666;`
Output:
```77 matching numbers:
1   2   4   6   8  12  16  18  20  24
28  30  32  36  40  42  48  54  56  60
64  66  72  78  80  84  88  90  96 100
104 108 112 120 126 128 132 140 144 150
156 160 162 168 176 180 192 196 198 200
204 208 210 216 220 224 228 234 240 252
256 260 264 270 272 276 280 288 294 300
304 306 308 312 320 324 330

666 is practical? True
6666 is practical? True
66666 is practical? False```

## Phix

Translation of: Python – (the composition of functions version)
```function sum_of_any_subset(integer n, sequence f)
-- return true if any subset of f sums to n.
if find(n,f) then return true end if
integer total = sum(f)
if n=total then return true
elsif n>total then return false end if
integer d = n-f[\$]
f = f[1..\$-1]
return find(d,f)
or (d>0 and sum_of_any_subset(d, f))
or sum_of_any_subset(n, f)
end function

function is_practical(integer n)
sequence f = factors(n,-1)
for i=1 to n-1 do
if not sum_of_any_subset(i,f) then return false end if
end for
return true
end function

sequence res = apply(true,sprintf,{{"%3d"},filter(tagset(333),is_practical)})
printf(1,"Found %d practical numbers:\n%s\n\n",{length(res),join(shorten(res,"",10),", ")})

procedure stretch(integer n)
printf(1,"is_practical(%d):%t\n",{n,is_practical(n)})
end procedure
papply({666,6666,66666,672,720},stretch)
```
Output:
```Found 77 practical numbers:
1,   2,   4,   6,   8,  12,  16,  18,  20,  24, ..., 288, 294, 300, 304, 306, 308, 312, 320, 324, 330

is_practical(666):true
is_practical(6666):true
is_practical(66666):false
is_practical(672):true
is_practical(720):true
```

## Python

### Python: Straight forward implementation

`from itertools import chain, cycle, accumulate, combinationsfrom typing import List, Tuple # %% Factors def factors5(n: int) -> List[int]:    """Factors of n, (but not n)."""    def prime_powers(n):        # c goes through 2, 3, 5, then the infinite (6n+1, 6n+5) series        for c in accumulate(chain([2, 1, 2], cycle([2,4]))):            if c*c > n: break            if n%c: continue            d,p = (), c            while not n%c:                n,p,d = n//c, p*c, d + (p,)            yield(d)        if n > 1: yield((n,))     r = [1]    for e in prime_powers(n):        r += [a*b for a in r for b in e]    return r[:-1] # %% Powerset def powerset(s: List[int]) -> List[Tuple[int, ...]]:    """powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3) ."""    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1)) # %% Practical number def is_practical(x: int) -> bool:    """    Is x a practical number.     I.e. Can some selection of the proper divisors of x, (other than x), sum    to i for all i in the range 1..x-1 inclusive.    """    if x == 1:        return True    if x %2:        return False  # No Odd number more than 1    f = factors5(x)    ps = powerset(f)    found = {y for y in {sum(i) for i in ps}             if 1 <= y < x}    return len(found) == x - 1  if __name__ == '__main__':    n = 333    p = [x for x in range(1, n + 1) if is_practical(x)]    print(f"There are {len(p)} Practical numbers from 1 to {n}:")    print(' ', str(p[:10])[1:-1], '...', str(p[-10:])[1:-1])    x = 666    print(f"\nSTRETCH GOAL: {x} is {'not ' if not is_practical(x) else ''}Practical.")`
Output:
```There are 77 Practical numbers from 1 to 333:
1, 2, 4, 6, 8, 12, 16, 18, 20, 24 ... 288, 294, 300, 304, 306, 308, 312, 320, 324, 330

STRETCH GOAL: 666 is Practical.```

#### Python: Faster version

This version has an optimisation that proves much faster when testing a range of numbers for Practicality.

A number with a large number of factors, f has `2**len(f)` sets in its powerset. 672 for example has 23 factors and so 8_388_608 sets in its powerset.
Just taking the sets as they are generated and stopping when we first know that 672 is Practical needs just the first 28_826 or 0.34% of the sets. 720, another Practical number needs just 0.01% of its half a billion sets to prove it is Practical.

The inner loop is sensitive to the order of factors passed to the powerset generator and experimentation shows that reverse sorting the factors saves the most computation.
An extra check on the sum of all factors has a minor positive effect too.

`def is_practical5(x: int) -> bool:    """Practical number test with factor reverse sort and short-circuiting."""     if x == 1:        return True    if x % 2:        return False  # No Odd number more than 1    mult_4_or_6 = (x % 4 == 0) or (x % 6 == 0)    if x > 2 and not mult_4_or_6:        return False  # If > 2 then must be a divisor of 4 or 6     f = sorted(factors5(x), reverse=True)    if sum(f) < x - 1:        return False # Never get x-1    ps = powerset(f)     found = set()    for nps in ps:        if len(found) < x - 1:            y = sum(nps)            if 1 <= y < x:                found.add(y)        else:            break   # Short-circuiting the loop.     return len(found) == x - 1  if __name__ == '__main__':    n = 333    print("\n" + is_practical5.__doc__)    p = [x for x in range(1, n + 1) if is_practical5(x)]    print(f"There are {len(p)} Practical numbers from 1 to {n}:")    print(' ', str(p[:10])[1:-1], '...', str(p[-10:])[1:-1])    x = 666    print(f"\nSTRETCH GOAL: {x} is {'not ' if not is_practical(x) else ''}Practical.")    x = 5184    print(f"\nEXTRA GOAL: {x} is {'not ' if not is_practical(x) else ''}Practical.")`
Output:

Using the definition of factors5 from the simple case above then the following results are obtained.

```Practical number test with factor reverse sort and short-circuiting.
There are 77 Practical numbers from 1 to 333:
1, 2, 4, 6, 8, 12, 16, 18, 20, 24 ... 288, 294, 300, 304, 306, 308, 312, 320, 324, 330

STRETCH GOAL: 666 is Practical.

EXTRA GOAL: 5184 is Practical.```

5184, which is practical, has 34 factors!

A little further investigation shows that you need to get to 3850, for the first example of a number with 23 or more factors that is not Practical.

Around 1/8'th of the integers up to the 10_000'th Practical number have more than 22 factors but are not practical numbers themselves. (Some of these have 31 factors whilst being divisible by four or six so would need the long loop to complete)!

### Composition of pure functions

`'''Practical numbers''' from itertools import accumulate, chain, groupby, productfrom math import floor, sqrtfrom operator import mulfrom functools import reducefrom typing import Callable, List  def isPractical(n: int) -> bool:    '''True if n is a Practical number       (a member of OEIS A005153)    '''    ds = properDivisors(n)    return all(map(        sumOfAnySubset(ds),        range(1, n)    ))  def sumOfAnySubset(xs: List[int]) -> Callable[[int], bool]:    '''True if any subset of xs sums to n.    '''    def go(n):        if n in xs:            return True        else:            total = sum(xs)            if n == total:                return True            elif n < total:                h, *t = reversed(xs)                d = n - h                return d in t or (                    d > 0 and sumOfAnySubset(t)(d)                ) or sumOfAnySubset(t)(n)            else:                return False    return go  # ------------------------- TEST -------------------------def main() -> None:    '''Practical numbers in the range [1..333],       and the OEIS A005153 membership of 666.    '''     xs = [x for x in range(1, 334) if isPractical(x)]    print(        f'{len(xs)} OEIS A005153 numbers in [1..333]\n\n' + (            spacedTable(                chunksOf(10)([                    str(x) for x in xs                ])            )        )    )    print("\n")    for n in [666]:        print(            f'{n} is practical :: {isPractical(n)}'        )  # ----------------------- GENERIC ------------------------ def chunksOf(n: int) -> Callable[[List[str]], List[List[str]]]:    '''A series of lists of length n, subdividing the       contents of xs. Where the length of xs is not evenly       divible, the final list will be shorter than n.    '''    def go(xs):        return [            xs[i:n + i] for i in range(0, len(xs), n)        ] if 0 < n else None    return go  def primeFactors(n: int) -> List[int]:    '''A list of the prime factors of n.    '''    def f(qr):        r = qr[1]        return step(r), 1 + r     def step(x):        return 1 + (x << 2) - ((x >> 1) << 1)     def go(x):        root = floor(sqrt(x))         def p(qr):            q = qr[0]            return root < q or 0 == (x % q)         q = until(p)(f)(            (2 if 0 == x % 2 else 3, 1)        )[0]        return [x] if q > root else [q] + go(x // q)     return go(n)  def properDivisors(n: int) -> List[int]:    '''The ordered divisors of n, excluding n itself.    '''    def go(a, x):        return [a * b for a, b in product(            a,            accumulate(chain([1], x), mul)        )]    return sorted(        reduce(go, [            list(g) for _, g in groupby(primeFactors(n))        ], [1])    )[:-1] if 1 < n else []  def listTranspose(xss: List[List[str]]) -> List[List[str]]:    '''Transposed matrix'''    def go(xss):        if xss:            h, *t = xss            return (                [[h[0]] + [xs[0] for xs in t if xs]] + (                    go([h[1:]] + [xs[1:] for xs in t])                )            ) if h and isinstance(h, list) else go(t)        else:            return []    return go(xss)  def until(p: Callable[[int], bool]) -> Callable[[int], bool]:    '''The result of repeatedly applying f until p holds.       The initial seed value is x.    '''    def go(f):        def g(x):            v = x            while not p(v):                v = f(v)            return v        return g    return go  # ---------------------- FORMATTING ---------------------- def spacedTable(rows: List[List[str]]) -> str:    '''Tabulation with right-aligned cells'''    columnWidths = [        len(str(row[-1])) for row in listTranspose(rows)    ]     def aligned(s, w):        return s.rjust(w, ' ')     return '\n'.join(        ' '.join(            map(aligned, row, columnWidths)        ) for row in rows    )  # MAIN ---if __name__ == '__main__':    main()`
Output:
```77 OEIS A005153 numbers in [1..333]

1   2   4   6   8  12  16  18  20  24
28  30  32  36  40  42  48  54  56  60
64  66  72  78  80  84  88  90  96 100
104 108 112 120 126 128 132 140 144 150
156 160 162 168 176 180 192 196 198 200
204 208 210 216 220 224 228 234 240 252
256 260 264 270 272 276 280 288 294 300
304 306 308 312 320 324 330

666 is practical :: True```

## Raku

`use Prime::Factor:ver<0.3.0+>; sub is-practical (\$n) {   return True  if \$n == 1;   return False if \$n % 2;   my @proper = \$n.&proper-divisors :sort;   return True if all( @proper.rotor(2 => -1).map: { .[0] / .[1] >= .5 } );   my @proper-sums = @proper.combinations».sum.unique.sort;   +@proper-sums < \$n-1 ?? False !! @proper-sums[^\$n] eqv (^\$n).list ?? True !! False} say "{+\$_} matching numbers:\n{.batch(10)».fmt('%3d').join: "\n"}\n"    given [ (1..333).hyper(:32batch).grep: { is-practical(\$_) } ]; printf "%5s is practical? %s\n", \$_, .&is-practical for 666, 6666, 66666, 672, 720;`
Output:
```77 matching numbers:
1   2   4   6   8  12  16  18  20  24
28  30  32  36  40  42  48  54  56  60
64  66  72  78  80  84  88  90  96 100
104 108 112 120 126 128 132 140 144 150
156 160 162 168 176 180 192 196 198 200
204 208 210 216 220 224 228 234 240 252
256 260 264 270 272 276 280 288 294 300
304 306 308 312 320 324 330

666 is practical? True
6666 is practical? True
66666 is practical? False
672 is practical? True
720 is practical? True```

## Rust

Translation of: Phix
`fn sum_of_any_subset(n: isize, f: &[isize]) -> bool {    let len = f.len();    if len == 0 {        return false;    }    if f.contains(&n) {        return true;    }    let mut total = 0;    for i in 0..len {        total += f[i];    }    if n == total {        return true;    }    if n > total {        return false;    }    let d = n - f[len - 1];    let g = &f[0..len - 1];    (d > 0 && sum_of_any_subset(d, g)) || sum_of_any_subset(n, g)} fn proper_divisors(n: isize) -> Vec<isize> {    let mut f = vec![1];    let mut i = 2;    loop {        let i2 = i * i;        if i2 > n {            break;        }        if n % i == 0 {            f.push(i);            if i2 != n {                f.push(n / i);            }        }        i += 1;    }    f.sort();    f} fn is_practical(n: isize) -> bool {    let f = proper_divisors(n);    for i in 1..n {        if !sum_of_any_subset(i, &f) {            return false;        }    }    true} fn shorten(v: &[isize], n: usize) -> String {    let mut str = String::new();    let len = v.len();    let mut i = 0;    if n > 0 && len > 0 {        str.push_str(&v[i].to_string());        i += 1;    }    while i < n && i < len {        str.push_str(", ");        str.push_str(&v[i].to_string());        i += 1;    }    if len > i + n {        if n > 0 {            str.push_str(", ...");        }        i = len - n;    }    while i < len {        str.push_str(", ");        str.push_str(&v[i].to_string());        i += 1;    }    str} fn main() {    let from = 1;    let to = 333;    let mut practical = Vec::new();    for n in from..=to {        if is_practical(n) {            practical.push(n);        }    }    println!(        "Found {} practical numbers between {} and {}:\n{}",        practical.len(),        from,        to,        shorten(&practical, 10)    );    for n in &[666, 6666, 66666, 672, 720, 222222] {        if is_practical(*n) {            println!("{} is a practical number.", n);        } else {            println!("{} is not practical number.", n);        }    }}`
Output:
```Found 77 practical numbers between 1 and 333:
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, ..., 288, 294, 300, 304, 306, 308, 312, 320, 324, 330
666 is a practical number.
6666 is a practical number.
66666 is not practical number.
672 is a practical number.
720 is a practical number.
222222 is a practical number.
```

## Sidef

Built-in:

`say is_practical(2**128 + 1)   #=> falsesay is_practical(2**128 + 4)   #=> true`

Slow implementation (as the task requires):

`func is_practical(n) {     var set = Set()     n.divisors.grep { _ < n }.subsets {|*a|        set << a.sum    }     1..n-1 -> all { set.has(_) }} var from = 1var upto = 333 var list = (from..upto).grep { is_practical(_) } say "There are #{list.len} practical numbers in the range #{from}..#{upto}."say "#{list.first(10).join(', ')} ... #{list.last(10).join(', ')}\n" for n in ([666, 6666, 66666]) {    say "#{'%5s' % n } is practical? #{is_practical(n)}"}`

Efficient algorithm:

`func is_practical(n) {     n.is_odd && return (n == 1)    n.is_pos || return false     var p = 1    var f = n.factor_exp     f.each_cons(2, {|a,b|        p *= sigma(a.head**a.tail)        b.head > (1 + p) && return false    })     return true}`
Output:
```There are 77 practical numbers in the range 1..333.
1, 2, 4, 6, 8, 12, 16, 18, 20, 24 ... 288, 294, 300, 304, 306, 308, 312, 320, 324, 330

666 is practical? true
6666 is practical? true
66666 is practical? false
```

## Visual Basic .NET

Translation of: C#
`Imports System.Collections.Generic, System.Linq, System.Console Module Module1    Function soas(ByVal n As Integer, ByVal f As IEnumerable(Of Integer)) As Boolean        If n <= 0 Then Return False Else If f.Contains(n) Then Return True        Select Case n.CompareTo(f.Sum())            Case 1 : Return False : Case 0 : Return True            Case -1 : Dim rf As List(Of Integer) = f.Reverse().ToList() : Dim D as Integer = n - rf(0)                 rf.RemoveAt(0) : Return soas(d, rf) OrElse soas(n, rf)        End Select : Return true    End Function     Function ip(ByVal n As Integer) As Boolean        Dim f As IEnumerable(Of Integer) = Enumerable.Range(1, n >> 1).Where(Function(d) n Mod d = 0).ToList()        Return Enumerable.Range(1, n - 1).ToList().TrueForAll(Function(i) soas(i, f))    End Function     Sub Main()        Dim c As Integer = 0, m As Integer = 333, i As Integer = 1 : While i <= m            If ip(i) OrElse i = 1 Then c += 1 : Write("{0,3} {1}", i, If(c Mod 10 = 0, vbLf, ""))            i += If(i = 1, 1, 2) : End While        Write(vbLf & "Found {0} practical numbers between 1 and {1} inclusive." & vbLf, c, m)        Do : m = If(m < 500, m << 1, m * 10 + 6)            Write(vbLf & "{0,5} is a{1}practical number.", m, If(ip(m), " ", "n im")) : Loop While m < 1e4    End SubEnd Module`
Output:

Same as C#

## Wren

Library: Wren-math
`import "/math" for Int, Nums var powerset // recursivepowerset = Fn.new { |set|    if (set.count == 0) return [[]]    var head = set[0]    var tail = set[1..-1]    return powerset.call(tail) + powerset.call(tail).map { |s| [head] + s }} var isPractical = Fn.new { |n|   if (n == 1) return true   var divs = Int.properDivisors(n)   var subsets = powerset.call(divs)   var found = List.filled(n, false)   var count = 0   for (subset in subsets) {       var sum = Nums.sum(subset)       if (sum > 0 && sum < n && !found[sum]) {          found[sum] = true          count = count + 1          if (count == n - 1) return true       }   }   return false} System.print("In the range 1..333, there are:")var practical = []for (i in 1..333) {    if (isPractical.call(i)) {        practical.add(i)    }}System.print("  %(practical.count) practical numbers")System.print("  The first ten are %(practical[0..9])")System.print("  The final ten are %(practical[-10..-1])")System.print("\n666 is practical: %(isPractical.call(666))")`
Output:
```In the range 1..333, there are:
77 practical numbers
The first ten are [1, 2, 4, 6, 8, 12, 16, 18, 20, 24]
The final ten are [288, 294, 300, 304, 306, 308, 312, 320, 324, 330]

666 is practical: true
```