2> properdivs:longest(20000). With 79, Number 15120 has the most divisors re root of n% f%(count%)=n%/i% count%=count%+1 ENDIF ENDIF NEXT i% ENDIF ' RETURN count%-1 ENDFUNC

Output is:

```txt

Divisors for 1:
Divisors for 2: 1
Divisors for 3: 1
Divisors for 4: 1 2
Divisors for 5: 1
Divisors for 6: 1 2 3
Divisors for 7: 1
Divisors for 8: 1 2 4
Divisors for 9: 1 3
Divisors for 10: 1 2 5
Largest number of divisors is 79 for 15120

Go

{{trans|Kotlin}}

package main

import (
    "fmt"
    "strconv"
)

func listProperDivisors(limit int) {
    if limit < 1 {
        return
    }
    width := len(strconv.Itoa(limit))
    for i := 1; i <= limit; i++ {
        fmt.Printf("%*d -> ", width, i)
        if i == 1 {
            fmt.Println("(None)")
            continue
        }
        for j := 1; j <= i/2; j++ {
            if i%j == 0 {
                fmt.Printf(" %d", j)
            }
        }
        fmt.Println()
    }
}

func countProperDivisors(n int) int {
    if n < 2 {
        return 0
    }
    count := 0
    for i := 1; i <= n/2; i++ {
        if n%i == 0 {
            count++
        }
    }
    return count
}

func main() {
    fmt.Println("The proper divisors of the following numbers are :\n")
    listProperDivisors(10)
    fmt.Println()
    maxCount := 0
    most := []int{1}
    for n := 2; n <= 20000; n++ {
        count := countProperDivisors(n)
        if count == maxCount {
            most = append(most, n)
        } else if count > maxCount {
            maxCount = count
            most = most[0:1]
            most[0] = n
        }
    }
    fmt.Print("The following number(s) <= 20000 have the most proper divisors, ")
    fmt.Println("namely", maxCount, "\b\n")
    for _, n := range most {
        fmt.Println(n)
    }
}

{{out}}

The proper divisors of the following numbers are :

 1 -> (None)
 2 ->  1
 3 ->  1
 4 ->  1 2
 5 ->  1
 6 ->  1 2 3
 7 ->  1
 8 ->  1 2 4
 9 ->  1 3
10 ->  1 2 5

The following number(s) <= 20000 have the most proper divisors, namely 79

15120
18480

Haskell

import Data.Ord
import Data.List

divisors :: (Integral a) => a -> [a]
divisors n = filter ((0 ==) . (n `mod`)) [1 .. (n `div` 2)]

main :: IO ()
main = do
  putStrLn "divisors of 1 to 10:"
  mapM_ (print . divisors) [1 .. 10]
  putStrLn "a number with the most divisors within 1 to 20000 (number, count):"
  print $ maximumBy (comparing snd)
    [(n, length $ divisors n) | n <- [1 .. 20000]]

{{out}}

divisors of 1 to 10:
[]
[1]
[1]
[1,2]
[1]
[1,2,3]
[1]
[1,2,4]
[1,3]
[1,2,5]
a number with the most divisors within 1 to 20000 (number, count):
(18480,79)

Or, for a little more efficiency, filtering only up to the root, and deriving the higher proper divisors from the lower ones, as quotients:

import Data.List (maximumBy)
import Data.Ord (comparing)
import Data.Bool (bool)

properDivisors
  :: Integral a
  => a -> [a]
properDivisors n =
  let root = (floor . sqrt . fromIntegral) n
      lows = filter ((0 ==) . rem n) [1 .. root]
  in init (lows ++ bool id tail (n == root * root) (reverse (quot n <$> lows)))

main :: IO ()
main = do
  putStrLn "Proper divisors of 1 to 10:"
  mapM_ (print . properDivisors) [1 .. 10]
  mapM_
    putStrLn
    [ ""
    , "A number in the range 1 to 20,000 with the most proper divisors,"
    , "as (number, count of proper divisors):"
    , ""
    ]
  print $
    maximumBy (comparing snd) $
    (,) <*> (length . properDivisors) <$> [1 .. 20000]

{{Out}}

Proper divisors of 1 to 10:
[]
[1]
[1]
[1,2]
[1]
[1,2,3]
[1]
[1,2,4]
[1,3]
[1,2,5]

A number in the range 1 to 20,000 with the most proper divisors,
as (number, count of proper divisors):

(18480,79)

J

The proper divisors of an integer are the [[Factors of an integer]] without the integer itself.

So, borrowing from [[Factors of an integer#J|the J implementation]] of that related task:

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

Proper divisors of numbers 1 through 10:

   (,&": ' -- ' ,&": properDivisors)&>1+i.10
1 --
2 -- 1
3 -- 1
4 -- 1 2
5 -- 1
6 -- 1 2 3
7 -- 1
8 -- 1 2 4
9 -- 1 3
10 -- 1 2 5

Number(s) not exceeding 20000 with largest number of proper divisors (and the count of those divisors):

   (, #@properDivisors)&> 1+I.(= >./) #@properDivisors@> 1+i.20000
15120 79
18480 79

Note that it's a bit more efficient to simply count factors here, when selecting the candidate numbers.

      (, #@properDivisors)&> 1+I.(= >./) #@factors@> 1+i.20000
15120 79
18480 79

We could also arbitrarily toss either 15120 or 18480 (keeping the other number), if it were important that we produce only one result.

Java

{{works with|Java|1.5+}}

import java.util.Collections;
import java.util.LinkedList;
import java.util.List;

public class Proper{
    public static List<Integer> properDivs(int n){
        List<Integer> divs = new LinkedList<Integer>();
        if(n == 1) return divs;
        divs.add(1);
        for(int x = 2; x < n; x++){
            if(n % x == 0) divs.add(x);
        }

        Collections.sort(divs);

        return divs;
    }

    public static void main(String[] args){
        for(int x = 1; x <= 10; x++){
            System.out.println(x + ": " + properDivs(x));
        }

        int x = 0, count = 0;
        for(int n = 1; n <= 20000; n++){
            if(properDivs(n).size() > count){
                x = n;
                count = properDivs(n).size();
            }
        }
        System.out.println(x + ": " + count);
    }
}

{{out}}

1: []
2: [1]
3: [1]
4: [1, 2]
5: [1]
6: [1, 2, 3]
7: [1]
8: [1, 2, 4]
9: [1, 3]
10: [1, 2, 5]
15120: 79

JavaScript

ES5

(function () {

    // Proper divisors
    function properDivisors(n) {
        if (n < 2) return [];
        else {
            var rRoot = Math.sqrt(n),
                intRoot = Math.floor(rRoot),

                lows = range(1, intRoot).filter(function (x) {
                    return (n % x) === 0;
                });

            return lows.concat(lows.slice(1).map(function (x) {
                return n / x;
            }).reverse().slice((rRoot === intRoot) | 0));
        }
    }

    // [m..n]
    function range(m, n) {
        var a = Array(n - m + 1),
            i = n + 1;
        while (i--) a[i - 1] = i;
        return a;
    }

    var tblOneToTen = [
            ['Number', 'Proper Divisors', 'Count']
        ].concat(range(1, 10).map(function (x) {
            var ds = properDivisors(x);

            return [x, ds.join(', '), ds.length];
        })),

        dctMostBelow20k = range(1, 20000).reduce(function (a, x) {
            var lng = properDivisors(x).length;

            return lng > a.divisorCount ? {
                n: x,
                divisorCount: lng
            } : a;
        }, {
            n: 0,
            divisorCount: 0
        });


    // [[a]] -> bool -> s -> s
    function wikiTable(lstRows, blnHeaderRow, strStyle) {
        return '{| class="wikitable" ' + (
            strStyle ? 'style="' + strStyle + '"' : ''
        ) + lstRows.map(function (lstRow, iRow) {
            var strDelim = ((blnHeaderRow && !iRow) ? '!' : '|');

            return '\n|-\n' + strDelim + ' ' + lstRow.map(function (v) {
                return typeof v === 'undefined' ? ' ' : v;
            }).join(' ' + strDelim + strDelim + ' ');
        }).join('') + '\n|}';
    }

    return wikiTable(
        tblOneToTen,
        true
    ) + '\n\nMost proper divisors below 20,000:\n\n  ' + JSON.stringify(
        dctMostBelow20k
    );

})();

{{out}}

{| class="wikitable" |- ! Number !! Proper Divisors !! Count |- | 1 || || 0 |- | 2 || 1 || 1 |- | 3 || 1 || 1 |- | 4 || 1, 2 || 2 |- | 5 || 1 || 1 |- | 6 || 1, 2, 3 || 3 |- | 7 || 1 || 1 |- | 8 || 1, 2, 4 || 3 |- | 9 || 1, 3 || 2 |- | 10 || 1, 2, 5 || 3 |}

Most proper divisors below 20,000:

{"n":15120,"divisorCount":79}

ES6

(() => {

    // properDivisors :: Int -> [Int]
    let properDivisors = n => {
            let rRoot = Math.sqrt(n),
                intRoot = Math.floor(rRoot),
                blnPerfectSquare = rRoot === intRoot,

                lows = range(1, intRoot)
                .filter(x => (n % x) === 0);

            // for perfect squares, we can drop
            // the head of the 'highs' list
            return lows.concat(lows
                    .map(x => n / x)
                    .reverse()
                    .slice(blnPerfectSquare | 0)
                )
                .slice(0, -1); // except n itself
        },

        // range :: Int -> Int -> [Int]
        range = (m, n) => Array.from({
            length: (n - m) + 1
        }, (_, i) => m + i);



    return {
        properDivisorsOf1to10: range(1, 10)
            .reduce((a, x) => (
                a[x.toString()] = properDivisors(x),
                a
            ), {}),

        intMaxDivisorsUnder20k: range(1, 20000)
            .reduce((a, x) => {
                let intDivisors = properDivisors(x)
                    .length;

                return intDivisors >= a.divisors ? {
                    max: x,
                    divisors: intDivisors
                } : a;

            }, {
                max: 0,
                divisors: 0
            })
    };

})();

{{Out}}

{
  "properDivisorsOf1to10":{
    "1":[], "2":[1], "3":[1], "4":[1, 2], "5":[1],
    "6":[1, 2, 3], "7":[1], "8":[1, 2, 4], "9":[1, 3], "10":[1, 2, 5]
  },
  "intMaxDivisorsUnder20k":{"max":18480, "divisors":79}
}

jq

{{works with|jq|1.4}} In the following, proper_divisors returns a stream. In order to count the number of items in the stream economically, we first define "count(stream)":

def count(stream): reduce stream as $i (0; . + 1);

# unordered
def proper_divisors:
  . as $n
  | if $n > 1 then 1,
      ( range(2; 1 + (sqrt|floor)) as $i
        | if ($n % $i) == 0 then $i,
            (($n / $i) | if . == $i then empty else . end)
         else empty
	 end)
    else empty
    end;

# The first integer in 1 .. n inclusive
# with the maximal number of proper divisors in that range:
def most_proper_divisors(n):
  reduce range(1; n+1) as $i
    ( [null, 0];
      count( $i | proper_divisors ) as $count
      | if $count > .[1] then [$i, $count] else . end);

'''The tasks:'''

"The proper divisors of the numbers 1 to 10 inclusive are:",
(range(1;11) as $i | "\($i): \( [ $i | proper_divisors] )"),
"",
"The pair consisting of the least number in the range 1 to 20,000 with",
"the maximal number proper divisors together with the corresponding",
"count of proper divisors is:",
most_proper_divisors(20000)

{{out}}

$ jq -n -c -r -f /Users/peter/jq/proper_divisors.jq
The proper divisors of the numbers 1 to 10 inclusive are:
1: []
2: [1]
3: [1]
4: [1,2]
5: [1]
6: [1,2,3]
7: [1]
8: [1,2,4]
9: [1,3]
10: [1,2,5]

The pair consisting of the least number in the range 1 to 20,000 with
the maximal number proper divisors together with the corresponding
count of proper divisors is:
[15120,79]

Julia

Use factor to obtain the prime factorization of the target number. I adopted the argument handling style of factor in my properdivisors function.

function properdivisors{T<:Integer}(n::T)
    0 < n || throw(ArgumentError("number to be factored must be ≥ 0, got $n"))
    1 < n || return T[]
    !isprime(n) || return T[one(T), n]
    f = factor(n)
    d = T[one(T)]
    for (k, v) in f
        c = T[k^i for i in 0:v]
        d = d*c'
        d = reshape(d, length(d))
    end
    sort!(d)
    return d[1:end-1]
end

lo = 1
hi = 10
println("List the proper divisors for ", lo, " through ", hi, ".")
for i in lo:hi
    println(@sprintf("%4d", i), " ", properdivisors(i))
end

hi = 2*10^4
println("\nFind the numbers within [", lo, ",", hi, "] having the most divisors.")

maxdiv = 0
nlst = Int[]

for i in lo:hi
    ndiv = length(properdivisors(i))
    if ndiv > maxdiv
        maxdiv = ndiv
        nlst = [i]
    elseif ndiv == maxdiv
        push!(nlst, i)
    end
end

println(nlst, " have the maximum proper divisor count of ", maxdiv, ".")

{{out}}

List the proper divisors for 1 through 10.
   1 []
   2 [1,2]
   3 [1,3]
   4 [1,2]
   5 [1,5]
   6 [1,2,3]
   7 [1,7]
   8 [1,2,4]
   9 [1,3]
  10 [1,2,5]

Find the numbers within [1,20000] having the most divisors.
[15120,18480] have the maximum proper divisor count of 79.

Kotlin

// version 1.0.5-2

fun listProperDivisors(limit: Int) {
    if (limit < 1) return
    for(i in 1..limit) {
        print(i.toString().padStart(2) + " -> ")
        if (i == 1) {
            println("(None)")
            continue
        }
        (1..i/2).filter{ i % it == 0 }.forEach { print(" $it") }
        println()
    }
}

fun countProperDivisors(n: Int): Int {
    if (n < 2) return 0
    return (1..n/2).count { (n % it) == 0 }
}

fun main(args: Array<String>) {
    println("The proper divisors of the following numbers are :\n")
    listProperDivisors(10)
    println()
    var count: Int
    var maxCount = 0
    val most: MutableList<Int> = mutableListOf(1)
    for (n in 2..20000) {
        count = countProperDivisors(n)
        if (count == maxCount)
            most.add(n)
        else if (count > maxCount) {
            maxCount = count
            most.clear()
            most.add(n)
        }
    }
    println("The following number(s) have the most proper divisors, namely " + maxCount + "\n")
    for (n in most) println(n)
}

{{out}}

The proper divisors of the following numbers are :

 1 -> (None)
 2 ->  1
 3 ->  1
 4 ->  1 2
 5 ->  1
 6 ->  1 2 3
 7 ->  1
 8 ->  1 2 4
 9 ->  1 3
10 ->  1 2 5

The following number(s) have the most proper divisors, namely 79

15120
18480

Lua

-- Return a table of the proper divisors of n
function propDivs (n)
    if n < 2 then return {} end
    local divs, sqr = {1}, math.sqrt(n)
    for d = 2, sqr do
        if n % d == 0 then
            table.insert(divs, d)
            if d ~= sqr then table.insert(divs, n/d) end
        end
    end
    table.sort(divs)
    return divs
end

-- Show n followed by all values in t
function show (n, t)
    io.write(n .. ":\t")
    for _, v in pairs(t) do io.write(v .. " ") end
    print()
end

-- Main procedure
local mostDivs, numDivs, answer = 0
for i = 1, 10 do show(i, propDivs(i)) end
for i = 1, 20000 do
    numDivs = #propDivs(i)
    if numDivs > mostDivs then
        mostDivs = numDivs
        answer = i
    end
end
print(answer .. " has " .. mostDivs .. " proper divisors.")

{{out}}

1:
2:      1
3:      1
4:      1 2
5:      1
6:      1 2 3
7:      1
8:      1 2 4
9:      1 3
10:     1 2 5
15120 has 79 proper divisors.

=={{header|Mathematica}} / {{header|Wolfram Language}}==

A Function that yields the proper divisors of an integer n:

ProperDivisors[n_Integer /; n > 0] := Most@Divisors@n;

Proper divisors of n from 1 to 10:

Grid@Table[{n, ProperDivisors[n]}, {n, 1, 10}]

{{out}}

1	{}
2	{1}
3	{1}
4	{1,2}
5	{1}
6	{1,2,3}
7	{1}
8	{1,2,4}
9	{1,3}
10	{1,2,5}

The number with the most divisors between 1 and 20,000:

Fold[
 Last[SortBy[{#1, {#2, Length@ProperDivisors[#2]}}, Last]] &,
 {0, 0},
 Range[20000]]

{{out}}

{18480, 79}

An alternate way to find the number with the most divisors between 1 and 20,000:

Last@SortBy[
  Table[
    {n, Length@ProperDivisors[n]},
    {n, 1, 20000}],
  Last]

{{out}}

{15120, 79}

Matlab

function D=pd(N)
K=1:ceil(N/2);
D=K(~(rem(N, K)));

{{out}}

for I=1:10
   disp([num2str(I) ' : ' num2str(pd(I))])
end
1 : 1
2 : 1
3 : 1
4 : 1  2
5 : 1
6 : 1  2  3
7 : 1
8 : 1  2  4
9 : 1  3
10 : 1  2  5

maxL=0; maxI=0;
for I=1:20000
   L=length(pd(I));
   if L>maxL
      maxL=L; maxI=I;
   end
end
maxI

maxI =

       15120

maxL

maxL =

    79

=={{header|Modula-2}}==

MODULE ProperDivisors;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;

PROCEDURE WriteInt(n : INTEGER);
VAR buf : ARRAY[0..15] OF CHAR;
BEGIN
    FormatString("%i", buf, n);
    WriteString(buf)
END WriteInt;

PROCEDURE proper_divisors(n : INTEGER; print_flag : BOOLEAN) : INTEGER;
VAR count,i : INTEGER;
BEGIN
    count := 0;
    FOR i:=1 TO n-1 DO
        IF n MOD i = 0 THEN
            INC(count);
            IF print_flag THEN
                WriteInt(i);
                WriteString(" ")
            END
        END
    END;
    IF print_flag THEN WriteLn END;
    RETURN count;
END proper_divisors;

VAR
    buf : ARRAY[0..63] OF CHAR;
    i,max,max_i,v : INTEGER;
BEGIN
    FOR i:=1 TO 10 DO
        WriteInt(i);
        WriteString(": ");
        proper_divisors(i, TRUE)
    END;

    max := 0;
    max_i := 1;

    FOR i:=1 TO 20000 DO
        v := proper_divisors(i, FALSE);
        IF v>= max THEN
            max := v;
            max_i := i
        END
    END;

    FormatString("%i with %i divisors\n", buf, max_i, max);
    WriteString(buf);

    ReadChar
END ProperDivisors.

Objeck

use Collection;

class Proper{
  function : Main(args : String[]) ~ Nil {
    for(x := 1; x <= 10; x++;) {
      Print(x, ProperDivs(x));
    };

    x := 0;
    count := 0;

    for(n := 1; n <= 20000; n++;) {
      if(ProperDivs(n)->Size() > count) {
        x := n;
        count := ProperDivs(n)->Size();
      };
    };
    "{$x}: {$count}"->PrintLine();
  }

  function : ProperDivs(n : Int) ~ IntVector {
    divs := IntVector->New();

    if(n = 1) {
      return divs;
    };
    divs->AddBack(1);

    for(x := 2; x < n; x++;) {
      if(n % x = 0) {
        divs->AddBack(x);
      };
    };
    divs->Sort();

    return divs;
  }

  function : Print(x : Int, result : IntVector) ~ Nil {
    "{$x}: "->Print();
    result->ToArray()->ToString()->PrintLine();
  }
}

Output:

1: []
2: [1]
3: [1]
4: [1,2]
5: [1]
6: [1,2,3]
7: [1]
8: [1,2,4]
9: [1,3]
10: [1,2,5]
15120: 79

=={{header|Oberon-2}}==

MODULE ProperDivisors;
IMPORT
  Out;

CONST
    initialSize = 128;
TYPE
  Result* = POINTER TO ResultDesc;
  ResultDesc = RECORD
    found-: LONGINT; (* number of slots in pd *)
    pd-: POINTER TO ARRAY OF LONGINT;
    cap: LONGINT;   (* Capacity *)
  END;

VAR
  i,found,max,idxMx: LONGINT;
  mx: ARRAY 32 OF LONGINT;
  rs: Result;

  PROCEDURE (r: Result) Init(size: LONGINT);
  BEGIN
    r.found := 0;
    r.cap := size;
    NEW(r.pd,r.cap);
  END Init;

  PROCEDURE (r: Result) Add(n: LONGINT);
  BEGIN
    (* Out.String("--->");Out.LongInt(n,0);Out.String(" At: ");Out.LongInt(r.found,0);Out.Ln; *)
    IF (r.found < LEN(r.pd^) - 1) THEN
      r.pd[r.found] := n;
    ELSE
      (* expand pd for more room *)
    END;
    INC(r.found);
  END Add;

  PROCEDURE (r:Result) Show();
  VAR
    i: LONGINT;
  BEGIN
      Out.String("(Result:");Out.LongInt(r.found + 1,0);(* Out.String("/");Out.LongInt(r.cap,0);*)
      Out.String("-");
      IF r.found > 0 THEN
        FOR i:= 0 TO r.found - 1 DO
          Out.LongInt(r.pd[i],0);
          IF i = r.found - 1 THEN Out.Char(')') ELSE Out.Char(',') END
        END
      END;
      Out.Ln
  END Show;

  PROCEDURE (r:Result) Reset();
  BEGIN
    r.found := 0;
  END Reset;

  PROCEDURE GetFor(n: LONGINT;VAR rs: Result);
  VAR
    i: LONGINT;
  BEGIN
    IF n > 1 THEN
      rs.Add(1);i := 2;
      WHILE (i < n) DO
        IF (n MOD i) = 0 THEN rs.Add(i) END;
        INC(i)
      END
    END;
  END GetFor;

BEGIN
  NEW(rs);rs.Init(initialSize);
  FOR i := 1 TO 10 DO
    Out.LongInt(i,4);Out.Char(':');
    GetFor(i,rs);
    rs.Show();
    rs.Reset();
  END;
  Out.LongInt(100,4);Out.Char(':');GetFor(100,rs);rs.Show();rs.Reset();
  max := 0;idxMx := 0;found := 0;
  FOR i := 1 TO 20000 DO
    GetFor(i,rs);
    IF rs.found > max THEN
      idxMx:= 0;mx[idxMx] := i;max := rs.found
    ELSIF rs.found = max THEN
      INC(idxMx);mx[idxMx] := i
    END;
    rs.Reset()
  END;
  Out.String("Found: ");Out.LongInt(idxMx + 1,0);
  Out.String(" Numbers with most proper divisors ");
  Out.LongInt(max,0);Out.String(": ");Out.Ln;
  FOR i := 0 TO idxMx DO
    Out.LongInt(mx[i],0);Out.Ln
  END
END ProperDivisors.

{{out}}

   1:(Result:1-
   2:(Result:2-1)
   3:(Result:2-1)
   4:(Result:3-1,2)
   5:(Result:2-1)
   6:(Result:4-1,2,3)
   7:(Result:2-1)
   8:(Result:4-1,2,4)
   9:(Result:3-1,3)
  10:(Result:4-1,2,5)
 100:(Result:9-1,2,4,5,10,20,25,50)
Found: 2 Numbers with most proper divisors 79:
15120
18480

Oforth

Integer method: properDivs  self 2 / seq filter(#[ self swap mod 0 == ]) }

10 seq apply(#[ dup print " : " print properDivs println ])
20000 seq map(#[ dup properDivs size Pair new ]) reduce(#maxKey) println

{{out}}

1 : []
2 : [1]
3 : [1]
4 : [1, 2]
5 : [1]
6 : [1, 2, 3]
7 : [1]
8 : [1, 2, 4]
9 : [1, 3]
10 : [1, 2, 5]
[79, 15120]

PARI/GP

proper(n)=if(n==1, [], my(d=divisors(n)); d[2..#d]);
apply(proper, [1..10])
r=at=0; for(n=1,20000, t=numdiv(n); if(t>r, r=t; at=n)); [at, numdiv(t)-1]

{{out}}

%1 = [[], [2], [3], [2, 4], [5], [2, 3, 6], [7], [2, 4, 8], [3, 9], [2, 5, 10]]
%2 = [15120, 7]

Pascal

{{works with|Free Pascal}} Using prime factorisation

{$IFDEF FPC}{$MODE DELPHI}{$ELSE}{$APPTYPE CONSOLE}{$ENDIF}
uses
  sysutils;
const
  MAXPROPERDIVS = 1920;
type
  tRes = array[0..MAXPROPERDIVS] of LongWord;
  tPot = record
           potPrim,
           potMax :LongWord;
         end;

  tprimeFac = record
                 pfPrims : array[1..10] of tPot;
                 pfCnt,
                 pfNum   : LongWord;
               end;
  tSmallPrimes = array[0..6541] of longWord;

var
  SmallPrimes: tSmallPrimes;

procedure InitSmallPrimes;
var
  pr,testPr,j,maxprimidx: Longword;
  isPrime : boolean;
Begin
  maxprimidx := 0;
  SmallPrimes[0] := 2;
  pr := 3;
  repeat
    isprime := true;
    j := 0;
    repeat
      testPr := SmallPrimes[j];
      IF testPr*testPr > pr then
        break;
      If pr mod testPr = 0 then
      Begin
        isprime := false;
        break;
      end;
      inc(j);
    until false;

    if isprime then
    Begin
      inc(maxprimidx);
      SmallPrimes[maxprimidx]:= pr;
    end;
    inc(pr,2);
  until pr > 1 shl 16 -1;
end;

procedure PrimeFacOut(primeDecomp:tprimeFac);
var
  i : LongWord;
begin
  with primeDecomp do
  Begin
    write(pfNum,' = ');
    For i := 1 to pfCnt-1 do
      with pfPrims[i] do
        If potMax = 1 then
          write(potPrim,'*')
        else
          write(potPrim,'^',potMax,'*');
    with pfPrims[pfCnt] do
      If potMax = 1 then
        write(potPrim)
      else
        write(potPrim,'^',potMax);
  end;
end;

procedure PrimeDecomposition(n:LongWord;var res:tprimeFac);
var
  i,pr,cnt,quot{to minimize divisions} : LongWord;
Begin
  res.pfNum := n;
  res.pfCnt:= 0;
  i := 0;
  cnt := 0;
  repeat
    pr := SmallPrimes[i];
    IF pr*pr>n then
      Break;

    quot := n div pr;
    IF pr*quot = n then
      with res do
      Begin
        inc(pfCnt);
        with pfPrims[pfCnt] do
        Begin
          potPrim := pr;
          potMax := 0;
          repeat
            n := quot;
            quot := quot div pr;
            inc(potMax);
          until pr*quot <> n;
        end;
      end;
     inc(i);
  until false;
  //a big prime left over?
  IF n <> 1 then
    with res do
    Begin
      inc(pfCnt);
      with pfPrims[pfCnt] do
      Begin
        potPrim := n;
        potMax := 1;
      end;
    end;
end;

function CntProperDivs(const primeDecomp:tprimeFac):LongWord;
//count of proper divisors
var
   i: LongWord;
begin
  result := 1;
  with primeDecomp do
    For i := 1 to pfCnt do
      result := result*(pfPrims[i].potMax+1);
  //remove
  dec(result);
end;

function findProperdivs(n:LongWord;var res:TRes):LongWord;
//simple trial division to get a sorted list of all proper divisors
var
  i,j: LongWord;
Begin
  result := 0;
  i := 1;
  j := n;
  while j>i do
  begin
    j := n DIV i;
    IF i*j = n then
    Begin
      //smaller factor part at the beginning upwards
      res[result]:= i;
      IF i <> j then
        //bigger factor at the end downwards
        res[MAXPROPERDIVS-result]:= j
      else
        //n is square number
        res[MAXPROPERDIVS-result]:= 0;
      inc(result);
    end;
    inc(i);
  end;

  If result>0 then
  Begin
    //move close together
    i := result;
    j := MAXPROPERDIVS-result+1;
    result := 2*result-1;
    repeat
      res[i] := res[j];
      inc(j);
      inc(i);
    until i > result;

    if res[result-1] = 0 then
      dec(result);
  end;
end;

procedure AllFacsOut(n: Longword);
var
  res:TRes;
  i,k,j:LongInt;
Begin
   j := findProperdivs(n,res);
   write(n:5,' : ');
   For k := 0 to j-2 do write(res[k],',');
   IF j>=1 then
     write(res[j-1]);
   writeln;
end;

var
  primeDecomp: tprimeFac;
  rs : tRes;
  i,j,max,maxcnt: LongWord;
BEGIN
  InitSmallPrimes;
  For i := 1 to 10 do
    AllFacsOut(i);
  writeln;
  max    := 0;
  maxCnt := 0;
  For i := 1 to 20*1000 do
  Begin
    PrimeDecomposition(i,primeDecomp);
    j := CntProperDivs(primeDecomp);
    IF j> maxCnt then
    Begin
      maxcnt := j;
      max := i;
    end;
  end;
  PrimeDecomposition(max,primeDecomp);
  j := CntProperDivs(primeDecomp);

  PrimeFacOut(primeDecomp);writeln('  ',j:10,' factors'); writeln;
  //https://en.wikipedia.org/wiki/Highly_composite_number <= HCN
  //http://wwwhomes.uni-bielefeld.de/achim/highly.txt the first 1200 HCN
  max := 3491888400;
  PrimeDecomposition(max,primeDecomp);
  j := CntProperDivs(primeDecomp);
  PrimeFacOut(primeDecomp);writeln('  ',j:10,' factors'); writeln;
END.

{{Output}}

    1 :
    2 : 1
    3 : 1
    4 : 1,2
    5 : 1
    6 : 1,2,3
    7 : 1
    8 : 1,2,4
    9 : 1,3
   10 : 1,2,5

15120 = 2^4*3^3*5*7          79 factors

3491888400 = 2^4*3^3*5^2*7*11*13*17*19        1919 factors

real    0m0.004s

Perl

Using a module for divisors

{{libheader|ntheory}}

use ntheory qw/divisors/;
sub proper_divisors {
  my $n = shift;
  # Like Pari/GP, divisors(0) = (0,1) and divisors(1) = ()
  return 1 if $n == 0;
  my @d = divisors($n);
  pop @d;  # divisors are in sorted order, so last entry is the input
  @d;
}
say "$_: ", join " ", proper_divisors($_) for 1..10;
# 1. For the max, we can do a traditional loop.
my($max,$ind) = (0,0);
for (1..20000) {
  my $nd = scalar proper_divisors($_);
 ($max,$ind) = ($nd,$_) if $nd > $max;
}
say "$max $ind";
# 2. Or we can use List::Util's max with decoration (this exploits its implementation)
{
  use List::Util qw/max/;
  no warnings 'numeric';
  say max(map { scalar(proper_divisors($_)) . " $_" } 1..20000);
}

{{out}}

1:
2: 1
3: 1
4: 1 2
5: 1
6: 1 2 3
7: 1
8: 1 2 4
9: 1 3
10: 1 2 5
79 15120
79 18480

Note that the first code will choose the first max, while the second chooses the last.

Perl 6

{{Works with|rakudo|2018.10}} Once your threshold is over 1000, the maximum proper divisors will always include 2, 3 and 5 as divisors, so only bother to check multiples of 2, 3 and 5.

There really isn't any point in using concurrency for a limit of 20_000. The setup and bookkeeping drowns out any benefit. Really doesn't start to pay off until the limit is 50_000 and higher. Try swapping in the commented out race map iterator line below for comparison.

sub propdiv (\x) {
    my @l = 1 if x > 1;
    (2 .. x.sqrt.floor).map: -> \d {
        unless x % d { @l.push: d; my \y = x div d; @l.push: y if y != d }
    }
    @l
}

put "$_ [{propdiv($_)}]" for 1..10;

my @candidates;
loop (my int $c = 30; $c <= 20_000; $c += 30) {
#(30, *+30 …^ * > 500_000).race.map: -> $c {
    my \mx = +propdiv($c);
    next if mx < @candidates - 1;
    @candidates[mx].push: $c
}

say "max = {@candidates - 1}, candidates = {@candidates.tail}";

{{out}}

1 []
2 [1]
3 [1]
4 [1 2]
5 [1]
6 [1 2 3]
7 [1]
8 [1 2 4]
9 [1 3]
10 [1 2 5]
max = 79, candidates = 15120 18480

Phix

The factors routine is an auto-include. The actual implementation of it, from builtins\pfactors.e is

global function factors(atom n, integer include1=0)
-- returns a list of all integer factors of n
--  if include1 is 0 (the default), result does not contain either 1 or n
--  if include1 is 1, and n>1, the result contains 1 and n
--  if include1 is -1, and n>1, the result contains 1 but not n
sequence lfactors = {}, hfactors = {}
atom hfactor
integer p = 2,
        lim = floor(sqrt(n))

    if n!=1 and include1!=0 then
        lfactors = {1}
        if include1=1 then
            hfactors = {n}
        end if
    end if
    while p<=lim do
        if remainder(n,p)=0 then
            lfactors = append(lfactors,p)
            hfactor = n/p
            if hfactor=p then exit end if
            hfactors = prepend(hfactors,hfactor)
        end if
        p += 1
    end while
    return lfactors & hfactors
end function

The compiler knows where to find that, so the main program is just:

for i=1 to 10 do
    ?{i,factors(i,-1)}
end for

integer maxd = 0, k
sequence candidates = {}

    for i=1 to 20000 do
        k = length(factors(i,-1))
        if k>=maxd then
            if k=maxd then
                candidates &= i
            else
                candidates = {i}
                maxd = k
            end if
        end if
    end for

    printf(1,"%d divisors:", maxd)
    ?candidates
    {} = wait_key()

{{out}}

{1,{}}
{2,{1}}
{3,{1}}
{4,{1,2}}
{5,{1}}
{6,{1,2,3}}
{7,{1}}
{8,{1,2,4}}
{9,{1,3}}
{10,{1,2,5}}
79 divisors:{15120,18480}

PHP

<?php
function ProperDivisors($n) {
  yield 1;
  $large_divisors = [];
  for ($i = 2; $i <= sqrt($n); $i++) {
    if ($n % $i == 0) {
      yield $i;
      if ($i*$i != $n) {
        $large_divisors[] = $n / $i;
      }
    }
  }
  foreach (array_reverse($large_divisors) as $i) {
    yield $i;
  }
}

assert([1, 2, 4, 5, 10, 20, 25, 50] ==
        iterator_to_array(ProperDivisors(100)));

foreach (range(1, 10) as $n) {
  echo "$n =>";
  foreach (ProperDivisors($n) as $divisor) {
    echo " $divisor";
  }
  echo "\n";
}

$divisorsCount = [];
for ($i = 1; $i < 20000; $i++) {
  $divisorsCount[sizeof(iterator_to_array(ProperDivisors($i)))][] = $i;
}
ksort($divisorsCount);

echo "Numbers with most divisors: ", implode(", ", end($divisorsCount)), ".\n";
echo "They have ", key($divisorsCount), " divisors.\n";

Outputs:

1 => 1
2 => 1
3 => 1
4 => 1 2
5 => 1
6 => 1 2 3
7 => 1
8 => 1 2 4
9 => 1 3
10 => 1 2 5
Numbers with most divisors: 15120, 18480.
They have 79 divisors.

PicoLisp

# Generate all proper divisors.
(de propdiv (N)
   (head -1 (filter
      '((X) (=0 (% N X)))
      (range 1 N) )) )

# Obtaining the values from 1 to 10 inclusive.
(mapcar propdiv (range 1 10))
# Output:
# (NIL (1) (1) (1 2) (1) (1 2 3) (1) (1 2 4) (1 3) (1 2 5))

===Brute-force===

(de propdiv (N)
   (cdr
      (rot
         (make
            (for I N
               (and (=0 (% N I)) (link I)) ) ) ) ) )
(de countdiv (N)
   (let C -1
      (for I N
         (and (=0 (% N I)) (inc 'C)) )
      C ) )
(let F (-5 -8)
   (tab F "N" "LIST")
   (for I 10
      (tab F
         I
         (glue " + " (propdiv I)) ) ) )
(println
   (maxi
      countdiv
      (range 1 20000) ) )

Factorization

(de accu1 (Var Key)
   (if (assoc Key (val Var))
      (con @ (inc (cdr @)))
      (push Var (cons Key 2)) )
   Key )
(de factor (N)
   (let
      (R NIL
         D 2
         L (1 2 2 . (4 2 4 2 4 6 2 6 .))
         M (sqrt N) )
      (while (>= M D)
         (if (=0 (% N D))
            (setq M
               (sqrt (setq N (/ N (accu1 'R D)))) )
            (inc 'D (pop 'L)) ) )
      (accu1 'R N)
      (dec (apply * (mapcar cdr R))) ) )
(bench
   (println
      (maxi
         factor
         (range 1 20000) )
      @@ ) )

Output:

15120 79
0.081 sec

PL/I

*process source xref;
 (subrg):
 cpd: Proc Options(main);
 p9a=time();
 Dcl (p9a,p9b) Pic'(9)9';
 Dcl cnt(3) Bin Fixed(31) Init((3)0);
 Dcl x Bin Fixed(31);
 Dcl pd(300) Bin Fixed(31);
 Dcl sumpd   Bin Fixed(31);
 Dcl npd     Bin Fixed(31);
 Dcl hi      Bin Fixed(31) Init(0);
 Dcl (xl(10),xi) Bin Fixed(31);
 Dcl i       Bin Fixed(31);
 Do x=1 To 10;
   Call proper_divisors(x,pd,npd);
   Put Edit(x,' -> ',(pd(i) Do i=1 To npd))(Skip,f(2),a,10(f(2)));
   End;
 xi=0;
 Do x=1 To 20000;
   Call proper_divisors(x,pd,npd);
   Select;
     When(npd>hi) Do;
       xi=1;
       xl(1)=x;
       hi=npd;
       End;
     When(npd=hi) Do;
       xi+=1;
       xl(xi)=x;
       End;
     Otherwise;
     End;
   End;
 Put Edit(hi,' -> ',(xl(i) Do i=1 To xi))(Skip,f(3),a,10(f(6)));

 x= 166320; Call proper_divisors(x,pd,npd);
 Put Edit(x,' -> ',npd)(Skip,f(8),a,f(4));
 x=1441440; Call proper_divisors(x,pd,npd);
 Put Edit(x,' -> ',npd)(Skip,f(8),a,f(4));


 p9b=time();
 Put Edit((p9b-p9a)/1000,' seconds elapsed')(Skip,f(6,3),a);
 Return;

 proper_divisors: Proc(n,pd,npd);
 Dcl (n,pd(300),npd) Bin Fixed(31);
 Dcl (d,delta)       Bin Fixed(31);
 npd=0;
 If n>1 Then Do;
   If mod(n,2)=1 Then  /* odd number  */
     delta=2;
   Else                /* even number */
     delta=1;
   Do d=1 To n/2 By delta;
     If mod(n,d)=0 Then Do;
       npd+=1;
       pd(npd)=d;
       End;
     End;
   End;
 End;

 End;

{{out}}

 1 ->
 2 ->  1
 3 ->  1
 4 ->  1 2
 5 ->  1
 6 ->  1 2 3
 7 ->  1
 8 ->  1 2 4
 9 ->  1 3
10 ->  1 2 5
 79 ->  15120 18480
  166320 ->  159
 1441440 ->  287
 0.530 seconds elapsed

PowerShell

version 1

function proper-divisor ($n) {
    if($n -ge 2) {
        $lim = [Math]::Floor([Math]::Sqrt($n))
        $less, $greater = @(1), @()
        for($i = 2; $i -lt $lim; $i++){
            if($n%$i -eq 0) {
                $less += @($i)
                $greater = @($n/$i) + $greater
            }
        }
        if(($lim -ne 1) -and ($n%$lim -eq 0)) {$less += @($lim)}
        $($less + $greater)
    } else {@()}
}
"$(proper-divisor 100)"
"$(proper-divisor 496)"
"$(proper-divisor 2048)"

version 2

function proper-divisor ($n) {
    if($n -ge 2) {
        $lim = [Math]::Floor($n/2)+1
        $proper = @(1)
        for($i = 2; $i -lt $lim; $i++){
            if($n%$i -eq 0) {
                $proper += @($i)
            }
        }
        $proper
    } else {@()}
}
"$(proper-divisor 100)"
"$(proper-divisor 496)"
"$(proper-divisor 2048)"

version 3

function eratosthenes ($n) {
    if($n -gt 1){
        $prime = @(0..$n| foreach{$true})
        $m = [Math]::Floor([Math]::Sqrt($n))
        function multiple($i) {
            for($j = $i*$i; $j -le $n; $j += $i) {
                $prime[$j] = $false
            }
        }
        multiple 2
        for($i = 3; $i -le $m; $i += 2) {
            if($prime[$i]) {multiple $i}
        }
        2
        for($i = 3; $i -le $n; $i += 2) {
            if($prime[$i]) {$i}
        }

    } else {
        Write-Error "$n is not greater than 1"
    }
}
function prime-decomposition ($n) {
    $array = eratosthenes $n
    $prime = @()
    foreach($p in $array) {
        while($n%$p -eq 0) {
            $n /= $p
            $prime += @($p)
        }
    }
    $prime
}
function proper-divisor ($n) {
    if($n -ge 2) {
        $array = prime-decomposition $n
        $lim = $array.Count
        function state($res, $i){
            if($i -lt $lim) {
                state ($res) ($i + 1)
                state ($res*$array[$i]) ($i + 1)
            } elseif($res -lt $n) {$res}
        }
        state 1 0 | sort -Unique
    } else {@()}
}
"$(proper-divisor 100)"
"$(proper-divisor 496)"
"$(proper-divisor 2048)"

Output:

1 2 4 5 10 20 25 50
1 2 4 8 16 31 62 124 248
1 2 4 8 16 32 64 128 256 512 1024

Prolog

{{works with | SWI-Prolog 7}}

Taking a cue from [http://stackoverflow.com/a/171779 an SO answer]:

divisor(N, Divisor) :-
    UpperBound is round(sqrt(N)),
    between(1, UpperBound, D),
    0 is N mod D,
    (
        Divisor = D
     ;
        LargerDivisor is N/D,
        LargerDivisor =\= D,
        Divisor = LargerDivisor
    ).

proper_divisor(N, D) :-
    divisor(N, D),
    D =\= N.


%% Task 1
%

proper_divisors(N, Ds) :-
    setof(D, proper_divisor(N, D), Ds).


%% Task 2
%

show_proper_divisors_of_range(Low, High) :-
    findall( N:Ds,
             ( between(Low, High, N),
               proper_divisors(N, Ds) ),
             Results ),
    maplist(writeln, Results).


%% Task 3
%

proper_divisor_count(N, Count) :-
    proper_divisors(N, Ds),
    length(Ds, Count).

find_most_proper_divisors_in_range(Low, High, Result) :-
    aggregate_all( max(Count, N),
                   ( between(Low, High, N),
                     proper_divisor_count(N, Count) ),
                   max(MaxCount, Num) ),
    Result = (num(Num)-divisor_count(MaxCount)).

Output:

?- show_proper_divisors_of_range(1,10).
2:[1]
3:[1]
4:[1,2]
5:[1]
6:[1,2,3]
7:[1]
8:[1,2,4]
9:[1,3]
10:[1,2,5]
true.

?- find_most_proper_divisors_in_range(1,20000,Result).
Result = num(15120)-divisor_count(79).

PureBasic

EnableExplicit

Procedure ListProperDivisors(Number, List Lst())
  If Number < 2 : ProcedureReturn : EndIf
  Protected i
  For i = 1 To Number / 2
    If Number % i = 0
      AddElement(Lst())
      Lst() = i
    EndIf
  Next
EndProcedure

Procedure.i CountProperDivisors(Number)
  If Number < 2 : ProcedureReturn 0 : EndIf
  Protected i, count = 0
  For i = 1 To Number / 2
    If Number % i = 0
      count + 1
    EndIf
  Next
  ProcedureReturn count
EndProcedure

Define n, count, most = 1, maxCount = 0
If OpenConsole()
  PrintN("The proper divisors of the following numbers are : ")
  PrintN("")
  NewList lst()
  For n = 1 To 10
    ListProperDivisors(n, lst())
    Print(RSet(Str(n), 3) + " -> ")
    If ListSize(lst()) = 0
      Print("(None)")
    Else
      ForEach lst()
        Print(Str(lst()) + " ")
      Next
    EndIf
    ClearList(lst())
    PrintN("")
  Next
  For n = 2 To 20000
    count = CountProperDivisors(n)
    If count > maxCount
      maxCount = count
      most = n
    EndIf
  Next
  PrintN("")
  PrintN(Str(most) + " has the most proper divisors, namely " + Str(maxCount))
  PrintN("")
  PrintN("Press any key to close the console")
  Repeat: Delay(10) : Until Inkey() <> ""
  CloseConsole()
EndIf

{{out}}

The proper divisors of the following numbers are :

  1 -> (None)
  2 -> 1
  3 -> 1
  4 -> 1 2
  5 -> 1
  6 -> 1 2 3
  7 -> 1
  8 -> 1 2 4
  9 -> 1 3
 10 -> 1 2 5

15120 has the most proper divisors, namely 79

Python

Python: Literal

A very literal interpretation

 def proper_divs2(n):
...     return {x for x in range(1, (n + 1) // 2 + 1) if n % x == 0 and n != x}
...
>>> [proper_divs2(n) for n in range(1, 11)]
[set(), {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}, {1, 3}, {1, 2, 5}]
>>>
>>> n, length = max(((n, len(proper_divs2(n))) for n in range(1, 20001)), key=lambda pd: pd[1])
>>> n
15120
>>> length
79
>>>

Python: From prime factors

I found [http://stackoverflow.com/a/171784/10562 a reference] on how to generate factors from all the prime factors and the number of times each prime factor goes into N - its multiplicity.

For example, given N having prime factors P(i) with associated multiplicity M(i}) then the factors are given by:

for m[0] in range(M(0) + 1):
    for m[1] in range(M[1] + 1):
        ...
                for m[i - 1] in range(M[i - 1] + 1):
                    mult = 1
                    for j in range(i):
                        mult *= P[j] ** m[j]
                    yield mult

This version is over an order of magnitude faster for generating the proper divisors of the first 20,000 integers; at the expense of simplicity.

from math import sqrt
from functools import lru_cache, reduce
from collections import Counter
from itertools import product


MUL = int.__mul__


def prime_factors(n):
    'Map prime factors to their multiplicity for n'
    d = _divs(n)
    d = [] if d == [n] else (d[:-1] if d[-1] == d else d)
    pf = Counter(d)
    return dict(pf)

@lru_cache(maxsize=None)
def _divs(n):
    'Memoized recursive function returning prime factors of n as a list'
    for i in range(2, int(sqrt(n)+1)):
        d, m  = divmod(n, i)
        if not m:
            return [i] + _divs(d)
    return [n]


def proper_divs(n):
    '''Return the set of proper divisors of n.'''
    pf = prime_factors(n)
    pfactors, occurrences = pf.keys(), pf.values()
    multiplicities = product(*(range(oc + 1) for oc in occurrences))
    divs = {reduce(MUL, (pf**m for pf, m in zip(pfactors, multis)), 1)
            for multis in multiplicities}
    try:
        divs.remove(n)
    except KeyError:
        pass
    return divs or ({1} if n != 1 else set())


if __name__ == '__main__':
    rangemax = 20000

    print([proper_divs(n) for n in range(1, 11)])
    print(*max(((n, len(proper_divs(n))) for n in range(1, 20001)), key=lambda pd: pd[1]))

{{out}}

[set(), {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}, {1, 3}, {1, 2, 5}]
15120 79

R

{{Works with|R|3.3.2 and above}}

# Proper divisors. 12/10/16 aev
require(numbers);
V <- sapply(1:20000, Sigma, k = 0, proper = TRUE); ind <- which(V==max(V));
cat("  *** max number of divisors:", max(V), "\n"," *** for the following indices:",ind, "\n");

{{Output}}

Loading required package: numbers
  *** max number of divisors: 79
  *** for the following indices: 15120 18480

Racket

Short version

#lang racket
(require math)
(define (proper-divisors n) (drop-right (divisors n) 1))
(for ([n (in-range 1 (add1 10))])
  (printf "proper divisors of: ~a\t~a\n" n (proper-divisors n)))
(define most-under-20000
  (for/fold ([best '(1)]) ([n (in-range 2 (add1 20000))])
    (define divs (proper-divisors n))
    (if (< (length (cdr best)) (length divs)) (cons n divs) best)))
(printf "~a has ~a proper divisors\n"
        (car most-under-20000) (length (cdr most-under-20000)))

{{out}}

proper divisors of: 1	()
proper divisors of: 2	(1)
proper divisors of: 3	(1)
proper divisors of: 4	(1 2)
proper divisors of: 5	(1)
proper divisors of: 6	(1 2 3)
proper divisors of: 7	(1)
proper divisors of: 8	(1 2 4)
proper divisors of: 9	(1 3)
proper divisors of: 10	(1 2 5)
15120 has 79 proper divisors

Long version

The '''main''' module will only be executed when this file is executed. When used as a library, it will not be used.

#lang racket/base
(provide fold-divisors ; name as per "Abundant..."
         proper-divisors)

(define (fold-divisors v n k0 kons)
  (define n1 (add1 n))
  (cond
    [(>= n1 (vector-length v))
     (define rv (make-vector n1 k0))
     (for* ([n (in-range 1 n1)] [m (in-range (* 2 n) n1 n)])
       (vector-set! rv m (kons n (vector-ref rv m))))
     rv]
    [else v]))

(define proper-divisors
  (let ([p.d-v (vector)])
    (λ (n)
      (set! p.d-v (reverse (fold-divisors p.d-v n null cons)))
      (vector-ref p.d-v n))))

(module+ main
  (for ([n (in-range 1 (add1 10))])
    (printf "proper divisors of: ~a\t~a\n" n (proper-divisors n)))

  (define count-proper-divisors
    (let ([p.d-v (vector)])
      (λ(n) (set! p.d-v (fold-divisors p.d-v n 0 (λ (d n) (add1 n))))
            (vector-ref p.d-v n))))

  (void (count-proper-divisors 20000))

  (define-values [C I]
    (for*/fold ([C 0] [I 1])
               ([i (in-range 1 (add1 20000))]
                [c (in-value (count-proper-divisors i))]
                #:when [> c C])
      (values c i)))
  (printf "~a has ~a proper divisors\n" I C))

The output is the same as the short version above.

REXX

version 1

Call time 'R'
Do x=1 To 10
  Say x '->' proper_divisors(x)
  End

hi=1
Do x=1 To 20000
  /* If x//1000=0 Then Say x */
  npd=count_proper_divisors(x)
  Select
    When npd>hi Then Do
      list.npd=x
      hi=npd
      End
    When npd=hi Then
      list.hi=list.hi x
    Otherwise
      Nop
    End
  End

Say hi '->' list.hi

Say ' 166320 ->' count_proper_divisors(166320)
Say '1441440 ->' count_proper_divisors(1441440)

Say time('E') 'seconds elapsed'
Exit

proper_divisors: Procedure
Parse Arg n
If n=1 Then Return ''
pd=''
/* Optimization reduces 37 seconds to 28 seconds */
If n//2=1 Then  /* odd number  */
  delta=2
Else            /* even number */
  delta=1
Do d=1 To n%2 By delta
  If n//d=0 Then
    pd=pd d
  End
Return space(pd)

count_proper_divisors: Procedure
Parse Arg n
Return words(proper_divisors(n))

{{out}}

1 ->
2 -> 1
3 -> 1
4 -> 1 2
5 -> 1
6 -> 1 2 3
7 -> 1
8 -> 1 2 4
9 -> 1 3
10 -> 1 2 5
79 -> 15120 18480
 166320 -> 159
1441440 -> 287
28.342000 seconds elapsed

version 2

The following REXX version is an adaptation of the ''optimized'' version for the REXX language example for ''Factors of an integer''.

This REXX version handles all integers (negative, zero, positive) and automatically adjusts the precision (decimal digits).

It also allows the specification of the ranges (for display and for finding the maximum), and allows for extra numbers to be

specified.

With the (function) optimization, it's over '''20''' times faster.

/*REXX program finds proper divisors (and count) of integer ranges; finds the max count.*/
parse arg bot top inc range xtra                 /*obtain optional arguments from the CL*/
if   bot=='' |   bot==","  then    bot=     1    /*Not specified?  Then use the default.*/
if   top=='' |   top==","  then    top=    10    /* "      "         "   "   "     "    */
if   inc=='' |   inc==","  then    inc=     1    /* "      "         "   "   "     "    */
if range=='' | range==","  then  range= 20000    /* "      "         "   "   "     "    */
w= max( length(top), length(bot), length(range)) /*determine the biggest number of these*/
numeric digits max(9, w + 1)                     /*have enough digits for  //  operator.*/
@.= 'and'                                        /*a literal used to separate #s in list*/
      do n=bot  to top  by inc                   /*process the first range specified.   */
      q= Pdivs(n);    #= words(q)                /*get proper divs; get number of Pdivs.*/
      if q=='∞'  then #= q                       /*adjust number of Pdivisors for zero. */
      say right(n, max(20, w) )   'has'   center(#, 4)     "proper divisors: "    q
      end   /*n*/
m=0                                              /*M ≡ maximum number of Pdivs (so far).*/
      do r=1  for range;    q= Pdivs(r)          /*process the second range specified.  */
      #= words(q);          if #<m  then iterate /*get proper divs; get number of Pdivs.*/
      if #<m  then iterate                       /*Less then max?   Then ignore this #. */
      @.#= @.#  @.  r;      m=#                  /*add this Pdiv to max list; set new M.*/
      end   /*r*/                                /* [↑]   process 2nd range of integers.*/
say
say m  ' is the highest number of proper divisors in range 1──►'range,
       ", and it's for: "       subword(@.m, 3)
say                                              /* [↓]  handle any given extra numbers.*/
      do i=1  for words(xtra);  n= word(xtra, i) /*obtain an extra number from XTRA list*/
      w= max(w, 1 + length(n) )                  /*use maximum width for aligned output.*/
      numeric digits max(9, 1 + length(n) )      /*have enough digits for  //  operator.*/
      q= Pdivs(n);              #= words(q)      /*get proper divs; get number of Pdivs.*/
      say  right(n, max(20, w) )     'has'     center(#, 4)      "proper divisors."
      end   /*i*/                                /* [↑] support extra specified integers*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Pdivs: procedure; parse arg x,b;  x= abs(x);   if x==1  then return ''          /*unity?*/
       odd= x // 2;                            if x==0  then return '∞'         /*zero ?*/
       a= 1                                      /* [↓]  use all, or only odd #s.    ___*/
           do j=2+odd  by 1+odd  while j*j < x   /*divide by some integers up to    √ X */
           if x//j==0  then do;  a=a j;  b=x%j b /*if ÷, add both divisors to α & ß.    */
                            end
           end   /*j*/                           /* [↑]  %  is the REXX integer division*/
                                                 /* [↓]  adjust for a square.        ___*/
       if j*j==x  then  return  a j b            /*Was  X  a square?    If so, add  √ X */
                        return  a   b            /*return the divisors  (both lists).   */

{{out|output|text= when using the following input: 0 10 1 20000 166320 1441440 11796480000 }}

                   0 has  ∞   proper divisors:  ∞
                   1 has  0   proper divisors:
                   2 has  1   proper divisors:  1
                   3 has  1   proper divisors:  1
                   4 has  2   proper divisors:  1 2
                   5 has  1   proper divisors:  1
                   6 has  3   proper divisors:  1 2 3
                   7 has  1   proper divisors:  1
                   8 has  3   proper divisors:  1 2 4
                   9 has  2   proper divisors:  1 3
                  10 has  3   proper divisors:  1 2 5

79 is the highest number of proper divisors in range 1──►20000, and it's for:  15120 and 18480

              166320 has 159  proper divisors.
             1441440 has 287  proper divisors.
         11796480000 has 329  proper divisors.

version 3

When factoring 20,000 integers, this REXX version is about '''10%''' faster than the REXX version 2.

When factoring 200,000 integers, this REXX version is about '''30%''' faster.

When factoring 2,000,000 integers, this REXX version is about '''40%''' faster.

When factoring 20,000,000 integers, this REXX version is about '''38%''' faster.

It accomplishes a faster speed by incorporating the calculation of an ''integer square root'' of an integer (without using any floating point arithmetic).

/*REXX program finds proper divisors (and count) of integer ranges; finds the max count.*/
parse arg bot top inc range xtra                 /*obtain optional arguments from the CL*/
if   bot=='' |   bot==","  then    bot=    1     /*Not specified?  Then use the default.*/
if   top=='' |   top==","  then    top=   10     /* "      "         "   "   "     "    */
if   inc=='' |   inc==","  then    inc=    1     /* "      "         "   "   "     "    */
if range=='' | range==","  then  range=20000     /* "      "         "   "   "     "    */
w= max( length(top), length(bot), length(range)) /*determine the biggest number of these*/
numeric digits max(9, w + 1)                     /*have enough digits for  //  operator.*/
@.= 'and'                                        /*a literal used to separate #s in list*/
      do n=bot  to top  by inc                   /*process the first range specified.   */
      q= Pdivs(n);    #= words(q)                /*get proper divs; get number of Pdivs.*/
      if q=='∞'  then #= q                       /*adjust number of Pdivisors for zero. */
      say right(n, max(20, w) )   'has'   center(#, 4)     "proper divisors: "    q
      end   /*n*/
m=0                                              /*M ≡ maximum number of Pdivs (so far).*/
      do r=1  for range;    q= Pdivs(r)          /*process the second range specified.  */
      #= words(q);          if #<m  then iterate /*get proper divs; get number of Pdivs.*/
      if #<m  then iterate                       /*Less then max?   Then ignore this #. */
      @.#= @.#  @.  r;      m=#                  /*add this Pdiv to max list; set new M.*/
      end   /*r*/                                /* [↑]   process 2nd range of integers.*/
say
say m  ' is the highest number of proper divisors in range 1──►'range,
       ", and it's for: "       subword(@.m, 3)
say                                              /* [↓]  handle any given extra numbers.*/
      do i=1  for words(xtra);  n= word(xtra, i) /*obtain an extra number from XTRA list*/
      w= max(w, 1 + length(n) )                  /*use maximum width for aligned output.*/
      numeric digits max(9, 1 + length(n) )      /*have enough digits for  //  operator.*/
      q= Pdivs(n);              #= words(q)      /*get proper divs; get number of Pdivs.*/
      say  right(n, max(20, w) )     'has'     center(#, 4)      "proper divisors."
      end   /*i*/                                /* [↑] support extra specified integers*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Pdivs: procedure; parse arg x 1 z,b;  x= abs(x);   if x==1  then return ''      /*unity?*/
       odd= x // 2;                                if x==0  then return '∞'     /*zero ?*/
       r= 0;         q= 1                        /* [↓] ══integer square root══     ___ */
            do while q<=z; q=q*4; end            /*R:  an integer which will be    √ X  */
            do while q>1;  q=q%4; _= z-r-q;  r=r%2;  if _>=0  then  do;  z=_;  r=r+q;  end
            end   /*while q>1*/                  /* [↑]  compute the integer sqrt of  X.*/
       a=1                                       /* [↓]  use all, or only odd #s.   ___ */
           do j=2 +odd  by 1 +odd to r -(r*r==x) /*divide by some integers up to   √ X  */
           if x//j==0  then do;  a=a j;  b=x%j b /*if ÷, add both divisors to α & ß.    */
                            end
           end   /*j*/                           /* [↑]  %  is the REXX integer division*/
                                                 /* [↓]  adjust for a square.        ___*/
       if j*j==x  then  return  a j b            /*Was  X  a square?    If so, add  √ X */
                        return  a   b            /*return the divisors  (both lists).   */