Amicable pairs: Difference between revisions

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(→‎{{header|C}}: fix a possible integer overflow bug; faster)
(→‎{{header|REXX}}: added version 2.)
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=={{header|REXX}}==
=={{header|REXX}}==
===version 1===
<lang rexx>Call time 'R'
<lang rexx>Call time 'R'
Do x=1 To 20000
Do x=1 To 20000
Line 1,103: Line 1,104:
17296 18416 found after 185.600000 seconds
17296 18416 found after 185.600000 seconds
188.836000 seconds total search time </pre>
188.836000 seconds total search time </pre>

===version 2===
This REXX version allows the specification of the upper limit (for the searching of amicable pairs), &nbsp; some optimization was
<br>incorporated into the &nbsp; ''proper divisor'' &nbsp; subroutine as well as the search itself, &nbsp; the subroutine is a modified version of the
<br>subroutine taken from the REXX language entry for Rosetta code task &nbsp; ''integer factors''.

The generation/summation is about forty times faster, searching is about four times faster.
<lang rexx>/*REXX program finds/displays all amicable pairs up to a given number.*/
parse arg H .; if H=='' then H=20000 /*get optional arg (high limit).*/
w=length(H) ; H.=H || . /*for columnar aligned output. */
@.=0
do k=1 for H; _=Pdivs(k); #=words(_) /*gen proper divs.*/
do i=1 for #; @.k=@.k + word(_,i) /*gen Pdivs sums. */
end /*i*/ /* [↑] sum the proper divisors.*/
end /*k*/ /* [↑] process a range of ints.*/
#=0 /*number of amicable pairs found.*/
do m=220 for H-220+1 /*start search at lowest number. */
do n=m+1 for H-m
if m==@.n then if n==@.m then do; #=#+1 /*bump the counter.*/
say right(m,w) ' and ' right(n,w) " are amicable pairs."
end
end /*p*/
end /*n*/ /*DO loop FORs: faster than TOs.*/
say
say # 'amicable pairs found up to' H. /*display count of amicable pairs*/
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────PDIVS subroutine────────────────────*/
Pdivs: procedure; parse arg x,b; odd=x//2 /* [↑] modified for amicable*/
a=1 /* [↓] use only EVEN|ODD integers*/
do j=2+odd by 1+odd while j*j<x /*divide by all integers up to √x*/
if x//j==0 then do; a=a j; b=x%j b; end /*add divs to α&ß lists if ÷*/
end /*j*/ /* [↑] % is REXX integer divide*/
/* [↓] adjust for square. _ */
if j*j==x then return a j b /*Was X a square? If so, add √x.*/
return a b /*return divisors (both lists). */</lang>
'''output''' &nbsp; when the default input is used:
<pre>
220 and 284 are amicable pairs.
1184 and 1210 are amicable pairs.
2620 and 2924 are amicable pairs.
5020 and 5564 are amicable pairs.
6232 and 6368 are amicable pairs.
10744 and 10856 are amicable pairs.
12285 and 14595 are amicable pairs.
17296 and 18416 are amicable pairs.

8 amicable pairs found up to 20000.
</pre>


=={{header|Ruby}}==
=={{header|Ruby}}==

Revision as of 07:34, 26 January 2015

Task
Amicable pairs
You are encouraged to solve this task according to the task description, using any language you may know.

Two integers and are said to be amicable pairs if and the sum of the proper divisors of () as well as .

For example 1184 and 1210 are an amicable pair (with proper divisors 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 and 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 respectively).

Task

Calculate and show here the Amicable pairs below 20,000; (there are eight).

Cf.

AutoHotkey

<lang d>SetBatchLines -1 Loop, 20000 { m := A_index

; Getting factors loop % floor(sqrt(m)) { if ( mod(m, A_index) = 0 ) { if ( A_index ** 2 == m ) { sum += A_index continue } else if ( A_index != 1 ) { sum += A_index + m//A_index } else if ( A_index = 1 ) { sum += A_index } } } ; Factors obtained

; Checking factors of sum if ( sum > 1 ) { loop % floor(sqrt(sum)) { if ( mod(sum, A_index) = 0 ) { if ( A_index ** 2 == sum ) { sum2 += A_index continue } else if ( A_index != 1 ) { sum2 += A_index + sum//A_index } else if ( A_index = 1 ) { sum2 += A_index } } } if ( m = sum2 ) && ( m != sum ) && ( m < sum ) final .= m . ":" . sum . "`n" } ; Checked

sum := 0 sum2 := 0 } MsgBox % final ExitApp</lang>

Output:
220:284
1184:1210
2620:2924
5020:5564
6232:6368
10744:10856
12285:14595
17296:18416

C

The program will overflow and error in all sorts of ways when given a commandline argument >= UINT_MAX/2 (generally 2^31) <lang c>#include <stdio.h>

  1. include <stdlib.h>

typedef unsigned int uint;

int main(int argc, char **argv) { uint top = atoi(argv[1]); uint *divsum = malloc((top + 1) * sizeof(*divsum)); uint pows[32] = {1, 0};

for (uint i = 0; i <= top; i++) divsum[i] = 1;

// sieve for (uint p = 2; p <= top; p++) { if (divsum[p] > 1) continue; // p not prime

uint x; // highest power of p we need

// checking x <= top/y instead of x*y <= top to avoid overflow for (x = 1; pows[x - 1] <= top/p; x++) pows[x] = p*pows[x - 1];

for (uint n = p; n <= top; n += p) { uint s; for (uint i = s = 1; i < x && !(n%pows[i]); s += pows[i++]); divsum[n] *= s; } }

// subtract number itself from divisor sum ('proper') for (uint i = 0; i <= top; i++) divsum[i] -= i;

for (uint a = 1; a <= top; a++) { uint b = divsum[a]; if (b > a && b <= top && divsum[b] == a) printf("%u %u\n", a, b); }

return 0; }</lang>

Output:
% ./a.out 20000
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416

D

Translation of: Python

<lang d>void main() @safe /*@nogc*/ { 

   import std.stdio, std.algorithm, std.range, std.typecons, std.array;

   immutable properDivs = (in uint n) pure nothrow @safe /*@nogc*/ =>
       iota(1, (n + 1) / 2 + 1).filter!(x => n % x == 0);

   enum rangeMax = 20_000;
   auto n2d = iota(1, rangeMax + 1).map!(n => properDivs(n).sum);

   foreach (immutable n, immutable divSum; n2d.enumerate(1))
       if (n < divSum && divSum <= rangeMax && n2d[divSum - 1] == n)
           writefln("Amicable pair: %d and %d with proper divisors:\n    %s\n    %s",
                    n, divSum, properDivs(n), properDivs(divSum));

}</lang>

Output:
Amicable pair: 220 and 284 with proper divisors:
    [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110]
    [1, 2, 4, 71, 142]
Amicable pair: 1184 and 1210 with proper divisors:
    [1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592]
    [1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605]
Amicable pair: 2620 and 2924 with proper divisors:
    [1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310]
    [1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462]
Amicable pair: 5020 and 5564 with proper divisors:
    [1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510]
    [1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782]
Amicable pair: 6232 and 6368 with proper divisors:
    [1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116]
    [1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184]
Amicable pair: 10744 and 10856 with proper divisors:
    [1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372]
    [1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428]
Amicable pair: 12285 and 14595 with proper divisors:
    [1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095]
    [1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865]
Amicable pair: 17296 and 18416 with proper divisors:
    [1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648]
    [1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]

Haskell

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

main :: IO () main = do 

 let range = [1 .. 20000 :: Int]
     divs = zip range $ map (sum . divisors) range
     pairs = [(n, m) | (n, nd) <- divs, (m, md) <- divs,
              n < m, nd == m, md == n]
 print pairs</lang>
Output:
[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]

J

Proper Divisor implementation:

<lang J>factors=: [: /:~@, */&>@{@((^ i.@>:)&.>/)@q:~&__ properDivisors=: factors -. -.&1</lang>

Amicable pairs:

<lang J> 1+0 20000 #:I.,(</~@i.@#*(*|:))(=/ +/@properDivisors@>) 1+i.20000 

 220   284
1184  1210
2620  2924
5020  5564
6232  6368

10744 10856 12285 14595 17296 18416</lang>

jq

<lang jq># unordered def proper_divisors: 

 . as $n
 | if $n > 1 then 1,
     (sqrt|floor as $s
     | range(2; $s+1) as $i
     | if ($n % $i) == 0 then $i,
          (if $i * $i == $n then empty else ($n / $i) end)

else empty end) 

   else empty
   end;

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

def task(n): 

 (reduce range(0; n+1) as $n
   ( [];  . + [$n | addup(proper_divisors)] )) as $listing
 | range(1;n+1) as $j
 | range(1;$j) as $k
 | if $listing[$j] == $k and $listing[$k] == $j
   then "\($k) and \($j) are amicable"
   else empty
   end ;

task(20000)</lang>

Output:

<lang sh>$ jq -c -n -f amicable_pairs.jq 220 and 284 are amicable 1184 and 1210 are amicable 2620 and 2924 are amicable 5020 and 5564 are amicable 6232 and 6368 are amicable 10744 and 10856 are amicable 12285 and 14595 are amicable 17296 and 18416 are amicable</lang>

Mathematica / Wolfram Language

<lang Mathematica>amicableQ[n_] := 

Module[{sum = Total[Most@Divisors@n]},
 sum != n && n == Total[Most@Divisors@sum]]

Grid@Partition[Cases[Range[4, 20000], _?(amicableQ@# &)], 2]</lang>

Output:


220	284
1184	1210
2620	2924
5020	5564
6232	6368
10744	10856
12285	14595
17296	18416

PARI/GP

<lang parigp>for(x=1,20000,my(y=sigma(x)-x); if(y>x && x == sigma(y)-y,print(x" "y)))</lang>

Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416

Pascal

a "normal" Version. Nearly fast as perl using nTheory. <lang pascal>program AmicablePairs; {$IFDEF FPC} 

  {$MODE DELPHI}
  {$H+}

{$ELSE} 

 {$APPTYPE CONSOLE}

{$ENDIF} uses 

 sysutils;

const 

 MAX = 20000;

//MAX = 20*1000*1000; type 

 tValue = LongWord;
 tpValue = ^tValue;
 tPower = array[0..31] of tValue;
 tIndex = record
            idxI,
            idxS : Uint64;
          end;

var 

 Indices      : array[0..511] of tIndex;
 //primes up to 65536 enough until 2^32
 primes       : array[0..6542] of tValue;

procedure InitPrimes; // sieve of erathosthenes without multiples of 2 type 

 tSieve = array[0..(65536-1) div 2] of char;

var 

 ESieve : ^tSieve;
 idx,i,j,p : LongINt;

Begin 

 new(ESieve);
 fillchar(ESieve^[0],SizeOF(tSieve),#1);
 primes[0] := 2;
 idx := 1;

 //sieving
 j := 1;
 p := 2*j+1;
 repeat
   if Esieve^[j] = #1 then
   begin
     i := (2*j+2)*j;// i := (sqr(p) -1) div 2;
     if i > High(tSieve) then
       BREAK;
     repeat
       ESIeve^[i] := #0;
       inc(i,p);
     until i > High(tSieve);
   end;
   inc(j);
   inc(p,2);
 until j >High(tSieve);

 //collecting
 For i := 1 to High(tSieve) do
   IF Esieve^[i] = #1 then
   Begin
     primes[idx] := 2*i+1;
     inc(idx);
     IF idx>High(primes) then
       BREAK;
   end;
 dispose(Esieve);

end;

procedure Su_append(n,factor:tValue;var su:string); var 

 q,p : tValue;

begin 

 p := 0;
 repeat
   q := n div factor;
   IF q*factor<>n then
     Break;
   inc(p);
   n := q;
 until false;
 IF p > 0 then
   IF p= 1 then
     su:= su+IntToStr(factor)+'*'
   else
     su:= su+IntToStr(factor)+'^'+IntToStr(p)+'*';

end;

procedure ProperDivs(n: Uint64); //output of prime factorization var 

 su : string;
 primNo : tValue;
 p:tValue;

begin 

 str(n:8,su);
 su:= su +' [';
 primNo := 0;
 p := primes[0];
 repeat
   Su_Append(n,p,su);
   inc(primNo);
   p := primes[primNo];
 until (p=0) OR (p*p >= n);
 p := n;
 Su_Append(n,p,su);
 su[length(su)] := ']';
 writeln(su);

end;

procedure AmPairOutput(cnt:tValue); var 

 i : tValue;
 r_max,r_min,r : double;

begin 

 r_max := 1.0;
 r_min := 16.0;
 For i := 0 to cnt-1 do
   with Indices[i] do
   begin
     r := IdxS/IDxI;
     writeln(i+1:4,IdxI:16,IDxS:16,' ratio ',r:10:7);
     IF r < 1 then
     begin
       writeln(i);
       readln;
       halt;
     end;
     if r_max < r then
       r_max := r
     else
       if r_min > r then
         r_min := r;
   IF cnt < 20 then
     begin
       ProperDivs(IdxI);
       ProperDivs(IdxS);
     end;
   end;
 writeln(' min ratio ',r_min:12:10);  writeln(' max ratio ',r_max:12:10);

end;

procedure SumOFProperDiv(n: tValue;var SumOfProperDivs:tValue); // calculated by prime factorization var 

 i,q, primNo, Prime,pot : tValue;
 SumOfDivs: tValue;

begin 

 i := N;
 SumOfDivs := 1;
 primNo := 0;
 Prime := Primes[0];
 q := i DIV Prime;
 repeat
   if q*Prime = i then
   Begin
     pot := 1;
     repeat
       i := q;
       q := i div Prime;
       Pot := Pot * Prime+1;
     until q*Prime <> i;
     SumOfDivs := SumOfDivs * pot;
   end;
   Inc(primNo);
   Prime := Primes[primNo];
   q := i DIV Prime;

   {check if i already prime}
   if Prime > q then
   begin
     prime := i;
     q := 1;
   end;
 until i = 1;
 SumOfProperDivs := SumOfDivs - N;

end;

function Check:tValue; const 

 //going backwards
 DIV23 : array[0..5] of byte =
          //== 5,4,3,2,1,0
              (1,0,0,0,1,0);

var 

 i,s,k,n : tValue;
 idx : nativeInt;

begin 

 n := 0;
 idx := 3;
 For i := 2 to MAX do
 begin
   //must be divisble by 2 or 3 ( n < High(tValue) < 1e14 )
   IF DIV23[idx] = 0 then
   begin
     SumOFProperDiv(i,s);
     //only 24.7...%
     IF s>i then
     Begin
       SumOFProperDiv(s,k);
       IF k = i then
       begin
         With indices[n] do
         begin
           idxI := i;
           idxS := s;
         end;
         inc(n);
       end;
     end;
   end;
   dec(idx);
   IF idx < 0 then
     idx := high(DIV23);
 end;
 result := n;

end;

var 

 T2,T1: TDatetime;
 APcnt: tValue;

begin 

 InitPrimes;
 T1:= time;
 APCnt:= Check;
 T2:= time;
 AmPairOutput(APCnt);
 writeln('Time to find amicable pairs ',FormatDateTime('HH:NN:SS.ZZZ' ,T2-T1));
 {$IFNDEF UNIX} readln;{$ENDIF}

end.</lang> Output 

   1             220             284 ratio  1.2909091
     220 [2^2*5*11*220]
     284 [2^2*284]
   2            1184            1210 ratio  1.0219595
    1184 [2^5*1184]
    1210 [2*5*11^2*1210]
   3            2620            2924 ratio  1.1160305
    2620 [2^2*5*2620]
    2924 [2^2*17*43*2924]
   4            5020            5564 ratio  1.1083665
    5020 [2^2*5*5020]
    5564 [2^2*13*5564]
   5            6232            6368 ratio  1.0218228
    6232 [2^3*19*41*6232]
    6368 [2^5*6368]
   6           10744           10856 ratio  1.0104244
   10744 [2^3*17*79*10744]
   10856 [2^3*23*59*10856]
   7           12285           14595 ratio  1.1880342
   12285 [3^3*5*7*13*12285]
   14595 [3*5*7*14595]
   8           17296           18416 ratio  1.0647549
   17296 [2^4*23*47*17296]
   18416 [2^4*18416]

Alternative

about 25-times faster. This will not output the amicable number unless both! numbers are under the given limit.

So there will be differences to "Table of n, a(n) for n=1..39374" https://oeis.org/A002025/b002025.txt Up to 524'000'000 the pairs found are only correct by number up to no. 437 460122410 and only 442 out of 455 are found, because some pairs exceed the limit. The limits of the ratio between the numbers of the amicable pair up to 1E14 are, based on b002025.txt: 

No.    lower            upper         
31447  52326552030976  52326637800704 ratio  1.0000016 
52326552030976 [2^8*563*6079*59723]
52326637800704 [2^8*797*1439*178223]

38336  92371445691525 154378742017851 ratio  1.6712821
 92371445691525 [3^2*5^2*7^2*11*13^2*23*29^2*233]
154378742017851 [3^2*13^2*53*337*5682671]


The distance check is being corrected, the lower number is now not limited. The used method is not useful for very high limits.

n = p[1]^a[1]*p[2]^a[2]*...p[l]^a[l]

sum of divisors(n) = s(n) = (p[1]^(a[1]+1) -1) / (p[1] -1) * ... * (p[l]^(a[l]+1) -1) / (p[l] -1) with

p[k]^(a[k]+1) -1) / (p[k] -1) = sum (i= [1..a[k]])(p[k]^i)

Using "Sieve of Erathosthenes"-style

<lang pascal>program AmicablePairs; {find amicable pairs in a limited region 2..MAX beware that >both< numbers must be smaller than MAX there are 455 amicable pairs up to 524*1000*1000 correct up to

  1. 437 460122410

} //optimized for freepascal 2.6.4 32-Bit {$IFDEF FPC} 

  {$MODE DELPHI}
  {$OPTIMIZATION ON,peephole,cse,asmcse,regvar}
  {$CODEALIGN loop=1,proc=8}

{$ELSE} 

 {$APPTYPE CONSOLE}

{$ENDIF}

uses 

 sysutils;

const //MAX = 20000; {$IFDEF UNIX} MAX = 524*1000*1000;{$ELSE}MAX = 499*1000*1000;{$ENDIF} type 

 tValue = LongWord;
 tpValue = ^tValue;
 tPower = array[0..31] of tValue;
 tIndex = record
            idxI,
            idxS : tValue;
          end;
 tdpa   = array[0..2] of LongWord;

var 

 power        : tPower;
 PowerFac     : tPower;
 DivSumField  : array[0..MAX] of tValue;
 Indices      : array[0..511] of tIndex;
 DpaCnt       : tdpa;

procedure Init; var 

 i : LongInt;

begin 

 DivSumField[0]:= 0;
 For i := 1 to MAX do
   DivSumField[i]:= 1;

end;

procedure ProperDivs(n: tValue); //Only for output, normally a factorication would do var 

 su,so : string;
 i,q : tValue;

begin 

 su:= '1';
 so:= ;
 i := 2;
 while i*i <= n do
 begin
   q := n div i;
   IF q*i -n = 0 then
   begin
     su:= su+','+IntToStr(i);
     IF q <> i then
       so:= ','+IntToStr(q)+so;
   end;
   inc(i);
 end;
 writeln('  [',su+so,']');

end;

procedure AmPairOutput(cnt:tValue); var 

 i : tValue;
 r : double;

begin 

 r := 1.0;
 For i := 0 to cnt-1 do
 with Indices[i] do
 begin
   writeln(i+1:4,IdxI:12,IDxS:12,' ratio ',IdxS/IDxI:10:7);
   if r < IdxS/IDxI then
     r := IdxS/IDxI;
     IF cnt < 20 then
     begin
       ProperDivs(IdxI);
       ProperDivs(IdxS);
     end;
 end;
 writeln(' max ratio ',r:10:4);

end;

function Check:tValue; var 

 i,s,n : tValue;

begin 

 fillchar(DpaCnt,SizeOf(dpaCnt),#0);
 n := 0;
 For i := 1 to MAX do
 begin
   //s = sum of proper divs (I)  == sum of divs (I) - I
   s := DivSumField[i]-i;
   IF (s <=MAX) AND (s>i) then
   begin
     IF DivSumField[s]-s = i then
     begin
       With indices[n] do
       begin
         idxI := i;
         idxS := s;
       end;
       inc(n);
     end;
   end;
   inc(DpaCnt[Ord(s>=i)-Ord(s<=i)+1]);
 end;
 result := n;

end;

Procedure CalcPotfactor(prim:tValue); //PowerFac[k] = (prim^(k+1)-1)/(prim-1) == Sum (i=1..k) prim^i var 

 k: tValue;
 Pot,       //== prim^k
 PFac : Int64;

begin 

 Pot := prim;
 PFac := 1;
 For k := 0 to High(PowerFac) do
 begin
   PFac := PFac+Pot;
   IF (POT > MAX) then
     BREAK;
   PowerFac[k] := PFac;
   Pot := Pot*prim;
 end;

end;

procedure InitPW(prim:tValue); begin 

 fillchar(power,SizeOf(power),#0);
 CalcPotfactor(prim);

end;

function NextPotCnt(p: tValue):tValue;inline; //return the first power <> 0 //power == n to base prim var 

 i : tValue;

begin 

 result := 0;
 repeat
   i := power[result];
   Inc(i);
   IF i < p then
     BREAK
   else
   begin
     i := 0;
     power[result]  := 0;
     inc(result);
   end;
 until false;
 power[result] := i;

end;

function Sieve(prim: tValue):tValue; //simple version var 

 actNumber : tValue;

begin 

 while prim <= MAX do
 begin
   InitPW(prim);
   //actNumber = actual number = n*prim
   //power == n to base prim
   actNumber := prim;
   while actNumber < MAX do
   begin
     DivSumField[actNumber] := DivSumField[actNumber] *PowerFac[NextPotCnt(prim)];
     inc(actNumber,prim);
   end;
   //next prime
   repeat
     inc(prim);
   until (DivSumField[prim] = 1);
 end;
 result := prim;

end;

var 

 T2,T1,T0: TDatetime;
 APcnt: tValue;

begin 

 T0:= time;
 Init;
 Sieve(2);
 T1:= time;
 APCnt := Check;
 T2:= time;
 AmPairOutput(APCnt);
 writeln(DpaCnt[0]:10,' deficient');
 writeln(DpaCnt[1]:10,' perfect');
 writeln(DpaCnt[2]:10,' abundant');
 writeln(DpaCnt[2]/DpaCnt[0]:14:10,' ratio abundant/deficient ');
 writeln('Time to calc sum of divs    ',FormatDateTime('HH:NN:SS.ZZZ' ,T1-T0));
 writeln('Time to find amicable pairs ',FormatDateTime('HH:NN:SS.ZZZ' ,T2-T1));
 {$IFNDEF UNIX}
   readln;
 {$ENDIF}

end. </lang> output 

       220       284
  [1,2,4,5,10,11,20,22,44,55,110]
  [1,2,4,71,142]

      1184      1210
  [1,2,4,8,16,32,37,74,148,296,592]
  [1,2,5,10,11,22,55,110,121,242,605]

      2620      2924
  [1,2,4,5,10,20,131,262,524,655,1310]
  [1,2,4,17,34,43,68,86,172,731,1462]

      5020      5564
  [1,2,4,5,10,20,251,502,1004,1255,2510]
  [1,2,4,13,26,52,107,214,428,1391,2782]

      6232      6368
  [1,2,4,8,19,38,41,76,82,152,164,328,779,1558,3116]
  [1,2,4,8,16,32,199,398,796,1592,3184]

     10744     10856
  [1,2,4,8,17,34,68,79,136,158,316,632,1343,2686,5372]
  [1,2,4,8,23,46,59,92,118,184,236,472,1357,2714,5428]

     12285     14595
  [1,3,5,7,9,13,15,21,27,35,39,45,63,65,91,105,117,135,189,195,273,315,351,455,585,819,945,1365,1755,2457,4095]
  [1,3,5,7,15,21,35,105,139,417,695,973,2085,2919,4865]

     17296     18416
  [1,2,4,8,16,23,46,47,92,94,184,188,368,376,752,1081,2162,4324,8648]
  [1,2,4,8,16,1151,2302,4604,9208]

8 amicable numbers up to 20000
00:00:00.000

{Win7 nearly nothing else running.
 MAX = 499*1000*1000 

 435   460122410   484817110 ratio  1.0536698
 436   466389344   472453792 ratio  1.0130030
 max ratio     1.3537
 375440837 deficient
         5 perfect
 123559158 abundant
  0.3291042045 ratio abundant/deficient
Time to calc sum of divs    00:00:17.818
Time to find amicable pairs 00:00:04.493

Perl

Not particularly clever, but instant for this example, and does up to 20 million in 11 seconds.

Library: ntheory

<lang perl>use ntheory qw/divisor_sum/; for my $x (1..20000) { 

 my $y = divisor_sum($x)-$x;
 say "$x $y" if $y > $x && $x == divisor_sum($y)-$y;

}</lang>

Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416

Perl 6

<lang perl6>sub propdivsum (\x) { 

   [+] x > 1, gather for 2 .. x.sqrt.floor -> \d {
       my \y = x div d;
       if y * d == x { take d; take y unless y == d }
   }

}

for 1..20000 -> $i { 

   my $j = propdivsum($i);
   say "$i $j" if $j > $i and $i == propdivsum($j);

}</lang>

Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416

PL/I

Translation of: REXX

<lang pli>*process source xref; 

ami: Proc Options(main);
p9a=time();
Dcl (p9a,p9b,p9c) Pic'(9)9';
Dcl sumpd(20000) Bin Fixed(31);
Dcl pd(300) Bin Fixed(31);
Dcl npd     Bin Fixed(31);
Dcl (x,y)   Bin Fixed(31);

Do x=1 To 20000;
  Call proper_divisors(x,pd,npd);
  sumpd(x)=sum(pd,npd);
  End;
p9b=time();
Put Edit('sum(pd) computed in',(p9b-p9a)/1000,' seconds elapsed')
        (Skip,col(7),a,f(6,3),a);

Do x=1 To 20000;
  Do y=x+1 To 20000;
    If y=sumpd(x) &
       x=sumpd(y) Then
      Put Edit(x,y,' found after ',elapsed(),' seconds')
              (Skip,2(f(6)),a,f(6,3),a);
    End;
  End;
Put Edit(elapsed(),' seconds total search time')(Skip,f(6,3),a);

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;

sum: Proc(pd,npd) Returns(Bin Fixed(31));
Dcl (pd(300),npd) Bin Fixed(31);
Dcl sum Bin Fixed(31) Init(0);
Dcl i   Bin Fixed(31);
Do i=1 To npd;
  sum+=pd(i);
  End;
Return(sum);
End;

elapsed: Proc Returns(Dec Fixed(6,3));
p9c=time();
Return((p9c-p9b)/1000);
End;
End;</lang>
Output:
      sum(pd) computed in 0.510 seconds elapsed
   220   284 found after  0.010 seconds
  1184  1210 found after  0.060 seconds
  2620  2924 found after  0.110 seconds
  5020  5564 found after  0.210 seconds
  6232  6368 found after  0.260 seconds
 10744 10856 found after  2.110 seconds
 12285 14595 found after  2.150 seconds
 17296 18416 found after  2.240 seconds
 2.250 seconds total search time

Python

Importing Proper divisors from prime factors: <lang python>from proper_divisors import proper_divs

def amicable(rangemax=20000): 

   n2divsum = {n: sum(proper_divs(n)) for n in range(1, rangemax + 1)}
   for num, divsum in n2divsum.items():
       if num < divsum and divsum <= rangemax and n2divsum[divsum] == num:
           yield num, divsum

if __name__ == '__main__': 

   for num, divsum in amicable():
       print('Amicable pair: %i and %i With proper divisors:\n    %r\n    %r'
             % (num, divsum, sorted(proper_divs(num)), sorted(proper_divs(divsum))))</lang>
Output:
Amicable pair: 220 and 284 With proper divisors:
    [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110]
    [1, 2, 4, 71, 142]
Amicable pair: 1184 and 1210 With proper divisors:
    [1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592]
    [1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605]
Amicable pair: 2620 and 2924 With proper divisors:
    [1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310]
    [1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462]
Amicable pair: 5020 and 5564 With proper divisors:
    [1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510]
    [1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782]
Amicable pair: 6232 and 6368 With proper divisors:
    [1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116]
    [1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184]
Amicable pair: 10744 and 10856 With proper divisors:
    [1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372]
    [1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428]
Amicable pair: 12285 and 14595 With proper divisors:
    [1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095]
    [1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865]
Amicable pair: 17296 and 18416 With proper divisors:
    [1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648]
    [1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]

Racket

With Proper_divisors#Racket in place: <lang racket>#lang racket (require "proper-divisors.rkt") (define SCOPE 20000)

(define P 

 (let ((P-v (vector)))
   (λ (n)
     (set! P-v (fold-divisors P-v n 0 +))
     (vector-ref P-v n))))
returns #f if not an amicable number, amicable pairing otherwise

(define (amicable? n) 

 (define m (P n))
 (define m-sod (P m))
 (and (= m-sod n)
      (< m n) ; each pair exactly once, also eliminates perfect numbers
      m))

(void (amicable? SCOPE)) ; prime the memoisation

(for* ((n (in-range 1 (add1 SCOPE))) 

      (m (in-value (amicable? n)))
      #:when m)
 (printf #<<EOS

amicable pair: ~a, ~a 

 ~a: divisors: ~a
 ~a: divisors: ~a


EOS 

         n m n (proper-divisors n)  m (proper-divisors m)))

</lang>

Output:
amicable pair: 284, 220
  284: divisors: (1 2 4 71 142)
  220: divisors: (1 2 4 5 10 11 20 22 44 55 110)

amicable pair: 1210, 1184
  1210: divisors: (1 2 5 10 11 22 55 110 121 242 605)
  1184: divisors: (1 2 4 8 16 32 37 74 148 296 592)

amicable pair: 2924, 2620
  2924: divisors: (1 2 4 17 34 43 68 86 172 731 1462)
  2620: divisors: (1 2 4 5 10 20 131 262 524 655 1310)

amicable pair: 5564, 5020
  5564: divisors: (1 2 4 13 26 52 107 214 428 1391 2782)
  5020: divisors: (1 2 4 5 10 20 251 502 1004 1255 2510)

amicable pair: 6368, 6232
  6368: divisors: (1 2 4 8 16 32 199 398 796 1592 3184)
  6232: divisors: (1 2 4 8 19 38 41 76 82 152 164 328 779 1558 3116)

amicable pair: 10856, 10744
  10856: divisors: (1 2 4 8 23 46 59 92 118 184 236 472 1357 2714 5428)
  10744: divisors: (1 2 4 8 17 34 68 79 136 158 316 632 1343 2686 5372)

amicable pair: 14595, 12285
  14595: divisors: (1 3 5 7 15 21 35 105 139 417 695 973 2085 2919 4865)
  12285: divisors: (1 3 5 7 9 13 15 21 27 35 39 45 63 65 91 105 117 135 189 195 273 315 351 455 585 819 945 1365 1755 2457 4095)

amicable pair: 18416, 17296
  18416: divisors: (1 2 4 8 16 1151 2302 4604 9208)
  17296: divisors: (1 2 4 8 16 23 46 47 92 94 184 188 368 376 752 1081 2162 4324 8648)

REXX

version 1

<lang rexx>Call time 'R' Do x=1 To 20000 

 pd=proper_divisors(x)
 sumpd.x=sum(pd)
 End

Say 'sum(pd) computed in' time('E') 'seconds' Call time 'R' Do x=1 To 20000 

 /* If x//1000=0 Then Say x time() */
 Do y=x+1 To 20000
   If y=sumpd.x &,
      x=sumpd.y Then
   Say x y 'found after' time('E') 'seconds'
   End
 End

Say time('E') 'seconds total search time' Exit

proper_divisors: Procedure Parse Arg n Pd= If n=1 Then Return 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)

sum: Procedure Parse Arg list sum=0 Do i=1 To words(list) 

 sum=sum+word(list,i)
 End

Return sum</lang>

Output:
sum(pd) computed in 48.502000 seconds
220 284 found after 3.775000 seconds
1184 1210 found after 21.611000 seconds
2620 2924 found after 46.817000 seconds
5020 5564 found after 84.296000 seconds
6232 6368 found after 100.918000 seconds
10744 10856 found after 150.126000 seconds
12285 14595 found after 162.124000 seconds
17296 18416 found after 185.600000 seconds
188.836000 seconds total search time

version 2

This REXX version allows the specification of the upper limit (for the searching of amicable pairs),   some optimization was
incorporated into the   proper divisor   subroutine as well as the search itself,   the subroutine is a modified version of the
subroutine taken from the REXX language entry for Rosetta code task   integer factors.

The generation/summation is about forty times faster, searching is about four times faster. <lang rexx>/*REXX program finds/displays all amicable pairs up to a given number.*/ parse arg H .; if H== then H=20000 /*get optional arg (high limit).*/ w=length(H)  ; H.=H || . /*for columnar aligned output. */ @.=0 

    do   k=1  for H;    _=Pdivs(k);   #=words(_)     /*gen proper divs.*/
      do i=1  for #;    @.k=@.k + word(_,i)          /*gen Pdivs sums. */
      end   /*i*/                     /* [↑]   sum the proper divisors.*/
    end     /*k*/                     /* [↑]   process a range of ints.*/
  1. =0 /*number of amicable pairs found.*/
    do   m=220  for H-220+1           /*start search at lowest number. */
      do n=m+1  for H-m
      if m==@.n  then if n==@.m  then do;  #=#+1    /*bump the counter.*/
                                      say right(m,w) ' and ' right(n,w) " are amicable pairs."
                                      end
      end   /*p*/
    end     /*n*/                     /*DO loop FORs:  faster than TOs.*/

say say # 'amicable pairs found up to' H. /*display count of amicable pairs*/ exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────PDIVS subroutine────────────────────*/ Pdivs: procedure; parse arg x,b; odd=x//2 /* [↑] modified for amicable*/ a=1 /* [↓] use only EVEN|ODD integers*/ 

  do j=2+odd  by 1+odd  while j*j<x   /*divide by all integers up to √x*/
  if x//j==0  then do; a=a j; b=x%j b; end /*add divs to α&ß lists if ÷*/
  end   /*j*/                         /* [↑]  %  is REXX integer divide*/
                                      /* [↓]  adjust for square.     _ */

if j*j==x then return a j b /*Was X a square? If so, add √x.*/ 

                return  a   b         /*return divisors  (both lists). */</lang>

output   when the default input is used: 

  220  and    284  are amicable pairs.
 1184  and   1210  are amicable pairs.
 2620  and   2924  are amicable pairs.
 5020  and   5564  are amicable pairs.
 6232  and   6368  are amicable pairs.
10744  and  10856  are amicable pairs.
12285  and  14595  are amicable pairs.
17296  and  18416  are amicable pairs.

8 amicable pairs found up to 20000.

Ruby

With proper_divisors#Ruby in place: <lang ruby>h = {} (1..20_000).each{|n| h[n] = n.proper_divisors.inject(:+)} h.select{|k,v| h[v] == k && k < v}.each do |key,val| # k<v filters out doubles and perfects 

 puts "#{key} and #{val}"

end </lang>

Output:


220 and 284
1184 and 1210
2620 and 2924
5020 and 5564
6232 and 6368
10744 and 10856
12285 and 14595
17296 and 18416

Swift

<lang Swift>import func Darwin.sqrt

func sqrt(x:Int) -> Int { return Int(sqrt(Double(x))) }

func properDivs(n: Int) -> [Int] { 

   if n == 1 { return [] }
   
   var result = [Int]()
   
   for div in filter (1...sqrt(n), { n % $0 == 0 }) {
       
       result.append(div)

       if n/div != div && n/div != n { result.append(n/div) }
   }
   
   return sorted(result)

}


func sumDivs(n:Int) -> Int { 

   struct Cache { static var sum = [Int:Int]() }
   
   if let sum = Cache.sum[n] { return sum }
   
   let sum = properDivs(n).reduce(0) { $0 + $1 }
   
   Cache.sum[n] = sum
   
   return sum

}

func amicable(n:Int, m:Int) -> Bool { 

   if n == m { return false }
   
   if sumDivs(n) != m || sumDivs(m) != n { return false }
   
   return true

}

var pairs = [(Int, Int)]()

for n in 1 ..< 20_000 { 

   for m in n+1 ... 20_000 {
       if amicable(n, m) {
           pairs.append(n, m)
           println("\(n, m)")
       }
   }

}</lang>

Output:
(220, 284)
(1184, 1210)
(2620, 2924)
(5020, 5564)
(6232, 6368)
(10744, 10856)
(12285, 14595)
(17296, 18416)

Tcl

<lang tcl>proc properDivisors {n} { 

   if {$n == 1} return
   set divs 1
   set sum 1
   for {set i 2} {$i*$i <= $n} {incr i} {

if {!($n % $i)} { lappend divs $i incr sum $i if {$i*$i < $n} { lappend divs [set d [expr {$n / $i}]] incr sum $d } } 

   }
   return [list $sum $divs]

}

proc amicablePairs {limit} { 

   set result {}
   set sums [set divs {{}}]
   for {set n 1} {$n < $limit} {incr n} {

lassign [properDivisors $n] sum d lappend sums $sum lappend divs [lsort -integer $d] 

   }
   for {set n 1} {$n < $limit} {incr n} {

set nsum [lindex $sums $n] for {set m 1} {$m < $n} {incr m} { if {$n==[lindex $sums $m] && $m==$nsum} { lappend result $m $n [lindex $divs $m] [lindex $divs $n] } } 

   }
   return $result

}

foreach {m n md nd} [amicablePairs 20000] { 

   puts "$m and $n are an amicable pair with these proper divisors"
   puts "\t$m : $md"
   puts "\t$n : $nd"

}</lang>

Output:
220 and 284 are an amicable pair with these proper divisors
	220 : 1 2 4 5 10 11 20 22 44 55 110
	284 : 1 2 4 71 142
1184 and 1210 are an amicable pair with these proper divisors
	1184 : 1 2 4 8 16 32 37 74 148 296 592
	1210 : 1 2 5 10 11 22 55 110 121 242 605
2620 and 2924 are an amicable pair with these proper divisors
	2620 : 1 2 4 5 10 20 131 262 524 655 1310
	2924 : 1 2 4 17 34 43 68 86 172 731 1462
5020 and 5564 are an amicable pair with these proper divisors
	5020 : 1 2 4 5 10 20 251 502 1004 1255 2510
	5564 : 1 2 4 13 26 52 107 214 428 1391 2782
6232 and 6368 are an amicable pair with these proper divisors
	6232 : 1 2 4 8 19 38 41 76 82 152 164 328 779 1558 3116
	6368 : 1 2 4 8 16 32 199 398 796 1592 3184
10744 and 10856 are an amicable pair with these proper divisors
	10744 : 1 2 4 8 17 34 68 79 136 158 316 632 1343 2686 5372
	10856 : 1 2 4 8 23 46 59 92 118 184 236 472 1357 2714 5428
12285 and 14595 are an amicable pair with these proper divisors
	12285 : 1 3 5 7 9 13 15 21 27 35 39 45 63 65 91 105 117 135 189 195 273 315 351 455 585 819 945 1365 1755 2457 4095
	14595 : 1 3 5 7 15 21 35 105 139 417 695 973 2085 2919 4865
17296 and 18416 are an amicable pair with these proper divisors
	17296 : 1 2 4 8 16 23 46 47 92 94 184 188 368 376 752 1081 2162 4324 8648
	18416 : 1 2 4 8 16 1151 2302 4604 9208

VBScript

Not at all optimal. :-( <lang VBScript>start = Now Set nlookup = CreateObject("Scripting.Dictionary") Set uniquepair = CreateObject("Scripting.Dictionary")

For i = 1 To 20000 sum = 0 For n = 1 To 20000 If n < i Then If i Mod n = 0 Then sum = sum + n End If End If Next nlookup.Add i,sum Next

For j = 1 To 20000 sum = 0 For m = 1 To 20000 If m < j Then If j Mod m = 0 Then sum = sum + m End If End If Next If nlookup.Exists(sum) And nlookup.Item(sum) = j And j <> sum _ And uniquepair.Exists(sum) = False Then uniquepair.Add j,sum End If Next

For Each key In uniquepair.Keys WScript.Echo key & ":" & uniquepair.Item(key) Next

WScript.Echo "Execution Time: " & DateDiff("s",Start,Now) & " seconds"</lang>

Output:
220:284
1184:1210
2620:2924
5020:5564
6232:6368
10744:10856
12285:14595
17296:18416
Execution Time: 162 seconds

zkl

Slooooow <lang zkl>fcn properDivs(n){ [1.. (n + 1)/2 + 1].filter('wrap(x){ n%x==0 and n!=x }) } const N=20000; sums:=[1..N].pump(T(-1),fcn(n){ properDivs(n).sum(0) }); [0..].zip(sums).filter('wrap([(n,s)]){ (n<s<=N) and sums[s]==n }).println();</lang>

Output:
L(L(220,284),L(1184,1210),L(2620,2924),L(5020,5564),L(6232,6368),L(10744,10856),L(12285,14595),L(17296,18416))
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