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# Amicable pairs

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

Two integers ${\displaystyle N}$ and ${\displaystyle M}$ are said to be amicable pairs if ${\displaystyle N\neq M}$ and the sum of the proper divisors of ${\displaystyle N}$ (${\displaystyle \mathrm {sum} (\mathrm {propDivs} (N))}$) ${\displaystyle =M}$ as well as ${\displaystyle \mathrm {sum} (\mathrm {propDivs} (M))=N}$.

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.

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

## 11l

F sum_proper_divisors(n)   R I n < 2 {0} E sum((1 .. n I/ 2).filter(it -> (@n % it) == 0)) L(n) 1..20000   V m = sum_proper_divisors(n)   I m > n & sum_proper_divisors(m) == n      print(n"\t"m)

	org	100h	;;;	Calculate proper divisors of 2..20000	lxi	h,pdiv + 4	; 2 bytes per entry	lxi	d,19999		; [2 .. 20000] means 19999 entries	lxi	b,1		; Initialize each entry to 1init:	mov	m,c	inx	h	mov	m,b	inx	h	dcx	d	mov	a,d	ora	e	jnz	init		lxi	b,1		; BC = outer loop variableiouter:	inx	b	lxi	h,-10001	; Are we there yet?	dad	b	jc	idone		; If so, we've calculated all of them	mov	h,b	mov	l,c	dad	h	xchg			; DE = inner loop variableiinner:	push	d		; save DE	xchg	dad 	h		; calculate *pdiv[DE]	lxi	d,pdiv	dad	d	mov	e,m		; DE = pdiv[DE]	inx	h	mov	d,m	xchg			; pdiv[DE] += BC	dad 	b	xchg			; store it back	mov	m,d	dcx	h	mov	m,e	pop	h		; restore DE (into HL)	dad	b		; add BC	lxi	d,-20001	; are we there yet?	dad	d	jc	iouter		; then continue with outer loop	lxi	d,20001		; otherwise continue with inner loop	dad	d	xchg	jmp	iinneridone:	lxi	b,1		; BC = outer loop variabletouter:	inx	b	lxi	h,-20001	; Are we there yet?	dad	b	rc			; If so, stop	mov	d,b		; DE = outer loop variable	mov	e,ctinner:	inx	d	lxi	h,-20001	; Are we there yet?	dad	d	jc	touter		; If so continue with outer loop	push	d		; Store the variables	push	b	mov	h,b		; find *pdiv[BC]	mov	l,c	dad 	b 	lxi	b,pdiv	dad	b	mov	a,m		; Compare low byte (to E) 	cmp	e	jnz	tnext1		; Not equal = not amicable	inx	h	mov	a,m	cmp	d		; Compare high byte (to B) 	jnz	tnext1		; Not equal = not amicable	pop	b		; Restore BC	xchg			; find *pdiv[DE]	dad	h	lxi	d,pdiv	dad	d	mov	a,m		; Compare low byte (to C)	cmp	c	jnz	tnext2		; Not equal = not amicable	inx	h	mov	a,m		; Compare high byte (to B) 	cmp 	b	jnz	tnext2		; Not equal = not amicable	pop	d		; Restore DE	push	d		; Save them both on the stack again	push	b	push 	d	mov	h,b		; Print the first number	mov	l,c	call	prhl	pop 	h		; And the second number	call	prhl	lxi	d,nl		; And a newline	mvi	c,9	call 	5tnext1:	pop	b		; Restore Btnext2:	pop	d		; Restore D	jmp	tinner		; Continue	;;;	Print the number in HLprhl:	lxi	d,nbuf		; Store buffer pointer on stack	push 	d	lxi	b,-10		; Divisorpdgt:	lxi	d,-1		; Quotientpdivlp:	inx	d	dad	b	jc	pdivlp	mvi	a,'0'+10	; Make ASCII digit	add	l	pop	h		; Store in output buffer 	dcx	h	mov	m,a	push	h	xchg			; Keep going with rest of number	mov	a,h		; if not zero	ora	l	jnz	pdgt	mvi	c,9		; CP/M call to print string	pop	d		; Get buffer pointer	jmp	5	db	'*****'nbuf:	db	' $'nl: db 13,10,'$'pdiv:	equ	$; base Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ## 8086 Assembly LIMIT: equ 20000 ; Maximum value cpu 8086 org 100hsection .text mov ax,final ; Set DS and ES to point just beyond the mov cl,4 ; program. We're just going to assume MS-DOS shr ax,cl ; gave us enough memory. (Generally the case, inc ax ; a .COM gets a 64K segment and we need ~40K.) mov cx,cs add ax,cx mov ds,ax mov es,axcalc: mov ax,1 ; Calculate proper divisors for 2..20000 mov di,4 ; Initially, set each entry to 1. mov cx,LIMIT-1 ; 2 to 20000 inclusive = 19999 entries rep stosw mov ax,2 ; AX = outer loop counter mov cl,2 mov dx,LIMIT*2 ; Keep inner loop limit ready in DX mov bp,LIMIT/2 ; And outer loop limit in BP.outer: mov bx,ax ; BX = inner loop counter (multiplied by two) shl bx,cl ; Each entry is 2 bytes wide.inner: add [bx],ax ; divsum[BX/2] += AX add bx,ax ; Advance to next entry add bx,ax ; Twice, because each entry is 2 bytes wide cmp bx,dx ; Are we there yet? jbe .inner ; If not, keep going inc ax cmp ax,bp ; Is the outer loop done yet? jbe .outer ; If not, keep goingshow: mov dx,LIMIT ; Keep limit ready in DX mov ax,2 ; AX = outer loop counter mov si,4 ; SI = address for outer loop.outer: mov cx,ax ; CX = inner loop counter inc cx mov di,cx ; DI = address for inner loop shl di,1 mov bx,[si] ; Preload divsum[AX].inner: cmp cx,bx ; CX == divsum[AX]? jne .next ; If not, the pair is not amicable cmp ax,[di] ; AX == divsum[CX]? jne .next ; If not, the pair is not amicable push ax ; Keep the registers push bx push cx push dx push cx ; And CX twice because we need to print it call prax ; Print the first number pop ax call prax ; And the second number mov dx,nl ; And a newline call pstr pop dx ; Restore the registers pop cx pop bx pop ax.next: inc di ; Increment inner loop variable and address inc di ; Address twice because each entry has 2 bytes inc cx cmp cx,dx ; Are we done yet? jbe .inner ; If not, keep going inc si ; Increment outer loop variable and address inc si ; Address twice because each entry has 2 bytes inc ax cmp ax,dx ; Are we done yet? jbe .outer ; If not, keep going. ret ;;; Print the number in AX. Destroys AX, BX, CX, DX.prax: mov cx,10 ; Divisor mov bx,nbuf ; Buffer pointer.digit: xor dx,dx div cx ; Divide by 10 and extract digit add dl,'0' ; Add ASCII 0 to digit dec bx mov [cs:bx],dl ; Store in string test ax,ax ; Any more? jnz .digit ; If so, keep going mov dx,bx ; If not, print the result ;;; Print string from CS.pstr: push ds ; Save DS mov ax,cs ; Set DS to CS mov ds,ax mov ah,9 ; Print string using MS-DOS int 21h pop ds ; Restore DS ret db '*****'nbuf: db '$'nl:	db	13,10,'$'final: equ$
Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416

This solution uses the package Generic_Divisors from the Proper Divisors task [[1]].

with Ada.Text_IO, Generic_Divisors; use Ada.Text_IO; procedure Amicable_Pairs is    function Same(P: Positive) return Positive is (P);    package Divisor_Sum is new Generic_Divisors     (Result_Type => Natural, None => 0, One => Same, Add =>  "+");    Num2 : Integer;begin   for Num1 in 4 .. 20_000 loop      Num2 := Divisor_Sum.Process(Num1);      if Num1 < Num2 then	 if Num1 = Divisor_Sum.Process(Num2) then	   Put_Line(Integer'Image(Num1) & "," & Integer'Image(Num2));	 end if;      end if;   end loop;end Amicable_Pairs;
Output:
 220, 284
1184, 1210
2620, 2924
5020, 5564
6232, 6368
10744, 10856
12285, 14595
17296, 18416


## ALGOL 60

Works with: A60
 begin comment - return p mod q;integer procedure mod(p, q);  value p, q; integer p, q;begin  mod := p - q * entier(p / q);end; comment - return sum of the proper divisors of n;integer procedure sumf(n);  value n; integer n;begin  integer sum, f1, f2;  sum := 1;  f1 := 2;  for f1 := f1 while (f1 * f1) < n do    begin      if mod(n, f1) = 0 then        begin           sum := sum + f1;           f2 := n / f1;           if f2 > f1 then sum := sum + f2;        end;      f1 := f1 + 1;  end;  sumf := sum;end; comment - main program begins here;integer a, b, c, found;outstring(1,"Searching up to 20000 for amicable pairs\n");found := 0;for a := 2 step 1 until 20000 do  begin    b := sumf(a);    if b > a then       begin         c := sumf(b);         if a = c then            begin              found := found + 1;              outinteger(1,a);              outinteger(1,b);              outstring(1,"\n");            end;       end;  end;outinteger(1,found);outstring(1,"pairs were found"); end
Output:
Searching up to 20000 for amicable pairs
220  284
1184  1210
2620  2924
5020  5564
6232  6368
10744  10856
12285  14595
17296  18416
8 pairs were found


## ALGOL 68

# returns the sum of the proper divisors of n                    ## if n = 1, 0 or -1, we return 0                                  #PROC sum proper divisors = ( INT n )INT:     BEGIN         INT result := 0;         INT abs n = ABS n;         IF abs n > 1 THEN             FOR d FROM ENTIER sqrt( abs n ) BY -1 TO 2 DO                 IF abs n MOD d = 0 THEN                     # found another divisor                      #                     result +:= d;                     IF d * d /= n THEN                         # include the other divisor              #                         result +:= n OVER d                     FI                 FI             OD;             # 1 is always a proper divisor of numbers > 1        #             result +:= 1         FI;         result     END # sum proper divisors # ; # construct a table of the sum of the proper divisors of numbers  ## up to 20 000                                                    #INT max number = 20 000;[ 1 : max number ]INT proper divisor sum;FOR n TO UPB proper divisor sum DO proper divisor sum[ n ] := sum proper divisors( n ) OD; # returns TRUE if n1 and n2 are an amicable pair FALSE otherwise  ##         n1 and n2 are amicable if the sum of the proper diviors ##         n1 = n2 and the sum of the proper divisors of n2 = n1   #PROC is an amicable pair = ( INT n1, n2 )BOOL:     ( proper divisor sum[ n1 ] = n2 AND proper divisor sum[ n2 ] = n1 ); # find the amicable pairs up to 20 000                            #FOR p1 TO max number DO    FOR p2 FROM p1 + 1 TO max number DO        IF is an amicable pair( p1, p2 ) THEN            print( ( whole( p1, -6 ), " and ", whole( p2, -6 ), " are a amicable pair", newline ) )        FI    ODOD
Output:
   220 and    284 are a amicable pair
1184 and   1210 are a amicable pair
2620 and   2924 are a amicable pair
5020 and   5564 are a amicable pair
6232 and   6368 are a amicable pair
10744 and  10856 are a amicable pair
12285 and  14595 are a amicable pair
17296 and  18416 are a amicable pair


## ANSI Standard BASIC

Translation of: GFA Basic
100 DECLARE EXTERNAL FUNCTION sum_proper_divisors110 CLEAR120 !130 DIM f(20001)      ! sum of proper factors for each n140 FOR i=1 TO 20000150    LET f(i)=sum_proper_divisors(i)160 NEXT i170 ! look for pairs180 FOR i=1 TO 20000190    FOR j=i+1 TO 20000200       IF f(i)=j AND i=f(j) THEN210          PRINT "Amicable pair ";i;" ";j220       END IF230    NEXT j240 NEXT i250 !260 PRINT270 PRINT "-- found all amicable pairs"280 END290 !300 ! Compute the sum of proper divisors of given number310 !320 EXTERNAL FUNCTION sum_proper_divisors(n)330 !340 IF n>1 THEN ! n must be 2 or larger350    LET sum=1 ! start with 1360    LET root=SQR(n)    ! note that root is an integer370    ! check possible factors, up to sqrt380    FOR i=2 TO root390       IF MOD(n,i)=0 THEN400          LET sum=sum+i     ! i is a factor410          IF i*i<>n THEN    ! check i is not actual square root of n420             LET sum=sum+n/i  ! so n/i will also be a factor430          END IF440       END IF450    NEXT i460 END IF470 LET sum_proper_divisors = sum480 END FUNCTION

## AppleScript

Translation of: JavaScript
-- AMICABLE PAIRS ------------------------------------------------------------ -- amicablePairsUpTo :: Int -> Inton amicablePairsUpTo(max)     -- amicable :: [Int] -> Int -> Int -> [Int] -> [Int]    script amicable        on |λ|(a, m, n, lstSums)            if (m > n) and (m ≤ max) and ((item m of lstSums) = n) then                a & [[n, m]]            else                a            end if        end |λ|    end script     -- divisorsSummed :: Int -> Int    script divisorsSummed        -- sum :: Int -> Int -> Int        script sum            on |λ|(a, b)                a + b            end |λ|        end script         on |λ|(n)            foldl(sum, 0, properDivisors(n))        end |λ|    end script     foldl(amicable, {}, ¬        map(divisorsSummed, enumFromTo(1, max)))end amicablePairsUpTo  -- TEST ----------------------------------------------------------------------on run     amicablePairsUpTo(20000) end run  -- PROPER DIVISORS ----------------------------------------------------------- -- properDivisors :: Int -> [Int]on properDivisors(n)     -- isFactor :: Int -> Bool     script isFactor        on |λ|(x)            n mod x = 0        end |λ|    end script     -- integerQuotient :: Int -> Int    script integerQuotient        on |λ|(x)            (n / x) as integer        end |λ|    end script     if n = 1 then        {1}    else        set realRoot to n ^ (1 / 2)        set intRoot to realRoot as integer        set blnPerfectSquare to intRoot = realRoot         -- Factors up to square root of n,        set lows to filter(isFactor, enumFromTo(1, intRoot))         -- and quotients of these factors beyond the square root,        -- excluding n itself (last item)        items 1 thru -2 of (lows & map(integerQuotient, ¬            items (1 + (blnPerfectSquare as integer)) thru -1 of reverse of lows))    end ifend properDivisors -- GENERIC FUNCTIONS --------------------------------------------------------- -- enumFromTo :: Int -> Int -> [Int]on enumFromTo(m, n)    if m > n then        set d to -1    else        set d to 1    end if    set lst to {}    repeat with i from m to n by d        set end of lst to i    end repeat    return lstend enumFromTo -- filter :: (a -> Bool) -> [a] -> [a]on filter(f, xs)    tell mReturn(f)        set lst to {}        set lng to length of xs        repeat with i from 1 to lng            set v to item i of xs            if |λ|(v, i, xs) then set end of lst to v        end repeat        return lst    end tellend filter -- foldl :: (a -> b -> a) -> a -> [b] -> aon foldl(f, startValue, xs)    tell mReturn(f)        set v to startValue        set lng to length of xs        repeat with i from 1 to lng            set v to |λ|(v, item i of xs, i, xs)        end repeat        return v    end tellend foldl -- map :: (a -> b) -> [a] -> [b]on map(f, xs)    tell mReturn(f)        set lng to length of xs        set lst to {}        repeat with i from 1 to lng            set end of lst to |λ|(item i of xs, i, xs)        end repeat        return lst    end tellend map -- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Scripton mReturn(f)    if class of f is script then        f    else        script            property |λ| : f        end script    end ifend mReturn
Output:
{{220, 284}, {1184, 1210}, {2620, 2924}, {5020, 5564},{6232, 6368}, {10744, 10856}, {12285, 14595}, {17296, 18416}}

## Arturo

properDivs: function [x] ->    (factors x) -- x amicable: function [x][    y: sum properDivs x     if and? x = sum properDivs y             x <> y         -> return @[x,y]    return ø] amicables: [] loop 1..20000 'n [    am: amicable n    if am <> ø         -> 'amicables ++ @[sort am]] print unique amicables
Output:
[220 284] [1184 1210] [2620 2924] [5020 5564] [6232 6368] [10744 10856] [12285 14595] [17296 18416]

 (* ****** ****** *)//#include"share/atspre_staload.hats"#include"share/HATS/atspre_staload_libats_ML.hats"//(* ****** ****** *)//funsum_list_vt  (xs: List_vt(int)): int =(  case+ xs of  | ~list_vt_nil() => 0  | ~list_vt_cons(x, xs) => x + sum_list_vt(xs))//(* ****** ****** *) funpropDivs(  x0: int) : List0_vt(int) =  loop(x0, 2, list_vt_sing(1)) where{//funloop(x0: int, i: int, res: List0_vt(int)) : List0_vt(int) =(if(i * i) > x0then list_vt_reverse(res)else(  if x0 % i != 0    then      loop(x0, i+1, res)    // end of [then]    else let      val res =        cons_vt(i, res)      // end of [val]      val res =      (        if i * i = x0 then res else cons_vt(x0 / i, res)      ) : List0_vt(int) // end of [val]    in      loop(x0, i+1, res)    end // end of [else]  // end of [if])) (* end of [loop] *)//} // end of [propDivs] (* ****** ****** *) funsum_propDivs(x: int): int = sum_list_vt(propDivs(x)) (* ****** ****** *) valtheNat2 = auxmain(2) where{funauxmain( n: int) : stream_vt(int) = $ldelay(stream_vt_cons(n, auxmain(n+1)))} (* ****** ****** *)//valtheAmicable =(stream_vt_takeLte(theNat2, 20000)).filter()(lam x =>let val x2 = sum_propDivs(x)in x < x2 && x = sum_propDivs(x2) end)//(* ****** ****** *) val () =theAmicable.foreach()( lam x => println! ("(", x, ", ", sum_propDivs(x), ")")) (* ****** ****** *) implement main0 () = () (* ****** ****** *)  Output: (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)  ## AutoHotkey SetBatchLines -1Loop, 20000{ m := A_index ; Getting factors loop % floor(sqrt(m)) { if ( mod(m, A_index) = 0 ) { if ( A_index ** 2 == m ) { sum += A_index continue } else if ( A_index != 1 ) { sum += A_index + m//A_index } else if ( A_index = 1 ) { sum += A_index } } } ; Factors obtained ; Checking factors of sum if ( sum > 1 ) { loop % floor(sqrt(sum)) { if ( mod(sum, A_index) = 0 ) { if ( A_index ** 2 == sum ) { sum2 += A_index continue } else if ( A_index != 1 ) { sum2 += A_index + sum//A_index } else if ( A_index = 1 ) { sum2 += A_index } } } if ( m = sum2 ) && ( m != sum ) && ( m < sum ) final .= m . ":" . sum . "n" } ; Checked sum := 0 sum2 := 0}MsgBox % finalExitApp Output: 220:284 1184:1210 2620:2924 5020:5564 6232:6368 10744:10856 12285:14595 17296:18416  ## AWK  #!/bin/awk -ffunction sumprop(num, i,sum,root) {if (num < 2) return 0sum=1root=sqrt(num)for ( i=2; i < root; i++) { if (num % i == 0 ) { sum = sum + i + num/i } } if (num % root == 0) { sum = sum + root } return sum } BEGIN{limit=20000print "Amicable pairs < ",limitfor (n=1; n < limit+1; n++) { m=sumprop(n) if (n == sumprop(m) && n < m) print n,m }}} Output: # ./amicable Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## BCPL get "libhdr" manifest$(    MAXIMUM = 20000$) // Calculate proper divisors for 1..Nlet propDivSums(n) = valof$(  let v = getvec(n)    for i = 1 to n do v!i := 1    for i = 2 to n/2 do    $( let j = i*2 while j < n do$(  v!j := v!j + i            j := j + i        $)$)    resultis v$) // Are A and B an amicable pair, given the list of sums of proper divisors?let amicable(pdiv, a, b) = a = pdiv!b & b = pdiv!a let start() be$(  let pds = propDivSums(MAXIMUM)    for x = 1 to MAXIMUM do        for y = x+1 to MAXIMUM do            if amicable(pds, x, y) do                writef("%N, %N*N", x, y)$) Output: 220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416 ## Befunge [email protected]#-*8*:"2":$_:#!2#*8#g*#6:#0*#!:#-*#<v>*/.55+,1>:28*:*:*%\28*:*:*/06p28*:*:*/\2v %%^:*:<>*v+|!:-1g60/*:*:*82::+**:*:<<>:#**#8:#<*^>.28*^8 ::v>>*:*%/\28*:*:*%+\v>8+#$^#_+#\:#0<:\1/*:*2#<2v^:*82\/*:*:*82:::_v#!%%*:*:*82\/*:*:*82::<_^#<>>06p:28*:*:**1+01-\>1+::28*:*:*/\28*:*:*%:*\!^ Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ## C Remark: Look at Pascal Alternative [[2]].You are using the same principle, so here too both numbers of the pair must be < top. The program will overflow and error in all sorts of ways when given a commandline argument >= UINT_MAX/2 (generally 2^31) #include <stdio.h>#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 // only sieve within lower half , the modification starts at 2*p for (uint p = 2; p+p <= top; p++) { if (divsum[p] > 1) { divsum[p] -= p;// subtract number itself from divisor sum ('proper') 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]; //counter where n is not a*p with a = ?*p, useful for most p. //think of p>31 seldom divisions or p>sqrt(top) than no division is needed //n = 2*p, so the prime itself is left unchanged => k=p-1 uint k= p-1; for (uint n = p+p; n <= top; n += p) { uint s=1+pows[1]; k--; // search the right power only if needed if ( k==0) { for (uint i = 2; i < x && !(n%pows[i]); s += pows[i++]); k = p; } divsum[n] *= s; } } //now correct the upper half for (uint p = (top >> 1)+1; p <= top; p++) { if (divsum[p] > 1){ divsum[p] -= p;} } uint cnt = 0; 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); cnt++;} } printf("\nTop %u count : %u\n",top,cnt); return 0;} Output: % ./a.out 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 Top 20000 count : 8 % ./a.out 524000000 .. 475838415 514823985 491373104 511419856 509379344 523679536 Top 524000000 count : 442 real 0m16.285s user 0m16.156s  ## C# using System;using System.Collections.Generic;using System.Linq; namespace RosettaCode.AmicablePairs{ internal static class Program { private const int Limit = 20000; private static void Main() { foreach (var pair in GetPairs(Limit)) { Console.WriteLine("{0} {1}", pair.Item1, pair.Item2); } } private static IEnumerable<Tuple<int, int>> GetPairs(int max) { List<int> divsums = Enumerable.Range(0, max + 1).Select(i => ProperDivisors(i).Sum()).ToList(); for(int i=1; i<divsums.Count; i++) { int sum = divsums[i]; if(i < sum && sum <= divsums.Count && divsums[sum] == i) { yield return new Tuple<int, int>(i, sum); } } } private static IEnumerable<int> ProperDivisors(int number) { return Enumerable.Range(1, number / 2) .Where(divisor => number % divisor == 0); } }} Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## C++  #include <vector>#include <unordered_map>#include <iostream> int main() { std::vector<int> alreadyDiscovered; std::unordered_map<int, int> divsumMap; int count = 0; for (int N = 1; N <= 20000; ++N) { int divSumN = 0; for (int i = 1; i <= N / 2; ++i) { if (fmod(N, i) == 0) { divSumN += i; } } // populate map of integers to the sum of their proper divisors if (divSumN != 1) // do not include primes divsumMap[N] = divSumN; for (std::unordered_map<int, int>::iterator it = divsumMap.begin(); it != divsumMap.end(); ++it) { int M = it->first; int divSumM = it->second; int divSumN = divsumMap[N]; if (N != M && divSumM == N && divSumN == M) { // do not print duplicate pairs if (std::find(alreadyDiscovered.begin(), alreadyDiscovered.end(), N) != alreadyDiscovered.end()) break; std::cout << "[" << M << ", " << N << "]" << std::endl; alreadyDiscovered.push_back(M); alreadyDiscovered.push_back(N); count++; } } } std::cout << count << " amicable pairs discovered" << std::endl;}  Output: [220, 284] [1184, 1210] [2620, 2924] [5020, 5564] [6232, 6368] [10744, 10856] [12285, 14595] [17296, 18416] 8 amicable pairs discovered  ## Clojure  (ns example (:gen-class)) (defn factors [n] " Find the proper factors of a number " (into (sorted-set) (mapcat (fn [x] (if (= x 1) [x] [x (/ n x)])) (filter #(zero? (rem n %)) (range 1 (inc (Math/sqrt n)))) ))) (def find-pairs (into #{} (for [n (range 2 20000) :let [f (factors n) ; Factors of n M (apply + f) ; Sum of factors g (factors M) ; Factors of sum N (apply + g)] ; Sum of Factors of sum :when (= n N) ; (sum(proDivs(N)) = M and sum(propDivs(M)) = N :when (not= M N)] ; N not-equal M (sorted-set n M)))) ; Found pair ;; Output Results(doseq [q find-pairs] (println q))  Output: #{220 284} #{6232 6368} #{1184 1210} #{5020 5564} #{2620 2924} #{12285 14595} #{17296 18416} #{10744 10856}  ## Common Lisp (let ((cache (make-hash-table))) (defun sum-proper-divisors (n) (or (gethash n cache) (setf (gethash n cache) (loop for x from 1 to (/ n 2) when (zerop (rem n x)) sum x))))) (defun amicable-pairs-up-to (n) (loop for x from 1 to n for sum-divs = (sum-proper-divisors x) when (and (< x sum-divs) (= x (sum-proper-divisors sum-divs))) collect (list x sum-divs))) (amicable-pairs-up-to 20000) Output: ((220 284) (1184 1210) (2620 2924) (5020 5564) (6232 6368) (10744 10856) (12285 14595) (17296 18416)) ## Cowgol include "cowgol.coh"; const LIMIT := 20000; # Calculate sums of proper divisorsvar divSum: uint16[LIMIT + 1];var i: @indexof divSum;var j: @indexof divSum; i := 2;while i <= LIMIT loop divSum[i] := 1; i := i + 1;end loop; i := 2;while i <= LIMIT/2 loop j := i * 2; while j <= LIMIT loop divSum[j] := divSum[j] + i; j := j + i; end loop; i := i + 1;end loop; # Test each pairi := 2;while i <= LIMIT loop j := i + 1; while j <= LIMIT loop if divSum[i] == j and divSum[j] == i then print_i32(i as uint32); print(", "); print_i32(j as uint32); print_nl(); end if; j := j + 1; end loop; i := i + 1;end loop; Output: 220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416 ## Crystal  MX = 524_000_000N = Math.sqrt(MX).to_u32x = Array(Int32).new(MX+1, 1) (2..N).each { |i| p = i*i x[p] += i k = i+i+1 (p+i..MX).step(i) { |j| x[j] += k k += 1 }} (4..MX).each { |m| n = x[m] if n < m && n != 0 && m == x[n] puts "#{n} #{m}" end}  Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ...... ...... ....... ....... 426191535 514780497 475838415 514823985 509379344 523679536  ## D Translation of: Python 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));} 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] ## Delphi See Pascal. ## EchoLisp  ;; using (sum-divisors) from math.lib (lib 'math)(define (amicable N)(define n 0) (for/list ((m (in-range 2 N))) (set! n (sum-divisors m)) #:continue (>= n (* 1.5 m)) ;; assume n/m < 1.5 #:continue (<= n m) ;; prevent perfect numbers #:continue (!= (sum-divisors n) m) (cons m n))) (amicable 20000) → ((220 . 284) (1184 . 1210) (2620 . 2924) (5020 . 5564) (6232 . 6368) (10744 . 10856) (12285 . 14595) (17296 . 18416)) (amicable 1_000_000) ;; 42 pairs → (... (802725 . 863835) (879712 . 901424) (898216 . 980984) (947835 . 1125765) (998104 . 1043096))  ## Ela Translation of: Haskell open monad io number list divisors n = filter ((0 ==) << (n mod)) [1..(n div 2)]range = [1 .. 20000]divs = zip range$ map (sum << divisors) rangepairs = [(n, m) \\ (n, nd) <- divs, (m, md) <- divs | n < m && nd == m && md == n] do putLn pairs ::: IO
Output:
[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]

## Elena

Translation of: C#

ELENA 5.0 :

import extensions;import system'routines; const int N = 20000; extension op{    ProperDivisors        = Range.new(1,self / 2).filterBy:(n => self.mod:n == 0);     get AmicablePairs()    {        var divsums := Range                         .new(0, self + 1)                         .selectBy:(i => i.ProperDivisors.summarize(Integer.new()))                         .toArray();         ^ 1.repeatTill(divsums.Length)             .filterBy:(i)            {                var ii := i;                  var sum := divsums[i];                ^ (i < sum) && (sum < divsums.Length) && (divsums[sum] == i)            }             .selectBy:(i => new { Item1 = i; Item2 = divsums[i]; })    }} public program(){    N.AmicablePairs.forEach:(pair)    {        console.printLine(pair.Item1, " ", pair.Item2)    }}
Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416


### Alternative variant using strong-typed closures

import extensions;import system'routines'stex;import system'collections; const int N = 20000; extension op : IntNumber{    Enumerator<int> ProperDivisors        = new Range(1,self / 2).filterBy:(int n => self.mod:n == 0);     get AmicablePairs()    {        auto divsums := new List<int>(            cast Enumerator<int>(                new Range(0, self).selectBy:(int i => i.ProperDivisors.summarize(0))));         ^ new Range(0, divsums.Length)            .filterBy:(int i)            {                auto sum := divsums[i];                ^ (i < sum) && (sum < divsums.Length) && (divsums[sum] == i)            }            .selectBy:(int i => new Tuple<int,int>(i,divsums[i]));    }} public program(){    N.AmicablePairs.forEach:(var Tuple<int,int> pair)    {        console.printLine(pair.Item1, " ", pair.Item2)    }}

### Alternative variant using yield enumerator

import extensions;import system'routines'stex;import system'collections; const int Limit = 20000; singleton ProperDivisors{    Enumerator<int> function(int number)        = Range.new(1, number / 2).filterBy:(int n => number.mod:n == 0);} public sealed AmicablePairs{    int max;     constructor(int max)    {        this max := max    }     yieldable Tuple<int, int> next()    {        List<int> divsums := Range.new(0, max + 1).selectBy:(int i => ProperDivisors(i).summarize(0));         for (int i := 1, i < divsums.Length, i += 1)        {            int sum := divsums[i];            if(i < sum && sum <= divsums.Length && divsums[sum] == i) {                yield:new Tuple<int, int>(i, sum);            }                    };         ^ nil    }   }  public program(){    auto e := new AmicablePairs(Limit);    for(auto pair := e.next(), pair != nil)    {        console.printLine(pair.Item1, " ", pair.Item2)    }}
Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416


## Elixir

Works with: Elixir version 1.2

With proper_divisors#Elixir in place:

defmodule Proper do  def divisors(1), do: []  def divisors(n), do: [1 | divisors(2,n,:math.sqrt(n))] |> Enum.sort   defp divisors(k,_n,q) when k>q, do: []  defp divisors(k,n,q) when rem(n,k)>0, do: divisors(k+1,n,q)  defp divisors(k,n,q) when k * k == n, do: [k | divisors(k+1,n,q)]  defp divisors(k,n,q)                , do: [k,div(n,k) | divisors(k+1,n,q)]end map = Map.new(1..20000, fn n -> {n, Proper.divisors(n) |> Enum.sum} end)Enum.filter(map, fn {n,sum} -> map[sum] == n and n < sum end)|> Enum.sort|> Enum.each(fn {i,j} -> IO.puts "#{i} and #{j}" end)
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


## Erlang

### Erlang, slow

Very slow solution. Same functions by and large as in proper divisors and co.

 -module(properdivs).                                                            -export([amicable/1,divs/1,sumdivs/1]). amicable(Limit) -> amicable(Limit,[],3,2).                                          amicable(Limit,List,_Current,Acc) when Acc >= Limit -> List;                       amicable(Limit,List,Current,Acc) when Current =< Acc/2  ->                             amicable(Limit,List,Acc,Acc+1);                                          amicable(Limit,List,Current,Acc) ->                                                    CS = sumdivs(Current),                                                             AS = sumdivs(Acc),                                                                 if                                                                                     CS == Acc andalso AS == Current andalso Acc =/= Current ->                           io:format("A: ~w, B: ~w, ~nL: ~w~w~n",  [Current,Acc,divs(Current),divs(Acc)]),                                                                                          NL = List ++ [{Current,Acc}],                                                      amicable(Limit,NL,Acc+1,Acc+1);                                                  true ->           amicable(Limit,List,Current-1,Acc) end.                             divs(0) -> [];                                                                  divs(1) -> [];                                                                  divs(N) -> lists:sort(divisors(1,N)).                                            divisors(1,N) ->                                                                     [1] ++ divisors(2,N,math:sqrt(N)).                                          divisors(K,_N,Q) when K > Q -> [];                                              divisors(K,N,_Q) when N rem K =/= 0 ->                                              [] ++ divisors(K+1,N,math:sqrt(N));                                         divisors(K,N,_Q) when K * K  == N ->                                                [K] ++ divisors(K+1,N,math:sqrt(N));                                        divisors(K,N,_Q) ->                                                                 [K, N div K] ++ divisors(K+1,N,math:sqrt(N)).                                sumdivs(N) -> lists:sum(divs(N)).                                                
Output:
3> properdivs:amicable(20000).
A: 220, B: 284,
L: [1,2,4,5,10,11,20,22,44,55,110][1,2,4,71,142]
A: 1184, B: 1210,
L: [1,2,4,8,16,32,37,74,148,296,592][1,2,5,10,11,22,55,110,121,242,605]
A: 2620, B: 2924,
L: [1,2,4,5,10,20,131,262,524,655,1310][1,2,4,17,34,43,68,86,172,731,1462]
A: 5020, B: 5564,
L: [1,2,4,5,10,20,251,502,1004,1255,2510][1,2,4,13,26,52,107,214,428,1391,2782]
A: 6232, B: 6368,
L: [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]
A: 10744, B: 10856,
L: [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]
A: 12285, B: 14595,
L: [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]
A: 17296, B: 18416,
L: [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]
[{220,284},
{1184,1210},
{2620,2924},
{5020,5564},
{6232,6368},
{10744,10856},
{12285,14595},
{17296,18416}]


### Erlang, faster

This is lazy AND depends on the fun fact that we're not really identifying pairs. They just happen to order. Probably, this answer is false in some sense. But a good deal faster :) As above with the additional function.

[See the talk section   amicable pairs, out of order   for this Rosetta Code task.]

 friendly(Limit) ->    List = [{X,properdivs:sumdivs(X)} || X <- lists:seq(3,Limit)],    Final = [ X ||         X <- lists:seq(3,Limit),         X == properdivs:sumdivs(proplists:get_value(X,List))         andalso X =/= proplists:get_value(X,List)],    io:format("L: ~w~n", [Final]).                                                     
Output:
45> properdivs:friendly(20000).
L: [220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,14595,17296,18416]
ok


We might answer a challenge by saying:

 friendly(Limit) ->                                                                     List = [{X,properdivs:sumdivs(X)} || X <- lists:seq(3,Limit)],                     Final = [ X || X <- lists:seq(3,Limit), X == properdivs:sumdivs(proplists:get_value(X,List))             andalso X =/= proplists:get_value(X,List)],    findfriendlies(Final,[]).  findfriendlies(List,Acc) when length(List) =< 0 -> Acc;findfriendlies(List,Acc) ->                                                            A = lists:nth(1,List),                                                             AS = sumdivs(A),                                                                   B = lists:nth(2,List),                                                             BS = sumdivs(B),                                                                   if                                                                                     AS == B andalso BS == A ->                                                           {_,BL} = lists:split(2,List),                                                      findfriendlies(BL,Acc++[{A,B}]);                                                 true -> false                                                                  end.                                                                             
Output:
94>  properdivs:friendly(20000).
[{220,284},
{1184,1210},
{2620,2924},
{5020,5564},
{6232,6368},
{10744,10856},
{12285,14595},
{17296,18416}]


In either case, it's a lot faster than the recursion in my first example.

## ERRE

PROGRAM AMICABLE CONST LIMIT=20000 PROCEDURE SUMPROP(NUM->M)  IF NUM<2 THEN M=0 EXIT PROCEDURE  SUM=1  ROOT=SQR(NUM)  FOR I=2 TO ROOT-1 DO     IF (NUM=I*INT(NUM/I)) THEN         SUM=SUM+I+NUM/I     END IF     IF (NUM=ROOT*INT(NUM/ROOT)) THEN         SUM=SUM+ROOT     END IF  END FOR  M=SUMEND PROCEDURE BEGIN  PRINT(CHR$(12);) ! CLS PRINT("Amicable pairs < ";LIMIT) FOR N=1 TO LIMIT DO SUMPROP(N->M1) SUMPROP(M1->M2) IF (N=M2 AND N<M1) THEN PRINT(N,M1) END FOREND PROGRAM Output: Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## F#  [2..20000 - 1]|> List.map (fun n-> n, ([1..n/2] |> List.filter (fun x->n % x = 0) |> List.sum))|> List.map (fun (a,b) ->if a<b then (a,b) else (b,a))|> List.groupBy id|> List.map snd|> List.filter (List.length >> ((=) 2))|> List.map List.head|> List.iter (printfn "%A")  Output: (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)  ## Factor This solution focuses on the language's namesake: factoring code into small words which are subsequently composed to form more powerful — yet just as simple — words. Using this approach, the final word naturally arrives at the solution. This is often referred to as the bottom-up approach, which is a way in which Factor (and other concatenative languages) commonly differs from other languages.  USING: grouping math.primes.factors math.ranges ; : pdivs ( n -- seq ) divisors but-last ;: dsum ( n -- sum ) pdivs sum ;: dsum= ( n m -- ? ) dsum = ;: both-dsum= ( n m -- ? ) [ dsum= ] [ swap dsum= ] 2bi and ;: amicable? ( n m -- ? ) [ both-dsum= ] [ = not ] 2bi and ;: drange ( -- seq ) 2 20000 [a,b) ;: dsums ( -- seq ) drange [ dsum ] map ;: is-am?-seq ( -- seq ) dsums drange [ amicable? ] 2map ;: am-nums ( -- seq ) t is-am?-seq indices ;: am-nums-c ( -- seq ) am-nums [ 2 + ] map ;: am-pairs ( -- seq ) am-nums-c 2 group ;: print-am ( -- ) am-pairs [ >array . ] each ; print-am  Output: { 220 284 } { 1184 1210 } { 2620 2924 } { 5020 5564 } { 6232 6368 } { 10744 10856 } { 12285 14595 } { 17296 18416 }  ## Fortran This version uses some latter-day facilities such as array assignment that could be replaced by an ordinary DO-loop, as could the FOR ALL statement that for two adds two to every second element, for three adds three to every third, etc. Each FORALL statement applies its DO-given increment to all the selected array elements potentially in any order or even simultaneously. Likewise, the "MODULE" protocol could be abandoned, which would mean that the KNOWNSUM array would have to be declared COMMON for access across routines - or the whole re-written as a single mainline. And if the PARAMETER statements were replaced appropriately, this source could be compiled using Fortran 77. Output: Perfect!! 6 Perfect!! 28 Amicable! 220 284 Perfect!! 496 Amicable! 1184 1210 Amicable! 2620 2924 Amicable! 5020 5564 Amicable! 6232 6368 Perfect!! 8128 Amicable! 10744 10856 Amicable! 12285 14595 Amicable! 17296 18416   MODULE FACTORSTUFF !This protocol evades the need for multiple parameters, or COMMON, or one shapeless main line...Concocted by R.N.McLean, MMXV. INTEGER LOTS,ILIMIT !Some bounds. PARAMETER (ILIMIT = 2147483647) !Computer arithmetic is not with real numbers. PARAMETER (LOTS = 22000) !Nor is computer storage infinite. INTEGER KNOWNSUM(LOTS) !Calculate these once as multiple references are expected. CONTAINS !Assistants. INTEGER FUNCTION SUMF(N) !Sum of the proper divisors of N. INTEGER N !The number in question. INTEGER S,F,F2,INC,BOOST !Assistants. IF (N.LE.LOTS) THEN !If we're within reach, SUMF = KNOWNSUM(N) !The result is to hand. ELSE !Otherwise, some on-the-spot effort ensues.Could use SUMF in place of S, but some compilers have been confused by such usage. S = 1 !1 is always a factor of N, but N is deemed not. F = 1 !Prepare a crude search for factors. INC = 1 !One by plodding one. IF (MOD(N,2) .EQ. 1) INC = 2!Ah, but an odd number cannot have an even number as a divisor. 1 F = F + INC !So half the time we can doubleplod. F2 = F*F !Up to F2 < N rather than F < SQRT(N) and worries over inexact arithmetic. IF (F2 .LT. N) THEN !F2 = N handled below. IF (MOD(N,F) .EQ. 0) THEN !Does F divide N? BOOST = F + N/F !Yes. The divisor and its counterpart. IF (S .GT. ILIMIT - BOOST) GO TO 666 !Would their augmentation cause an overflow? S = S + BOOST !No, so count in the two divisors just discovered. END IF !So much for a divisor discovered. GO TO 1 !Try for another. END IF !So much for the horde. IF (F2 .EQ. N) THEN !Special case: N may be a perfect square, not necessarily of a prime number. IF (S .GT. ILIMIT - F) GO TO 666 !It is. And it too might cause overflow. S = S + F !But if not, count F once only. END IF !All done. SUMF = S !This is the result. END IF !Whichever way obtained, RETURN !Done.Cannot calculate the sum, because it exceeds the integer limit. 666 SUMF = -666 !An expression of dismay that the caller will notice. END FUNCTION SUMF !Alternatively, find the prime factors, and combine them... SUBROUTINE PREPARESUMF !Initialise the KNOWNSUM array.Convert the Sieve of Eratoshenes to have each slot contain the sum of the proper divisors of its slot number.Changes to instead count the number of factors, or prime factors, etc. would be simple enough. INTEGER F !A factor for numbers such as 2F, 3F, 4F, 5F, ... KNOWNSUM(1) = 0 !Proper divisors of N do not include N. KNOWNSUM(2:LOTS) = 1 !So, although 1 is a proper divisor of all N, 1 is excluded for itself. DO F = 2,LOTS/2 !Step through all the possible divisors of numbers not exceeding LOTS. FOR ALL(I = F + F:LOTS:F) KNOWNSUM(I) = KNOWNSUM(I) + F !And augment each corresponding slot. END DO !Different divisors can hit the same slot. For instance, 6 by 2 and also by 3. END SUBROUTINE PREPARESUMF !Could alternatively generate all products of prime numbers. END MODULE FACTORSTUFF !Enough assistants. PROGRAM AMICABLE !Seek N such that SumF(SumF(N)) = N, for N up to 20,000. USE FACTORSTUFF !This should help. INTEGER I,N !Steppers. INTEGER S1,S2 !Sums of factors. CALL PREPARESUMF !Values for every N up to the search limit will be called for at least once.c WRITE (6,66) (I,KNOWNSUM(I), I = 1,48)c 66 FORMAT (10(I3,":",I5,"|")) DO N = 2,20000 !Step through the specified search space. S1 = SUMF(N) !Only even numbers appear in the results, but check every one anyway. IF (S1 .EQ. N) THEN !Catch a tight loop. WRITE (6,*) "Perfect!!",N !Self amicable! Would otherwise appear as Amicable! n,n. ELSE IF (S1 .GT. N) THEN !Look for a pair going upwards only. S2 = SUMF(S1) !Since otherwise each would appear twice. IF (S2.EQ.N) WRITE (6,*) "Amicable!",N,S1 !Aha! END IF !So much for that candidate. END DO !On to the next. END !Done.  ## FreeBASIC ### using Mod  ' FreeBASIC v1.05.0 win64 Function SumProperDivisors(number As Integer) As Integer If number < 2 Then Return 0 Dim sum As Integer = 0 For i As Integer = 1 To number \ 2 If number Mod i = 0 Then sum += i Next Return sumEnd Function Dim As Integer n, fDim As Integer sum(19999) For n = 1 To 19999 sum(n) = SumProperDivisors(n)Next Print "The pairs of amicable numbers below 20,000 are :"Print For n = 1 To 19998 ' f = SumProperDivisors(n) f = sum(n) If f <= n OrElse f < 1 OrElse f > 19999 Then Continue For If f = sum(n) AndAlso n = sum(f) Then Print Using "#####"; n; Print " and "; Using "#####"; sum(n) End IfNext PrintPrint "Press any key to exit the program"SleepEnd  Output: The pairs of amicable numbers below 20,000 are : 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ### using "Sieve of Erathosthenes" style ' version 04-10-2016' compile with: fbc -s console' replaced the function with 2 FOR NEXT loops #Define max 20000 ' test for pairs below max#Define max_1 max -1 Dim As String u_str = String(Len(Str(max))+1,"#")Dim As UInteger n, fDim Shared As UInteger sum(max_1) For n = 2 To max_1 sum(n) = 1Next For n = 2 To max_1 \ 2 For f = n * 2 To max_1 Step n sum(f) += n NextNext PrintPrint Using " The pairs of amicable numbers below" & u_str & ", are :"; maxPrint For n = 1 To max_1 -1 f = Sum(n) If f <= n OrElse f > max Then Continue For If f = sum(n) AndAlso n = sum(f) Then Print Using u_str & " and" & u_str ; n; f End IfNext ' empty keyboard bufferWhile Inkey <> "" : WendPrint : Print : Print " Hit any key to end program"SleepEnd  The pairs of amicable numbers below 20,000 are : 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416 ## Frink This example uses Frink's built-in efficient factorization algorithms. It can work for arbitrarily large numbers.  n = 1seen = new set do{ n = n + 1 if seen.contains[n] next sum = sum[allFactors[n, true, false, false]] if sum != n and sum[allFactors[sum, true, false, false]] == n { println["$n, $sum"] seen.put[sum] }} while n <= 20000  Output: 220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416  ## Futhark  This example does not show the output mentioned in the task description on this page (or a page linked to from here). Please ensure that it meets all task requirements and remove this message. Note that phrases in task descriptions such as "print and display" and "print and show" for example, indicate that (reasonable length) output be a part of a language's solution. This program is much too parallel and manifests all the pairs, which requires a giant amount of memory.  fun divisors(n: int): []int = filter (fn x => n%x == 0) (map (1+) (iota (n/2))) fun amicable((n: int, nd: int), (m: int, md: int)): bool = n < m && nd == m && md == n fun getPair (divs: [upper](int, int)) (flat_i: int): ((int,int), (int,int)) = let i = flat_i / upper let j = flat_i % upper in unsafe (divs[i], divs[j]) fun main(upper: int): [][2]int = let range = map (1+) (iota upper) let divs = zip range (map (fn n => reduce (+) 0 (divisors n)) range) let amicable = filter amicable (map (getPair divs) (iota (upper*upper))) in map (fn (np,mp) => [#1 np, #1 mp]) amicable  ## GFA Basic  OPENW 1CLEARW 1'DIM f%(20001) ! sum of proper factors for each nFOR i%=1 TO 20000 f%(i%)[email protected]_proper_divisors(i%)NEXT i%' look for pairsFOR i%=1 TO 20000 FOR j%=i%+1 TO 20000 IF f%(i%)=j% AND i%=f%(j%) PRINT "Amicable pair ";i%;" ";j% ENDIF NEXT j%NEXT i%'PRINTPRINT "-- found all amicable pairs"~INP(2)CLOSEW 1'' Compute the sum of proper divisors of given number'FUNCTION sum_proper_divisors(n%) LOCAL i%,sum%,root% ' IF n%>1 ! n% must be 2 or larger sum%=1 ! start with 1 root%=SQR(n%) ! note that root% is an integer ' check possible factors, up to sqrt FOR i%=2 TO root% IF n% MOD i%=0 sum%=sum%+i% ! i% is a factor IF i%*i%<>n% ! check i% is not actual square root of n% sum%=sum%+n%/i% ! so n%/i% will also be a factor ENDIF ENDIF NEXT i% ENDIF RETURN sum%ENDFUNC  Output is: Amicable pair: 220 284 Amicable pair: 1184 1210 Amicable pair: 2620 2924 Amicable pair: 5020 5564 Amicable pair: 6232 6368 Amicable pair: 10744 10856 Amicable pair: 12285 14595 Amicable pair: 17296 18416 -- found all amicable pairs  ## Go package main import "fmt" func pfacSum(i int) int { sum := 0 for p := 1; p <= i/2; p++ { if i%p == 0 { sum += p } } return sum} func main() { var a[20000]int for i := 1; i < 20000; i++ { a[i] = pfacSum(i) } fmt.Println("The amicable pairs below 20,000 are:") for n := 2; n < 19999; n++ { m := a[n] if m > n && m < 20000 && n == a[m] { fmt.Printf(" %5d and %5d\n", n, m) } }} Output: The amicable pairs below 20,000 are: 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## 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
Output:
[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]

Or, deriving proper divisors above the square root as cofactors (for better performance)

Output:
$jq -c -n -f amicable_pairs.jq220 and 284 are amicable1184 and 1210 are amicable2620 and 2924 are amicable5020 and 5564 are amicable6232 and 6368 are amicable10744 and 10856 are amicable12285 and 14595 are amicable17296 and 18416 are amicable ## Julia Given factor, it is not necessary to calculate the individual divisors to compute their sum. See Abundant, deficient and perfect number classifications for the details. It is safe to exclude primes from consideration; their proper divisor sum is always 1. This code also uses a minor trick to ensure that none of the numbers identified are above the limit. All numbers in the range are checked for an amicable partner, but the pair is cataloged only when the greater member is reached. using Primes, Printf function pcontrib(p::Int64, a::Int64) n = one(p) pcon = one(p) for i in 1:a n *= p pcon += n end return pconend function divisorsum(n::Int64) dsum = one(n) for (p, a) in factor(n) dsum *= pcontrib(p, a) end dsum -= nend function amicables(L = 2*10^7) acnt = 0 println("Amicable pairs not greater than ", L) for i in 2:L !isprime(i) || continue j = divisorsum(i) j < i && divisorsum(j) == i || continue acnt += 1 println(@sprintf("%4d", acnt), " => ", j, ", ", i) endend amicables()  Output: Amicable pairs not greater than 20000000 1 => 220, 284 2 => 1184, 1210 3 => 2620, 2924 4 => 5020, 5564 5 => 6232, 6368 6 => 10744, 10856 7 => 12285, 14595 8 => 17296, 18416 9 => 66928, 66992 10 => 67095, 71145 11 => 63020, 76084 12 => 69615, 87633 13 => 79750, 88730 14 => 122368, 123152 15 => 100485, 124155 16 => 122265, 139815 17 => 141664, 153176 18 => 142310, 168730 19 => 171856, 176336 20 => 176272, 180848 21 => 196724, 202444 22 => 185368, 203432 23 => 280540, 365084 24 => 308620, 389924 25 => 356408, 399592 26 => 319550, 430402 27 => 437456, 455344 28 => 469028, 486178 29 => 503056, 514736 30 => 522405, 525915 31 => 643336, 652664 32 => 600392, 669688 33 => 609928, 686072 34 => 624184, 691256 35 => 635624, 712216 36 => 667964, 783556 37 => 726104, 796696 38 => 802725, 863835 39 => 879712, 901424 40 => 898216, 980984 41 => 998104, 1043096 42 => 1077890, 1099390 43 => 947835, 1125765 44 => 1154450, 1189150 45 => 1185376, 1286744 46 => 1156870, 1292570 47 => 1280565, 1340235 48 => 1175265, 1438983 49 => 1392368, 1464592 50 => 1328470, 1483850 51 => 1358595, 1486845 52 => 1511930, 1598470 53 => 1466150, 1747930 54 => 1468324, 1749212 55 => 1798875, 1870245 56 => 1669910, 2062570 57 => 2082464, 2090656 58 => 2236570, 2429030 59 => 2723792, 2874064 60 => 2739704, 2928136 61 => 2652728, 2941672 62 => 2802416, 2947216 63 => 2728726, 3077354 64 => 2803580, 3716164 65 => 3276856, 3721544 66 => 3606850, 3892670 67 => 3805264, 4006736 68 => 3786904, 4300136 69 => 4238984, 4314616 70 => 4259750, 4445050 71 => 4246130, 4488910 72 => 4482765, 5120595 73 => 4604776, 5162744 74 => 5459176, 5495264 75 => 5123090, 5504110 76 => 5357625, 5684679 77 => 5232010, 5799542 78 => 5385310, 5812130 79 => 5147032, 5843048 80 => 5730615, 6088905 81 => 4532710, 6135962 82 => 5726072, 6369928 83 => 6329416, 6371384 84 => 6377175, 6680025 85 => 6993610, 7158710 86 => 6955216, 7418864 87 => 7275532, 7471508 88 => 5864660, 7489324 89 => 7489112, 7674088 90 => 7677248, 7684672 91 => 7800544, 7916696 92 => 7850512, 8052488 93 => 7288930, 8221598 94 => 8262136, 8369864 95 => 7577350, 8493050 96 => 9363584, 9437056 97 => 9071685, 9498555 98 => 9199496, 9592504 99 => 8619765, 9627915 100 => 9339704, 9892936 101 => 9660950, 10025290 102 => 8826070, 10043690 103 => 10254970, 10273670 104 => 8666860, 10638356 105 => 9206925, 10791795 106 => 10572550, 10854650 107 => 8754130, 10893230 108 => 10533296, 10949704 109 => 9491625, 10950615 110 => 9478910, 11049730 111 => 10596368, 11199112 112 => 9773505, 11791935 113 => 11498355, 12024045 114 => 10992735, 12070305 115 => 11252648, 12101272 116 => 11545616, 12247504 117 => 11693290, 12361622 118 => 12397552, 13136528 119 => 11173460, 13212076 120 => 11905504, 13337336 121 => 13921528, 13985672 122 => 10634085, 14084763 123 => 12707704, 14236136 124 => 13813150, 14310050 125 => 14311688, 14718712 126 => 15002464, 15334304 127 => 13671735, 15877065 128 => 14443730, 15882670 129 => 16137628, 16150628 130 => 15363832, 16517768 131 => 14654150, 16817050 132 => 15938055, 17308665 133 => 17257695, 17578785 134 => 17908064, 18017056 135 => 14426230, 18087818 136 => 18056312, 18166888 137 => 17041010, 19150222 138 => 18655744, 19154336 139 => 16871582, 19325698 140 => 17844255, 19895265 141 => 17754165, 19985355  ## K  propdivs:{1+&0=x!'1+!x%2} (8,2)#[email protected]&{(x=+/propdivs[a])&~x=a:+/propdivs[x]}' v:1+!20000(220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416)  ## Kotlin // version 1.1 fun sumProperDivisors(n: Int): Int { if (n < 2) return 0 return (1..n / 2).filter{ (n % it) == 0 }.sum()} fun main(args: Array<String>) { val sum = IntArray(20000, { sumProperDivisors(it) } ) println("The pairs of amicable numbers below 20,000 are:\n") for(n in 2..19998) { val m = sum[n] if (m > n && m < 20000 && n == sum[m]) { println(n.toString().padStart(5) + " and " + m.toString().padStart(5)) } }} Output: The pairs of amicable numbers below 20,000 are: 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Lua 0.02 of a second in 16 lines of code. The vital trick is to just set m to the sum of n's proper divisors each time. That way you only have to test the reverse, dividing your run time by half the loop limit (ie. 10,000)! function sumDivs (n) local sum = 1 for d = 2, math.sqrt(n) do if n % d == 0 then sum = sum + d sum = sum + n / d end end return sumend for n = 2, 20000 do m = sumDivs(n) if m > n then if sumDivs(m) == n then print(n, m) end endend Output:  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## MAD  NORMAL MODE IS INTEGER DIMENSION DIVS(20000) PRINT COMMENT$ AMICABLE PAIRS$R CALCULATE SUM OF DIVISORS OF N INTERNAL FUNCTION(N) ENTRY TO DIVSUM. DS = 0 THROUGH SUMMAT, FOR DIVC=1, 1, DIVC.GE.NSUMMAT WHENEVER N/DIVC*DIVC.E.N, DS = DS+DIVC FUNCTION RETURN DS END OF FUNCTION R CALCULATE SUM OF DIVISORS FOR ALL NUMBERS 1..20000 THROUGH MEMO, FOR I=1, 1, I.GE.20000MEMO DIVS(I) = DIVSUM.(I) R FIND ALL MATCHING PAIRS THROUGH CHECK, FOR I=1, 1, I.GE.20000 THROUGH CHECK, FOR J=1, 1, J.GE.ICHECK WHENEVER DIVS(I).E.J .AND. DIVS(J).E.I, 0 PRINT FORMAT AMI,I,J VECTOR VALUES AMI =$I6,I6*$END OF PROGRAM Output: AMICABLE PAIRS 284 220 1210 1184 2924 2620 5564 5020 6368 6232 10856 10744 14595 12285 18416 17296  ## Maple  This example does not show the output mentioned in the task description on this page (or a page linked to from here). Please ensure that it meets all task requirements and remove this message. Note that phrases in task descriptions such as "print and display" and "print and show" for example, indicate that (reasonable length) output be a part of a language's solution.  with(NumberTheory):pairs:=[];for i from 1 to 20000 do for j from i+1 to 20000 do sum1:=SumOfDivisors(j)-j; sum2:=SumOfDivisors(i)-i; if sum1=i and sum2=j and i<>j then pairs:=[op(pairs),[i,j]]; printf("%a", pairs); end if; end do;end do;pairs;  ## Mathematica / Wolfram Language amicableQ[n_] := Module[{sum = Total[[email protected]@n]}, sum != n && n == Total[[email protected]@sum]] [email protected][Cases[Range[4, 20000], _?([email protected]# &)], 2] Output:  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## MATLAB function amicable tic N=2:1:20000; aN=[]; N(isprime(N))=[]; %erase prime numbers I=1; a=N(1); b=sum(pd(a)); while length(N)>1 if a==b %erase perfect numbers; N(N==a)=[]; a=N(1); b=sum(pd(a)); elseif b<a %the first member of an amicable pair is abundant not defective N(N==a)=[]; a=N(1); b=sum(pd(a)); elseif ~ismember(b,N) %the other member was previously erased N(N==a)=[]; a=N(1); b=sum(pd(a)); else c=sum(pd(b)); if a==c aN(I,:)=[I a b]; I=I+1; N(N==b)=[]; else if ~ismember(c,N) %the other member was previously erased N(N==b)=[]; end end N(N==a)=[]; a=N(1); b=sum(pd(a)); clear c end end disp(array2table(aN,'Variablenames',{'N','Amicable1','Amicable2'})) tocend function D=pd(x) K=1:ceil(x/2); D=K(~(rem(x, K)));end Output:  N Amicable1 Amicable2 _ _________ _________ 1 220 284 2 1184 1210 3 2620 2924 4 5020 5564 5 6232 6368 6 10744 10856 7 12285 14595 8 17296 18416 Elapsed time is 8.958720 seconds.  ## Nim Being a novice, I submitted my code to the Nim community for review and received much feedback and advice. They were instrumental in fine-tuning this code for style and readability, I can't thank them enough.  from math import sqrt const N = 524_000_000.int32 proc sumProperDivisors(someNum: int32, chk4less: bool): int32 = result = 1 let maxPD = sqrt(someNum.float).int32 let offset = someNum mod 2 for divNum in countup(2 + offset, maxPD, 1 + offset): if someNum mod divNum == 0: result += divNum + someNum div divNum if chk4less and result >= someNum: return 0 for n in countdown(N, 2): let m = sumProperDivisors(n, true) if m != 0 and n == sumProperDivisors(m, false): echo$n, " ", $m  Output: 523679536 509379344 511419856 491373104 514823985 475838415 ...... ...... ..... ..... 18416 17296 14595 12285 10856 10744 6368 6232 5564 5020 2924 2620 1210 1184 284 220  Total number of pairs is 442, on my machine the code takes ~389 minutes to run. Here's a second version that uses a large amount of memory but runs in 2m32seconds. Again, thanks to the Nim community  from math import sqrt const N = 524_000_000.int32var x = newSeq[int32](N+1) for i in 2..sqrt(N.float).int32: var p = i*i x[p] += i var j = i + i while (p += i; p <= N): j.inc x[p] += j for m in 4..N: let n = x[m] + 1 if n < m and n != 0 and m == x[n] + 1: echo n, " ", m  Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ..... ..... ...... ...... 426191535 514780497 475838415 514823985 509379344 523679536  ## Oberon-2  MODULE AmicablePairs;IMPORT Out;CONST max = 20000; VAR i,j: INTEGER; pd: ARRAY max + 1 OF LONGINT; PROCEDURE ProperDivisorsSum(n: LONGINT): LONGINT;VAR i,sum: LONGINT;BEGIN sum := 0; IF n > 1 THEN INC(sum,1);i := 2; WHILE (i < n) DO IF (n MOD i) = 0 THEN INC(sum,i) END; INC(i) END END; RETURN sumEND ProperDivisorsSum; BEGIN FOR i := 0 TO max DO pd[i] := ProperDivisorsSum(i) END; FOR i := 2 TO max DO FOR j := i + 1 TO max DO IF (pd[i] = j) & (pd[j] = i) THEN Out.Char('[');Out.Int(i,0);Out.Char(',');Out.Int(j,0);Out.Char("]");Out.Ln END END ENDEND AmicablePairs.  Output: [220,284] [1184,1210] [2620,2924] [5020,5564] [6232,6368] [10744,10856] [12285,14595] [17296,18416]  ## Oforth Using properDivs implementation tasks without optimization (calculating proper divisors sum without returning a list for instance) : import: mapping Integer method: properDivs -- [] #[ self swap mod 0 == ] self 2 / seq filter ; : amicables| i j | Array new 20000 loop: i [ i properDivs sum dup ->j i <= if continue then j properDivs sum i <> if continue then [ i, j ] over add ]; Output: amicables . [[220, 284], [1184, 1210], [2620, 2924], [5020, 5564], [6232, 6368], [10744, 10856], [12285, 14595], [17296, 18416]]  ## PARI/GP for(x=1,20000,my(y=sigma(x)-x); if(y>x && x == sigma(y)-y,print(x" "y))) Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ## Pascal ### Direct approach Works with: Turbo Pascal Works with: Free Pascal This version mutates the Sieve of Eratoshenes from striking out factors into summing factors. The Pascal source compiles with Turbo Pascal (7, patched to avoid the zero divide problem for cpu speeds better than ~150MHz) except that the array limit is too large: 15,000 works but does not reach 20,000. The Free Pascal compiler however can handle an array of 20,000 elements. Because the sum of factors of N can exceed N an ad-hoc SumF procedure is provided, thus the search could continue past the table limit, but at a cost in calculation time. Output is Chasing Chains of Sums of Factors of Numbers. Perfect!! 6, Perfect!! 28, Amicable! 220,284, Perfect!! 496, Amicable! 1184,1210, Amicable! 2620,2924, Amicable! 5020,5564, Amicable! 6232,6368, Perfect!! 8128, Amicable! 10744,10856, Amicable! 12285,14595, Sociable: 12496,14288,15472,14536,14264, Sociable: 14316,19116,31704,47616,83328,177792,295488,629072,589786,294896,358336,418904,366556,274924,275444,243760,376736,381028,285778,152990,122410,97946,48976,45946,22976,22744,19916,17716, Amicable! 17296,18416,  Source file:  Program SumOfFactors; uses crt; {Perpetrated by R.N.McLean, December MCMXCV}//{$DEFINE ShowOverflow}{$IFDEF FPC} {$MODE DELPHI}//tested with lots = 524*1000*1000 takes 75 secs generating KnownSum{$ENDIF} var outf: text; const Limit = 2147483647; const lots = 20000; {This should be much bigger, but problems apply.} var KnownSum: array[1..lots] of longint; Function SumF(N: Longint): Longint; var f,f2,s,ulp: longint; Begin if n <= lots then SumF:=KnownSum[N] {Hurrah!} else begin {This is really crude...} s:=1; {1 is always a factor, but N is not.} f:=2; f2:=f*f; while f2 < N do begin if N mod f = 0 then begin {We have a divisor, and its friend.} ulp:=f + (N div f); if s > Limit - ulp then begin SumF:=-666; exit; end; s:=s + ulp; end; f:=f + 1; f2:=f*f; end; if f2 = N then {A perfect square gets its factor in once only.} if s <= Limit - f then s:=s + f else begin SumF:=-667; exit; end; SumF:=s; end; End; var i,j,l,sf,fs: LongInt; const enuff = 666; {Only so much sociability.} var trail: array[0..enuff] of longint; BEGIN ClrScr; WriteLn('Chasing Chains of Sums of Factors of Numbers.'); for i:=1 to lots do KnownSum[i]:=1; {Sigh. KnownSum:=1;} {start summing every divisor } for i:=2 to lots do begin j:=i + i; While j <= lots do {Sigh. For j:=i + i:Lots:i do KnownSum[j]:=KnownSum[j] + i;} begin KnownSum[j]:=KnownSum[j] + i; j:=j + i; end; end; {Enough preparation.} Assign(outf,'Factors.txt'); ReWrite(Outf); WriteLn(Outf,'Chasing Chains of Sums of Factors of Numbers.'); for i:=2 to lots do {Search.} begin l:=0; sf:=SumF(i); while (sf > i) and (l < enuff) do begin l:=l + 1; trail[l]:=sf; sf:=SumF(sf); end; if l >= enuff then writeln('Rope ran out! ',i);{$IFDEF ShowOverflow}     if sf < 0 then writeln('Overflow with ',i);{$ENDIF} if i = sf then {A loop?} begin {Yes. Reveal its members.} trail[0]:=i; {The first.} if l = 0 then write('Perfect!! ') else if l = 1 then write('Amicable! ') else write('Sociable: '); for j:=0 to l do Write(Trail[j],','); WriteLn; if l = 0 then write(outf,'Perfect!! ') else if l = 1 then write(outf,'Amicable! ') else write(outf,'Sociable: '); for j:=0 to l do write(outf,Trail[j],','); WriteLn(outf); end; end; Close (outf); END. ### More expansive. a "normal" Version. Nearly fast as perl using nTheory. program AmicablePairs;{$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 2type tSieve = array[0..(65536-1) div 2] of ansichar;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 factorizationvar 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 factorizationvar 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;{$ENDIF}end. Output  1 220 284 ratio 1.2909091 220 [2^2*5*11*220] 284 [2^2*284] 2 1184 1210 ratio 1.0219595 1184 [2^5*1184] 1210 [2*5*11^2*1210] 3 2620 2924 ratio 1.1160305 2620 [2^2*5*2620] 2924 [2^2*17*43*2924] 4 5020 5564 ratio 1.1083665 5020 [2^2*5*5020] 5564 [2^2*13*5564] 5 6232 6368 ratio 1.0218228 6232 [2^3*19*41*6232] 6368 [2^5*6368] 6 10744 10856 ratio 1.0104244 10744 [2^3*17*79*10744] 10856 [2^3*23*59*10856] 7 12285 14595 ratio 1.1880342 12285 [3^3*5*7*13*12285] 14595 [3*5*7*14595] 8 17296 18416 ratio 1.0647549 17296 [2^4*23*47*17296] 18416 [2^4*18416] ### Alternative about 25-times faster. This will not output the amicable number unless both! numbers are under the given limit. So there will be differences to "Table of n, a(n) for n=1..39374" https://oeis.org/A002025/b002025.txt Up to 524'000'000 the pairs found are only correct by number up to no. 437 460122410 and only 442 out of 455 are found, because some pairs exceed the limit. The limits of the ratio between the numbers of the amicable pair up to 1E14 are, based on b002025.txt: No. lower upper 31447 52326552030976 52326637800704 ratio 1.0000016 52326552030976 [2^8*563*6079*59723] 52326637800704 [2^8*797*1439*178223] 38336 92371445691525 154378742017851 ratio 1.6712821 92371445691525 [3^2*5^2*7^2*11*13^2*23*29^2*233] 154378742017851 [3^2*13^2*53*337*5682671]  The distance check is being corrected, the lower number is now not limited. The used method is not useful for very high limits. n = p[1]^a[1]*p[2]^a[2]*...p[l]^a[l] sum of divisors(n) = s(n) = (p[1]^(a[1]+1) -1) / (p[1] -1) * ... * (p[l]^(a[l]+1) -1) / (p[l] -1) with p[k]^(a[k]+1) -1) / (p[k] -1) = sum (i= [1..a[k]])(p[k]^i) Using "Sieve of Erathosthenes"-style program AmicPair;{find amicable pairs in a limited region 2..MAXbeware that >both< numbers must be smaller than MAXthere are 455 amicable pairs up to 524*1000*1000correct up to#437 460122410}//optimized for freepascal 2.6.4 32-Bit{$IFDEF FPC}   {$MODE DELPHI} {$OPTIMIZATION ON,peephole,cse,asmcse,regvar}   {$CODEALIGN loop=1,proc=8}{$ELSE}  {$APPTYPE CONSOLE}{$ENDIF} uses  sysutils; type  tValue = LongWord;  tpValue = ^tValue;  tDivSum = array[0..0] of tValue;// evil, but dynamic arrays are slower  tpDivSum = ^tDivSum;  tPower = array[0..31] of tValue;  tIndex = record             idxI,             idxS : tValue;           end;var  power,  PowerFac     : tPower;  ds           : array of tValue;  Indices      : array[0..511] of tIndex;  DivSumField  : tpDivSum;  MAX : tValue; 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 dovar  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  n := 0;  For i := 1 to MAX do  begin    //s = sum of proper divs (I)  == sum of divs (I) - I    s := DivSumField^[i];    IF (s <=MAX) AND (s>i) AND (DivSumField^[s]= i)then    begin      With indices[n] do      begin        idxI := i;        idxS := s;      end;      inc(n);    end;  end;  result := n;end; Procedure CalcPotfactor(prim:tValue);//PowerFac[k] = (prim^(k+1)-1)/(prim-1) == Sum (i=0..k) prim^ivar  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;//return the first power <> 0//power == n to base primvar  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; procedure Sieve(prim: tValue);var  actNumber,idx : tValue;begin  //sieve with "small" primes  while prim*prim <= MAX do  begin    InitPW(prim);    Begin      //actNumber = actual number = n*prim      actNumber := prim;      idx := prim;      while actNumber <= MAX do      begin        dec(idx);        IF idx > 0 then          DivSumField^[actNumber] *= PowerFac[0]        else        Begin          DivSumField^[actNumber] *= PowerFac[NextPotCnt(prim)+1];          idx := Prim;        end;        inc(actNumber,prim);      end;    end;    //next prime    repeat      inc(prim);    until DivSumField^[prim]= 1;//(DivSumField[prim] = 1);  end;   //sieve with "big" primes, only one factor is possible  while 2*prim <= MAX do  begin    InitPW(prim);    Begin      actNumber := prim;      idx := PowerFac[0];      while actNumber <= MAX do      begin        DivSumField^[actNumber] *= idx;        inc(actNumber,prim);      end;    end;    repeat      inc(prim);    until DivSumField^[prim]= 1;  end;   For idx := 2 to MAX do    dec(DivSumField^[idx],idx);end; var  T2,T1,T0: TDatetime;  APcnt: tValue;  i: NativeInt;begin  MAX := 20000;  IF  ParamCount > 0 then    MAX := StrToInt(ParamStr(1));  setlength(ds,MAX);  DivSumField := @ds[0];  T0:= time;  For i := 1 to 1 do  Begin    Init;    Sieve(2);  end;  T1:= time;   APCnt := Check;  T2:= time;  AmPairOutput(APCnt);  writeln(APCnt,' amicable pairs til ',MAX);  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));  setlength(ds,0);  {$IFNDEF UNIX} readln; {$ENDIF}end.

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

.... Test with 524*1000*1000 Linux32, FPC 3.0.1, i4330 3.5 Ghz //Win32 swaps first to allocate 2 GB )
440   475838415   514823985 ratio  1.0819303
441   491373104   511419856 ratio  1.0407974
442   509379344   523679536 ratio  1.0280738
max ratio     1.3537
442 amicable pairs til 524000000
Time to calc sum of divs    00:00:12.601
Time to find amicable pairs 00:00:02.557


## Perl

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

Library: ntheory
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)-$y;} Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ## Phix integer n for m=1 to 20000 do n = sum(factors(m,-1)) if m<n and m=sum(factors(n,-1)) then ?{m,n} end if end for  Output: {220,284} {1184,1210} {2620,2924} {5020,5564} {6232,6368} {10744,10856} {12285,14595} {17296,18416}  ## Phixmonti def sumDivs var n 1 var sum n sqrt 2 swap 2 tolist for var d n d mod not if sum d + n d / + var sum endif endfor sumenddef 2 20000 2 tolist for var i i sumDivs var m m i > if m sumDivs i == if i print "\t" print m print nl endif endifendfor nl msec print " s" print ## PHP <?php function sumDivs ($n) {    $sum = 1; for ($d = 2; $d <= sqrt($n); $d++) { if ($n % $d == 0)$sum += $n /$d + $d; } return$sum;} for ($n = 2;$n < 20000; $n++) {$m = sumDivs($n); if ($m > $n) { if (sumDivs($m) == $n) echo$n."&ensp;".$m."<br />"; }} ?> Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ## PicoLisp (de accud (Var Key) (if (assoc Key (val Var)) (con @ (inc (cdr @))) (push Var (cons Key 1)) ) Key )(de **sum (L) (let S 1 (for I (cdr L) (inc 'S (** (car L) I)) ) S ) )(de factor-sum (N) (if (=1 N) 0 (let (R NIL D 2 L (1 2 2 . (4 2 4 2 4 6 2 6 .)) M (sqrt N) N1 N S 1 ) (while (>= M D) (if (=0 (% N1 D)) (setq M (sqrt (setq N1 (/ N1 (accud 'R D)))) ) (inc 'D (pop 'L)) ) ) (accud 'R N1) (for I R (setq S (* S (**sum I))) ) (- S N) ) ) )(bench (for I 20000 (let X (factor-sum I) (and (< I X) (= I (factor-sum X)) (println I X) ) ) ) ) Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 0.101 sec  ## PL/I Translation of: REXX *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; 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 ## PL/M 100H:/* CP/M CALLS */BDOS: PROCEDURE (FN, ARG); DECLARE FN BYTE, ARG ADDRESS; GO TO 5; END BDOS;EXIT: PROCEDURE; CALL BDOS(0,0); END EXIT;PRINT: PROCEDURE (S); DECLARE S ADDRESS; CALL BDOS(9,S); END PRINT; /* PRINT A NUMBER */PRINT$NUMBER: PROCEDURE (N);    DECLARE S (6) BYTE INITIAL ('.....$'); DECLARE (N, P) ADDRESS, C BASED P BYTE; P = .S(5);DIGIT: P = P - 1; C = N MOD 10 + '0'; N = N / 10; IF N > 0 THEN GO TO DIGIT; CALL PRINT(P);END PRINT$NUMBER; /* CALCULATE SUMS OF PROPER DIVISORS */DECLARE DIV$SUM (20$001) ADDRESS;DECLARE (I, J) ADDRESS; DO I=2 TO 20$000; DIV$SUM(I) = 1; END;DO I=2 TO 10$000; DO J = I*2 TO 20$000 BY I;        DIV$SUM(J) = DIV$SUM(J) + I;    END;END; /* TEST EACH PAIR */DO I=2 TO 20$000; DO J=I+1 TO 20$000;        IF DIV$SUM(I)=J AND DIV$SUM(J)=I THEN DO;            CALL PRINT$NUMBER(I); CALL PRINT(.',$');            CALL PRINT$NUMBER(J); CALL PRINT(.(13,10,'$'));        END;    END;END; CALL EXIT;EOF
Output:
220, 284
1184, 1210
2620, 2924
5020, 5564
6232, 6368
10744, 10856
12285, 14595
17296, 18416

## PowerShell

Works with: PowerShell version 2
 function Get-ProperDivisorSum ( [int]$N ) {$Sum = 1    If ( $N -gt 3 ) {$SqrtN = [math]::Sqrt( $N ) ForEach ($Divisor1 in 2..$SqrtN ) {$Divisor2 = $N /$Divisor1            If ( $Divisor2 -is [int] ) {$Sum += $Divisor1 +$Divisor2 }            }        If ( $SqrtN -is [int] ) {$Sum -= $SqrtN } } return$Sum    } function Get-AmicablePairs ( $N = 300 ) { ForEach ($X in 1..$N ) {$Sum = Get-ProperDivisorSum $X If ($Sum -gt $X -and$X -eq ( Get-ProperDivisorSum $Sum ) ) { "$X, $Sum" } } } Get-AmicablePairs 20000  Output: 220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416  ## Prolog Works with: SWI-Prolog 7 With some guidance from other solutions here: 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. assoc_num_divsSum_in_range(Low, High, Assoc) :- findall( Num-DivSum, ( between(Low, High, Num), aggregate_all( sum(D), proper_divisor(Num, D), DivSum )), Pairs ), list_to_assoc(Pairs, Assoc). get_amicable_pair(Assoc, M-N) :- gen_assoc(M, Assoc, N), M < N, get_assoc(N, Assoc, M). amicable_pairs_under_20000(Pairs) :- assoc_num_divsSum_in_range(1,20000, Assoc), findall(P, get_amicable_pair(Assoc, P), Pairs). Output: ?- amicable_pairs_under_20000(R).R = [220-284, 1184-1210, 2620-2924, 5020-5564, 6232-6368, 10744-10856, 12285-14595, 17296-18416]. ## PureBasic  EnableExplicit Procedure.i SumProperDivisors(Number) If Number < 2 : ProcedureReturn 0 : EndIf Protected i, sum = 0 For i = 1 To Number / 2 If Number % i = 0 sum + i EndIf Next ProcedureReturn sumEndProcedure Define n, fDefine Dim sum(19999) If OpenConsole() For n = 1 To 19999 sum(n) = SumProperDivisors(n) Next PrintN("The pairs of amicable numbers below 20,000 are : ") PrintN("") For n = 1 To 19998 f = sum(n) If f <= n Or f < 1 Or f > 19999 : Continue : EndIf If f = sum(n) And n = sum(f) PrintN(RSet(Str(n),5) + " and " + RSet(Str(sum(n)), 5)) EndIf Next PrintN("") PrintN("Press any key to close the console") Repeat: Delay(10) : Until Inkey() <> "" CloseConsole()EndIf  Output: The pairs of amicable numbers below 20,000 are : 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Python Importing Proper divisors from prime factors: 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)))) 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] Or, supplying our own properDivisors function, and defining the harvest in terms of a generic concatMap: '''Amicable pairs''' from itertools import chainfrom math import sqrt # amicablePairsUpTo :: Int -> [(Int, Int)]def amicablePairsUpTo(n): '''List of all amicable pairs of integers below n. ''' sigma = compose(sum)(properDivisors) def amicable(x): y = sigma(x) return [(x, y)] if (x < y and x == sigma(y)) else [] return concatMap(amicable)( enumFromTo(1)(n) ) # TEST ----------------------------------------------------# main :: IO ()def main(): '''Amicable pairs of integers up to 20000''' for x in amicablePairsUpTo(20000): print(x) # GENERIC ------------------------------------------------- # compose (<<<) :: (b -> c) -> (a -> b) -> a -> cdef compose(g): '''Right to left function composition.''' return lambda f: lambda x: g(f(x)) # concatMap :: (a -> [b]) -> [a] -> [b]def concatMap(f): '''A concatenated list or string over which a function f has been mapped. The list monad can be derived by using an (a -> [b]) function which wraps its output in a list (using an empty list to represent computational failure). ''' return lambda xs: (''.join if isinstance(xs, str) else list)( chain.from_iterable(map(f, xs)) ) # enumFromTo :: Int -> Int -> [Int]def enumFromTo(m): '''Enumeration of integer values [m..n]''' def go(n): return list(range(m, 1 + n)) return lambda n: go(n) # properDivisors :: Int -> [Int]def properDivisors(n): '''Positive divisors of n, excluding n itself''' root_ = sqrt(n) intRoot = int(root_) blnSqr = root_ == intRoot lows = [x for x in range(1, 1 + intRoot) if 0 == n % x] return lows + [ n // x for x in reversed( lows[1:-1] if blnSqr else lows[1:] ) ] # MAIN ---if __name__ == '__main__': main() Output: (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416) ## Quackery properdivisors is defined at Proper divisors#Quackery.  [ properdivisors dup size 0 = iff [ drop 0 ] done behead swap witheach + ] is spd ( n --> n ) [ dup dup spd dup spd rot = unrot > and ] is largeamicable ( n --> b ) [ [] swap times [ i^ largeamicable if [ i^ dup spd swap join nested join ] ] ] is amicables ( n --> [ ) 20000 amicables witheach [ witheach [ echo sp ] cr ] Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## R  divisors <- function (n) { Filter( function (m) 0 == n %% m, 1:(n/2) )} table = sapply(1:19999, function (n) sum(divisors(n)) ) for (n in 1:19999) { m = table[n] if ((m > n) && (m < 20000) && (n == table[m])) cat(n, " ", m, "\n")}  Output: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Racket With Proper_divisors#Racket in place: #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 #<<EOSamicable pair: ~a, ~a ~a: divisors: ~a ~a: divisors: ~a EOS n m n (proper-divisors n) m (proper-divisors m)))  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)  ## Raku (formerly Perl 6) Works with: Rakudo version 2019.03.1 sub propdivsum (\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 } } sum @l} (1..20000).race.map: ->$i {    my $j = propdivsum($i);    say "$i$j" if $j >$i and $i == propdivsum($j);}
Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416

## REBOL

;- based on Lua code ;-) sum-of-divisors: func[n /local sum][    sum: 1    ; using to-integer for compatibility with Rebol2    for d 2 (to-integer square-root n) 1 [        if 0 = remainder n d [ sum: n / d + sum + d ]    ]    sum] for n 2 20000 1 [    if n < m: sum-of-divisors n [        if n = sum-of-divisors m [ print [n tab m] ]    ]]
Output:
220      284
1184     1210
2620     2924
5020     5564
6232     6368
10744    10856
12285    14595
17296    18416


## REXX

### version 1, with factoring

 /*REXX*/ Call time 'R'Do x=1 To 20000  pd=proper_divisors(x)  sumpd.x=sum(pd)  EndSay '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  EndSay time('E') 'seconds total search time'Exit proper_divisors: ProcedureParse Arg nPd=''If n=1 Then Return ''If n//2=1 Then  /* odd number  */  delta=2Else            /* even number */  delta=1Do d=1 To n%2 By delta  If n//d=0 Then    pd=pd d  EndReturn space(pd) sum: ProcedureParse Arg listsum=0Do i=1 To words(list)  sum=sum+word(list,i)  EndReturn sum
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, using SIGMA function

This REXX version allows the specification of the upper limit (for the searching of amicable pairs).

Some optimization was incorporated by using a   sigma   function,   which was a re-coded   proper divisors   (Pdivs)   function,
which was taken from the REXX language entry for Rosetta Code task   integer factors.

Other optimizations were incorporated which took advantage of several well-known generalizations about amicable pairs.

The generation/summation is about   5,000%   times faster than the 1st REXX version;   searching is about   10,000%   times faster.

CPU time consumption note:   for every doubling of   H   (the upper limit for searches),   the CPU time consumed triples.

/*REXX program  calculates and displays all   amicable pairs   up to  a given number.   */parse arg H .;  if H=='' | H==","  then H= 20000 /*get optional arguments  (high limit).*/w= length(H) ;  low= 220                         /*W: used for columnar output alignment*/@.=.                                             /* [↑]  LOW is lowest amicable number. */     do k=low  for H-low;     _= sigma(k)        /*generate sigma sums for a range of #s*/     if _>=low  then @.k= _                      /*only keep the pertinent sigma sums.  */     end   /*k*/                                 /* [↑]   process a range of integers.  */#= 0                                             /*number of amicable pairs found so far*/     do   m=low  to  H;       n= @.m             /*start the search at the lowest number*/     if [email protected].n  then do                          /*If equal, might be an amicable number*/                     if m==n  then iterate       /*Is this a perfect number?  Then skip.*/                     #= # + 1                    /*bump the  amicable pair  counter.    */                     say right(m,w)    ' and '     right(n,w)     " are an amicable pair."                     m= n                        /*start   M   (DO index)  from  N.     */                     end     end    /*m*/saysay #    ' amicable pairs found up to '     H    /*display count of the amicable pairs. */exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/sigma: procedure; parse arg x;     od= x // 2    /*use either  EVEN  or  ODD  integers. */       s= 1                                      /*set initial sigma sum to unity.   ___*/               do j=2+od  by 1+od  while  j*j<x  /*divide by all integers up to the √ X */               if x//j==0  then  s= s + j + x%j  /*add the two divisors to the sum.     */               end   /*j*/                       /* [↑]  %  is REXX integer division.   */                                                 /*                                 ___ */       if j*j==x  then  return s + j             /*Was  X  a square?   If so, add  √ X  */                        return s                 /*return (sigma) sum of the divisors.  */
output   when using the default input:
  220  and    284  are an amicable pair.
1184  and   1210  are an amicable pair.
2620  and   2924  are an amicable pair.
5020  and   5564  are an amicable pair.
6232  and   6368  are an amicable pair.
10744  and  10856  are an amicable pair.
12285  and  14595  are an amicable pair.
17296  and  18416  are an amicable pair.

8  amicable pairs found up to  20000


### version 3, SIGMA with limited searches

This REXX version is optimized to take advantage of the lowest ending-single-digit amicable number,   and
also incorporates the search of amicable numbers into the generation of the sigmas of the integers.

The optimization makes it about another   30%   faster when searching for amicable numbers up to one million.

/*REXX program  calculates and displays all  amicable pairs  up to  a given number.     */parse arg H .;   if H=='' | H==","  then H=20000 /*get optional arguments  (high limit).*/w=length(H)  ;   low=220                         /*W: used for columnar output alignment*/x= 220 34765731 6232 87633 284 12285 10856 36939357 6368 5684679          /*S  minimums.*/   do i=0 for 10;   $.i= word(x, i + 1); end /*minimum amicable #s for last dec dig.*/@.= /* [↑] LOW is lowest amicable number. */#= 0 /*number of amicable pairs found so far*/ do k=low for H-low /*generate sigma sums for a range of #s*/ parse var k '' -1 D /*obtain last decimal digit of K. */ if k<$.D    then iterate                    /*if no need to compute, then skip it. */          _= sigma(k)                            /*generate sigma sum for the number  K.*/     @.k= _                                      /*only keep the pertinent sigma sums.  */     if [email protected]._  then do                          /*is it a possible amicable number ?   */                     if _==k  then iterate       /*Is it a perfect number?  Then skip it*/                     #= # + 1                    /*bump the amicable pair counter.      */                     say right(_, w)    ' and '     right(k, w)   " are an amicable pair."                     end     end   /*k*/                                 /* [↑]   process a range of integers.  */saysay #   'amicable pairs found up to'    H        /*display the count of amicable pairs. */exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/sigma: procedure; parse arg x;   od= x // 2      /*use either  EVEN  or  ODD  integers. */       s= 1                                      /*set initial sigma sum to unity.   ___*/             do j=2+od  by 1+od  while  j*j<x    /*divide by all integers up to the √ x */             if x//j==0  then  s= s + j + x%j    /*add the two divisors to the sum.     */             end   /*j*/                         /* [↑]  %  is REXX integer division.   */                                                 /*                                 ___ */       if j*j==x  then  return s + j             /*Was  X  a square?   If so, add  √ X  */                        return s                 /*return (sigma) sum of the divisors.  */
output   is identical to the 2nd REXX version.

### version 4, SIGMA using integer SQRT

This REXX version is optimized to use the   integer square root of X   in the   sigma   function   (instead of
computing the square of   J   to see if that value exceeds   X).

The optimization makes it about another   20%   faster when searching for amicable numbers up to one million.

/*REXX program  calculates and displays all  amicable pairs  up to  a given number.     */parse arg H .;   if H=='' | H==","  then H=20000 /*get optional arguments  (high limit).*/w= length(H)  ;  low= 220                        /*W: used for columnar output alignment*/x= 220 34765731 6232 87633 284 12285 10856 36939357 6368 5684679          /*S  minimums.*/   do i=0  for 10;   $.i= word(x, i + 1); end /*minimum amicable #s for last dec dig.*/@.= /* [↑] LOW is lowest amicable number. */#= 0 /*number of amicable pairs found so far*/ do k=low for H-low /*generate sigma sums for a range of #s*/ parse var k '' -1 D /*obtain last decimal digit of K. */ if k<$.D    then iterate                    /*if no need to compute, then skip it. */          _= sigma(k)                            /*generate sigma sum for the number  K.*/     @.k= _                                      /*only keep the pertinent sigma sums.  */     if [email protected]._  then do                          /*is it a possible amicable number ?   */                     if _==k  then iterate       /*Is it a perfect number?  Then skip it*/                     #= # + 1                    /*bump the amicable pair counter.      */                     say right(_, w)    ' and '    right(k, w)    " are an amicable pair."                     end     end   /*k*/                                 /* [↑]   process a range of integers.  */saysay #   'amicable pairs found up to'    H        /*display the count of amicable pairs. */exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/iSqrt: procedure; parse arg x;  r= 0;  q= 1;           do while q<=x;    q= q * 4;    end                  do while q>1; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q; end; end       return r/*──────────────────────────────────────────────────────────────────────────────────────*/sigma: procedure; parse arg x;   od= x // 2      /*use either  EVEN  or  ODD  integers. */       s= 1                                      /*set initial sigma sum to unity.   ___*/                do j=2+od  by 1+od  to iSqrt(x)  /*divide by all integers up to the √ x */                if x//j==0  then  s= s + j + x%j /*add the two divisors to the sum.     */                end   /*j*/                      /* [↑]  % is the REXX integer division.*/                                                 /*                                 ___ */       if j*j==x  then  return s + j             /*Was  X  a square?   If so, add  √ X  */                        return s                 /*return (sigma) sum of the divisors.  */
output   is identical to the 2nd REXX version.

### version 5, SIGMA (in-line code)

This REXX version is optimized by bringing the functions in-line   (which minimizes the overhead of invoking two
internal functions),   and it also pre-computes the powers of four   (for the integer square root code).

This method of coding has the disadvantage in that the code (logic) is less idiomatic and therefore less readable.

The optimization makes it about another   15%   faster when searching for amicable numbers up to one million.

/*REXX program  calculates and displays all  amicable pairs  up to  a given number.     */parse arg H .;   if H=='' | H==","  then H=20000 /*get optional arguments  (high limit).*/w= length(H)  ;  low= 220                        /*W: used for columnar output alignment*/x= 220 34765731 6232 87633 284 12285 10856 36939357 6368 5684679          /*S  minimums.*/   do i=0  for 10;   $.i= word(x, i + 1); end /*minimum amicable #s for last dec dig.*/@.= /* [↑] LOW is lowest amicable number. */#= 0 /*number of amicable pairs found so far*/ do k=low for H-low /*generate sigma sums for a range of #s*/ parse var k '' -1 D /*obtain last decimal digit of K. */ if k<$.D    then iterate                    /*if no need to compute, then skip it. */          _= sigma(k)                            /*generate sigma sum for the number  K.*/     @.k= _                                      /*only keep the pertinent sigma sums.  */     if [email protected]._  then do                          /*is it a possible amicable number ?   */                     if _==k  then iterate       /*Is it a perfect number?  Then skip it*/                     #= # + 1                    /*bump the amicable pair counter.      */                     say right(_, w)    ' and '    right(k, w)    " are an amicable pair."                     end     end   /*k*/                                 /* [↑]   process a range of integers.  */saysay #   'amicable pairs found up to'    H        /*display the count of amicable pairs. */exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/iSqrt: procedure; parse arg x;  r= 0;  q= 1;           do while q<=x;    q= q * 4;    end                  do while q>1; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q; end; end       return r/*──────────────────────────────────────────────────────────────────────────────────────*/sigma: procedure; parse arg x;   od= x // 2      /*use either  EVEN  or  ODD  integers. */       s= 1                                      /*set initial sigma sum to unity.   ___*/                do j=2+od  by 1+od  to iSqrt(x)  /*divide by all integers up to the √ x */                if x//j==0  then  s= s + j + x%j /*add the two divisors to the sum.     */                end   /*j*/                      /* [↑]  % is the REXX integer division.*/                                                 /*                                 ___ */       if j*j==x  then  return s + j             /*Was  X  a square?   If so, add  √ X  */                        return s                 /*return (sigma) sum of the divisors.  */
output   is identical to the 2nd REXX version.

## Ring

 size = 18500for n = 1 to size    m = amicable(n)    if m>n and amicable(m)=n       see "" + n + " and " + m + nl oknextsee "OK" + nl func amicable nr     sum = 1     for d = 2 to sqrt(nr)         if nr % d = 0             sum = sum + d            sum = sum + nr / d ok     next     return sum 

## Ruby

With proper_divisors#Ruby in place:

h = {}(1..20_000).each{|n| h[n] = n.proper_divisors.sum }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 
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



## Run BASIC

size = 18500for n = 1 to size    m = amicable(n)    if m > n and amicable(m) = n then print  n ; " and " ; mnext function amicable(nr)     amicable = 1     for d = 2 to sqr(nr)         if nr mod d = 0 then amicable = amicable + d + nr / d     next end function
220 and 284
1184 and 1210
2620 and 2924
5020 and 5564
6232 and 6368
10744 and 10856
12285 and 14595
17296 and 18416


## Rust

fn sum_of_divisors(val: u32) -> u32 {    (1..val/2+1).filter(|n| val % n == 0)                .fold(0, |sum, n| sum + n)} fn main() {    let iter = (1..20_000).map(|i| (i, sum_of_divisors(i)))                          .filter(|&(i, div_sum)| i > div_sum);     for (i, sum1) in iter {        if sum_of_divisors(sum1) == i {           println!("{} {}", i, sum1);        }    }}
Output:
284 220
1210 1184
2924 2620
5564 5020
6368 6232
10856 10744
14595 12285
18416 17296


## Scala

def properDivisors(n: Int) = (1 to n/2).filter(i => n % i == 0)val divisorsSum = (1 to 20000).map(i => i -> properDivisors(i).sum).toMapval result = divisorsSum.filter(v => v._1 < v._2 && divisorsSum.get(v._2) == Some(v._1)) println( result mkString ", " )
Output:
5020 -> 5564, 220 -> 284, 6232 -> 6368, 17296 -> 18416, 2620 -> 2924, 10744 -> 10856, 12285 -> 14595, 1184 -> 1210

## Scheme

 (import (scheme base)        (scheme inexact)        (scheme write)        (only (srfi 1) fold)) ;; return a list of the proper-divisors of n(define (proper-divisors n)  (let ((root (sqrt n)))    (let loop ((divisors (list 1))               (i 2))      (if (> i root)        divisors        (loop (if (zero? (modulo n i))                 (if (= (square i) n)                  (cons i divisors)                  (append (list i (quotient n i)) divisors))                divisors)              (+ 1 i)))))) (define (sum-proper-divisors n)  (if (< n 2)    0    (fold + 0 (proper-divisors n)))) (define *max-n* 20000) ;; hold sums of proper divisors in a cache, to avoid recalculating(define *cache* (make-vector (+ 1 *max-n*)))(for-each (lambda (i) (vector-set! *cache* i (sum-proper-divisors i)))          (iota *max-n* 1)) (define (amicable-pair? i j)  (and (not (= i j))       (= i (vector-ref *cache* j))       (= j (vector-ref *cache* i)))) ;; double loop to *max-n*, displaying all amicable pairs(let loop-i ((i 1))  (when (<= i *max-n*)    (let loop-j ((j i))      (when (<= j *max-n*)        (when (amicable-pair? i j)          (display (string-append "Amicable pair: "                                  (number->string i)                                  " "                                  (number->string j)))          (newline))        (loop-j (+ 1 j))))    (loop-i (+ 1 i)))) 
Output:
Amicable pair: 220 284
Amicable pair: 1184 1210
Amicable pair: 2620 2924
Amicable pair: 5020 5564
Amicable pair: 6232 6368
Amicable pair: 10744 10856
Amicable pair: 12285 14595
Amicable pair: 17296 18416


## Sidef

func propdivsum(n) {    n.sigma - n} for i in (1..20000) {    var j = propdivsum(i)    say "#{i} #{j}" if (j>i && i==propdivsum(j))}
Output:
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416


## 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)") } }} ### Alternative about 800 times faster. import func Darwin.sqrt func sqrt(x:Int) -> Int { return Int(sqrt(Double(x))) } func sigma(n: Int) -> Int { if n == 1 { return 0 } // definition of aliquot sum var result = 1 let root = sqrt(n) for var div = 2; div <= root; ++div { if n % div == 0 { result += div + n/div } } if root*root == n { result -= root } return (result)} func amicables (upTo: Int) -> () { var aliquot = Array(count: upTo+1, repeatedValue: 0) for i in 1 ... upTo { // fill lookup array aliquot[i] = sigma(i) } for i in 1 ... upTo { let a = aliquot[i] if a > upTo {continue} //second part of pair out-of-bounds if a == i {continue} //skip perfect numbers if i == aliquot[a] { print("\(i, a)") aliquot[a] = upTo+1 //prevent second display of pair } }} amicables(20_000) Output: (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)  ## tbas  dim sums(20000) sub sum_proper_divisors(n) dim sum = 0 dim i if n > 1 then for i = 1 to (n \ 2) if n %% i = 0 then sum = sum + i end if next end if return sumend sub dim i, jfor i = 1 to 20000 sums(i) = sum_proper_divisors(i) for j = i-1 to 2 by -1 if sums(i) = j and sums(j) = i then print "Amicable pair:";sums(i);"-";sums(j) exit for end if next next  >tbas amicable_pairs.bas Amicable pair: 220 - 284 Amicable pair: 1184 - 1210 Amicable pair: 2620 - 2924 Amicable pair: 5020 - 5564 Amicable pair: 6232 - 6368 Amicable pair: 10744 - 10856 Amicable pair: 12285 - 14595 Amicable pair: 17296 - 18416  ## 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"}
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


## Transd

 #lang transd MainModule : {    _start: (lambda        (with sum 0 d 0 f Filter( from: 1 to: 20000 apply: (lambda                (= sum 1)                (for i in Range(2 (to-Int (sqrt @it))) do                     (if (not (mod @it i))                         (= d (/ @it i)) (+= sum i)                        (if (neq d i) (+= sum d))))                (ret sum)))            (with v (to-vector f)                (for i in v do                    (if (and (< i (size v))                              (eq (+ @idx 1) (get v (- i 1)))                              (< i (get v (- i 1))))                        (textout (+ @idx 1) ", " i "\n")                    )))))}
Output:
284, 220
1210, 1184
2924, 2620
5564, 5020
6368, 6232
10856, 10744
14595, 12285
18416, 17296


## uBasic/4tH

Input "Limit: ";lPrint "Amicable pairs < ";l For n = 1 To l  m = FUNC(_SumDivisors (n))-n  If m = 0 Then Continue               ' No division by zero, please  p = FUNC(_SumDivisors (m))-m  If (n=p) * (n<m) Then Print n;" and ";mNext End _LeastPower Param(2)  Local(1)   [email protected] = [email protected]  Do While ([email protected] % [email protected]) = 0    [email protected] = [email protected] * [email protected]  Loop Return ([email protected])  ' Return the sum of the proper divisors of [email protected] _SumDivisors Param(1)  Local(4)   [email protected] = [email protected]  [email protected] = 1   ' Handle two specially   [email protected] = FUNC(_LeastPower (2,[email protected]))  [email protected] = [email protected] * ([email protected] - 1)  [email protected] = [email protected] / ([email protected] / 2)   ' Handle odd factors   For [email protected] = 3 Step 2 While ([email protected]*[email protected]) < ([email protected]+1)    [email protected] = FUNC(_LeastPower ([email protected],[email protected]))    [email protected] = [email protected] * (([email protected] - 1) / ([email protected] - 1))    [email protected] = [email protected] / ([email protected] / [email protected])  Loop   ' At this point, t must be one or prime   If ([email protected] > 1) [email protected] = [email protected] * ([email protected]+1)Return ([email protected])
Output:
Limit: 20000
Amicable pairs < 20000
220 and 284
1184 and 1210
2620 and 2924
5020 and 5564
6232 and 6368
10744 and 10856
12285 and 14595
17296 and 18416

0 OK, 0:238

## UTFool

 ···http://rosettacode.org/wiki/Amicable_pairs···■ AmicablePairs  § static    ▶ main    • args⦂ String[]      ∀ n ∈ 1…20000        m⦂ int: sumPropDivs n        if m < n = sumPropDivs m           System.out.println "⸨m⸩ ; ⸨n⸩"     ▶ sumPropDivs⦂ int    • n⦂ int      m⦂ int: 1      ∀ i ∈ √n ⋯> 1        m +: n \ i = 0 ? i + (i = n / i ? 0 ! n / i) ! 0      ⏎ m 

## VBA

Option Explicit Public Sub AmicablePairs()Dim a(2 To 20000) As Long, c As New Collection, i As Long, j As Long, t#    t = Timer    For i = LBound(a) To UBound(a)        'collect the sum of the proper divisors        'of each numbers between 2 and 20000        a(i) = S(i)    Next    'Double Loops to test the amicable    For i = LBound(a) To UBound(a)        For j = i + 1 To UBound(a)            If i = a(j) Then                If a(i) = j Then                     On Error Resume Next                     c.Add i & " : " & j, CStr(i * j)                     On Error GoTo 0                     Exit For                End If            End If        Next    Next    'End. Return :    Debug.Print "Execution Time : " & Timer - t & " seconds."    Debug.Print "Amicable pairs below 20 000 are : "    For i = 1 To c.Count        Debug.Print c.Item(i)    Next iEnd Sub Private Function S(n As Long) As Long'returns the sum of the proper divisors of nDim j As Long    For j = 1 To n \ 2        If n Mod j = 0 Then S = j + S    NextEnd Function
Output:
Execution Time : 7,95703125 seconds.
Amicable pairs below 20 000 are :
220 : 284
1184 : 1210
2620 : 2924
5020 : 5564
6232 : 6368
10744 : 10856
12285 : 14595
17296 : 18416

## VBScript

Not at all optimal. :-(

start = NowSet 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,sumNext 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 IfNext For Each key In uniquepair.Keys	WScript.Echo key & ":" & uniquepair.Item(key)Next WScript.Echo "Execution Time: " & DateDiff("s",Start,Now) & " seconds"
Output:
220:284
1184:1210
2620:2924
5020:5564
6232:6368
10744:10856
12285:14595
17296:18416
Execution Time: 162 seconds

## Wren

Library: Wren-fmt
Library: Wren-math
import "/fmt" for Fmtimport "/math" for Int, Nums var a = List.filled(20000, 0)for (i in 1...20000) a[i] = Nums.sum(Int.properDivisors(i))System.print("The amicable pairs below 20,000 are:")for (n in 2...19999) {    var m = a[n]    if (m > n && m < 20000 && n == a[m]) {        System.print("  %(Fmt.d(5, n)) and %(Fmt.d(5, m))")    }}
Output:
The amicable pairs below 20,000 are:
220 and   284
1184 and  1210
2620 and  2924
5020 and  5564
6232 and  6368
10744 and 10856
12285 and 14595
17296 and 18416


## Yabasic

Translation of: Lua
sub sumDivs(n)    local sum, d     sum = 1     for d = 2 to sqrt(n)        if not mod(n, d) then            sum = sum + d            sum = sum + n / d        end if    next    return sumend sub for n = 2 to 20000    m = sumDivs(n)    if m > n then        if sumDivs(m) = n print n, "\t", m    end ifnext print : print peek("millisrunning"), " ms"

## zkl

Slooooow

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();
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))


## Zig

const MAXIMUM: u32 = 20_000; // Fill up a given array with arr[n] = sum(propDivs(n))pub fn calcPropDivs(divs: []u32) void {    for (divs) |*d| d.* = 1;    var i: u32 = 2;    while (i <= divs.len/2) : (i += 1) {        var j = i * 2;        while (j < divs.len) : (j += i)            divs[j] += i;    }} // Are (A, B) an amicable pair?pub fn amicable(divs: []const u32, a: u32, b: u32) bool {    return divs[a] == b and a == divs[b];} pub fn main() !void {    const stdout = @import("std").io.getStdOut().writer();     var divs: [MAXIMUM + 1]u32 = undefined;    calcPropDivs(divs[0..]);     var a: u32 = 1;    while (a < divs.len) : (a += 1) {        var b = a+1;        while (b < divs.len) : (b += 1) {            if (amicable(divs[0..], a, b))                try stdout.print("{d}, {d}\n", .{a, b});        }    }}
Output:
220, 284
1184, 1210
2620, 2924
5020, 5564
6232, 6368
10744, 10856
12285, 14595
17296, 18416

## ZX Spectrum Basic

Translation of: AWK
10 LET limit=2000020 PRINT "Amicable pairs < ";limit30 FOR n=1 TO limit40 LET num=n: GO SUB 100050 LET m=num60 GO SUB 100070 IF n=num AND n<m THEN PRINT n;" ";m80 NEXT n90 STOP 1000 REM sumprop1010 IF num<2 THEN LET num=0: RETURN 1020 LET sum=11030 LET root=SQR num1040 FOR i=2 TO root-.011050 IF num/i=INT (num/i) THEN LET sum=sum+i+num/i1060 NEXT i1070 IF num/root=INT (num/root) THEN LET sum=sum+root1080 LET num=sum1090 RETURN
Output:
Amicable pairs < 20000
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416`