Practical numbers

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.

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.

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.

Task

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


11lEdit

Translation of: Nim
F properDivisors(n)
   V result = [1]
   L(i) 2 .. Int(sqrt(n))
      I n % i == 0
         V j = n I/ i
         result.append(i)
         I i != j
            result.append(j)
   R result

F allSums(n)
   V divs = properDivisors(n)
   V currSet = Set[Int]()
   V result = Set[Int]()
   L(d) divs
      currSet = copy(result)
      L(sum) currSet
         result.add(sum + d)
      result.add(d)
   R result

F isPractical(n)
   R Set(Array(1 .< n)) <= allSums(n)

V count = 0
L(n) 1..333
   I isPractical(n)
      count++
      print(‘#3’.format(n), end' I count % 11 == 0 {"\n"} E ‘ ’)
print(‘Found ’count‘ practical numbers between 1 and 333.’)
print()
print(‘666 is ’(I isPractical(666) {‘’} E ‘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.

APLEdit

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#Edit

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++Edit

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.

DelphiEdit

See Pascal.

FreeBASICEdit

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 i
end 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 sum
end 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 true
end 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

GoEdit

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

JEdit

Borrowed from the Proper divisors#J page:

factors=: [: /:~@, */&>@{@((^ i.@>:)&.>/)@q:~&__
properDivisors=: factors -. ]

Borrowed from the Power set#J page:

ps=:  #~ 2 #:@i.@^ #

Implementation:

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

Task examples:

   +/ isPrac 1+i.333    NB. count practical numbers
77
   (#~ isPrac) 1+i.333  NB. list them
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 30...
   isPrac 666           NB. test
1

JavaEdit

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.

jqEdit

Adapted from Wren

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 order
def 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.

JuliaEdit

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 f
end

""" 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.

NimEdit

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 = 0
for 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.

PascalEdit

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 sorted
var
  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; //with
end;

function SumAllSetsForPractical(Limit: Uint32): boolean;
//mark sum and than shift by next divisor == add
//for practical numbers every sum must be marked
var
  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 %

alternativeEdit

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 %

PerlEdit

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

PhixEdit

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

PythonEdit

Python: Straight forward implementationEdit

from itertools import chain, cycle, accumulate, combinations
from 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 versionEdit

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 functionsEdit

'''Practical numbers'''

from itertools import accumulate, chain, groupby, product
from math import floor, sqrt
from operator import mul
from functools import reduce
from 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

RakuEdit

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

RustEdit

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.

SidefEdit

Built-in:

say is_practical(2**128 + 1)   #=> false
say 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 = 1
var 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 .NETEdit

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 Sub
End Module
Output:

Same as C#

WrenEdit

Library: Wren-math
import "/math" for Int, Nums

var powerset // recursive
powerset = 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