Amicable pairs
You are encouraged to solve this task according to the task description, using any language you may know.
Two integers and are said to be amicable pairs if and the sum of the proper divisors of () as well as .
- Example
1184 and 1210 are an amicable pair, with proper divisors:
- 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 and
- 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 respectively.
- Task
Calculate and show here the Amicable pairs below 20,000; (there are eight).
- Related tasks
- Proper divisors
- Abundant, deficient and perfect number classifications
- Aliquot sequence classifications and its amicable classification.
11l
<lang 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)</lang>
8080 Assembly
<lang 8080asm> 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 1 init: 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 variable iouter: 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 variable iinner: 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 iinner idone: lxi b,1 ; BC = outer loop variable touter: inx b lxi h,-20001 ; Are we there yet? dad b rc ; If so, stop mov d,b ; DE = outer loop variable mov e,c tinner: 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 5 tnext1: pop b ; Restore B tnext2: pop d ; Restore D jmp tinner ; Continue ;;; Print the number in HL prhl: lxi d,nbuf ; Store buffer pointer on stack push d lxi b,-10 ; Divisor pdgt: lxi d,-1 ; Quotient pdivlp: 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</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
8086 Assembly
<lang asm>LIMIT: equ 20000 ; Maximum value cpu 8086 org 100h section .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,ax calc: 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 going show: 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 $</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
AArch64 Assembly
<lang AArch64 Assembly> /* ARM assembly AARCH64 Raspberry PI 3B */ /* program amicable64.s */
/*******************************************/ /* Constantes file */ /*******************************************/ /* for this file see task include a file in language AArch64 assembly*/ .include "../includeConstantesARM64.inc"
.equ NMAXI, 20000 .equ TABMAXI, 100
/*********************************/ /* Initialized data */ /*********************************/ .data sMessResult: .asciz " @ : @\n" szCarriageReturn: .asciz "\n" szMessErr1: .asciz "Array too small !!" /*********************************/ /* UnInitialized data */ /*********************************/ .bss sZoneConv: .skip 24 tResult: .skip 8 * TABMAXI /*********************************/ /* code section */ /*********************************/ .text .global main main: // entry of program
ldr x3,qNMaxi // load limit mov x4,#2 // number begin
1:
mov x0,x4 // number bl decFactor // compute sum factors cmp x0,x4 // equal ? beq 2f mov x2,x0 // factor sum 1 bl decFactor cmp x0,x4 // equal number ? bne 2f mov x0,x4 // yes -> search in array mov x1,x2 // and store sum bl searchRes cmp x0,#0 // find ? bne 2f // yes mov x0,x4 // no -> display number ans sum mov x1,x2 bl displayResult
2:
add x4,x4,#1 // increment number cmp x4,x3 // end ? ble 1b
100: // standard end of the program
mov x0, #0 // return code mov x8, #EXIT // request to exit program svc #0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn qNMaxi: .quad NMAXI /***************************************************/ /* display message number */ /***************************************************/ /* x0 contains number 1 */ /* x1 contains number 2 */ displayResult:
stp x1,lr,[sp,-16]! // save registers stp x2,x3,[sp,-16]! // save registers mov x2,x1 ldr x1,qAdrsZoneConv bl conversion10 // call décimal conversion ldr x0,qAdrsMessResult ldr x1,qAdrsZoneConv // insert conversion in message bl strInsertAtCharInc mov x3,x0 mov x0,x2 ldr x1,qAdrsZoneConv bl conversion10 // call décimal conversion mov x0,x3 ldr x1,qAdrsZoneConv // insert conversion in message bl strInsertAtCharInc bl affichageMess // display message ldp x2,x3,[sp],16 // restaur 2 registers ldp x1,lr,[sp],16 // restaur 2 registers ret // return to address lr x30
qAdrsMessResult: .quad sMessResult qAdrsZoneConv: .quad sZoneConv /***************************************************/ /* compute factors sum */ /***************************************************/ /* x0 contains the number */ decFactor:
stp x1,lr,[sp,-16]! // save registers stp x2,x3,[sp,-16]! // save registers stp x4,x5,[sp,-16]! // save registers mov x4,#1 // init sum mov x1,#2 // start factor -> divisor
1:
udiv x2,x0,x1 msub x3,x2,x1,x0 // remainder cmp x1,x2 // divisor > quotient ? bgt 3f cmp x3,#0 // remainder = 0 ? bne 2f add x4,x4,x1 // add divisor to sum cmp x1,x2 // divisor = quotient ? beq 3f // yes -> end add x4,x4,x2 // no -> add quotient to sum
2:
add x1,x1,#1 // increment factor b 1b // and loop
3:
mov x0,x4 // return sum ldp x4,x5,[sp],16 // restaur 2 registers ldp x2,x3,[sp],16 // restaur 2 registers ldp x1,lr,[sp],16 // restaur 2 registers ret // return to address lr x30
/***************************************************/ /* search and store result in array */ /***************************************************/ /* x0 contains the number */ /* x1 contains factors sum */ /* x0 return 1 if find 0 else -1 if error */ searchRes:
stp x1,lr,[sp,-16]! // save registers stp x2,x3,[sp,-16]! // save registers stp x4,x5,[sp,-16]! // save registers ldr x4,qAdrtResult // array address mov x2,#0 // indice begin
1:
ldr x3,[x4,x2,lsl #3] // load one result array cmp x3,#0 // if 0 store new result beq 2f cmp x3,x0 // equal ? beq 3f // find -> return 1 add x2,x2,#1 // increment indice cmp x2,#TABMAXI // maxi array ? blt 1b ldr x0,qAdrszMessErr1 // error bl affichageMess mov x0,#-1 b 100f
2:
str x1,[x4,x2,lsl #3] mov x0,#0 // not find -> store and retun 0 b 100f
3:
mov x0,#1
100:
ldp x4,x5,[sp],16 // restaur 2 registers ldp x2,x3,[sp],16 // restaur 2 registers ldp x1,lr,[sp],16 // restaur 2 registers ret // return to address lr x30
qAdrtResult: .quad tResult qAdrszMessErr1: .quad szMessErr1 /********************************************************/ /* File Include fonctions */ /********************************************************/ /* for this file see task include a file in language AArch64 assembly */ .include "../includeARM64.inc" </lang>
220 : 284 1184 : 1210 2620 : 2924 5020 : 5564 6232 : 6368 10744 : 10856 12285 : 14595 17296 : 18416
Action!
Calculations on a real Atari 8-bit computer take quite long time. It is recommended to use an emulator capable with increasing speed of Atari CPU.
<lang Action!>INCLUDE "H6:SIEVE.ACT"
CARD FUNC SumDivisors(CARD x)
CARD i,max,sum
sum=1 i=2 max=x
WHILE i<max
DO
IF x MOD i=0 THEN
max=x/i
IF i<max THEN
sum==+i+max
ELSEIF i=max THEN
sum==+i
FI
FI
i==+1
OD
RETURN (sum)
PROC Main()
DEFINE MAXNUM="20000" BYTE ARRAY primes(MAXNUM+1) CARD m,n
Put(125) PutE() ;clear the screen
Sieve(primes,MAXNUM+1)
FOR m=1 TO MAXNUM-1
DO
IF primes(m)=0 THEN
n=SumDivisors(m)
IF n<MAXNUM AND primes(n)=0 AND n>m THEN
IF m=SumDivisors(n) THEN
PrintF("%U %U%E",m,n)
FI
FI
FI
OD
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Ada
This solution uses the package Generic_Divisors from the Proper Divisors task [[1]].
<lang Ada>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;</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
ALGOL 60
<lang algol60> begin
comment - return n mod m; integer procedure mod(n, m);
value n, q; integer n, m;
begin
mod := n - m * entier(n / m);
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, 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
if a = sumf(b) 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 </lang>
- 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
<lang algol68># 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 an amicable pair", newline ) )
FI
OD
OD</lang>
- Output:
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
ANSI Standard BASIC
<lang ANSI Standard BASIC>100 DECLARE EXTERNAL FUNCTION sum_proper_divisors 110 CLEAR 120 ! 130 DIM f(20001) ! sum of proper factors for each n 140 FOR i=1 TO 20000 150 LET f(i)=sum_proper_divisors(i) 160 NEXT i 170 ! look for pairs 180 FOR i=1 TO 20000 190 FOR j=i+1 TO 20000 200 IF f(i)=j AND i=f(j) THEN 210 PRINT "Amicable pair ";i;" ";j 220 END IF 230 NEXT j 240 NEXT i 250 ! 260 PRINT 270 PRINT "-- found all amicable pairs" 280 END 290 ! 300 ! Compute the sum of proper divisors of given number 310 ! 320 EXTERNAL FUNCTION sum_proper_divisors(n) 330 ! 340 IF n>1 THEN ! n must be 2 or larger 350 LET sum=1 ! start with 1 360 LET root=SQR(n) ! note that root is an integer 370 ! check possible factors, up to sqrt 380 FOR i=2 TO root 390 IF MOD(n,i)=0 THEN 400 LET sum=sum+i ! i is a factor 410 IF i*i<>n THEN ! check i is not actual square root of n 420 LET sum=sum+n/i ! so n/i will also be a factor 430 END IF 440 END IF 450 NEXT i 460 END IF 470 LET sum_proper_divisors = sum 480 END FUNCTION</lang>
AppleScript
<lang AppleScript>-- AMICABLE PAIRS ------------------------------------------------------------
-- amicablePairsUpTo :: Int -> Int on 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 if
end 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 lst
end 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 tell
end filter
-- foldl :: (a -> b -> a) -> a -> [b] -> a on 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 tell
end 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 tell
end map
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn</lang>
- Output:
<lang AppleScript>{{220, 284}, {1184, 1210}, {2620, 2924}, {5020, 5564}, {6232, 6368}, {10744, 10856}, {12285, 14595}, {17296, 18416}}</lang>
ARM Assembly
<lang ARM Assembly> /* ARM assembly Raspberry PI or android with termux */ /* program amicable.s */
/* REMARK 1 : this program use routines in a include file see task Include a file language arm assembly for the routine affichageMess conversion10 see at end of this program the instruction include */
/* for constantes see task include a file in arm assembly */ /************************************/ /* Constantes */ /************************************/ .include "../constantes.inc"
.equ NMAXI, 20000 .equ TABMAXI, 100
/*********************************/ /* Initialized data */ /*********************************/ .data sMessResult: .asciz " @ : @\n" szCarriageReturn: .asciz "\n" szMessErr1: .asciz "Array too small !!" /*********************************/ /* UnInitialized data */ /*********************************/ .bss sZoneConv: .skip 24 tResult: .skip 4 * TABMAXI /*********************************/ /* code section */ /*********************************/ .text .global main main: @ entry of program
ldr r3,iNMaxi @ load limit mov r4,#2 @ number begin
1:
mov r0,r4 @ number bl decFactor @ compute sum factors cmp r0,r4 @ equal ? beq 2f mov r2,r0 @ factor sum 1 bl decFactor cmp r0,r4 @ equal number ? bne 2f mov r0,r4 @ yes -> search in array mov r1,r2 @ and store sum bl searchRes cmp r0,#0 @ find ? bne 2f @ yes mov r0,r4 @ no -> display number ans sum mov r1,r2 bl displayResult
2:
add r4,#1 @ increment number cmp r4,r3 @ end ? ble 1b
100: @ standard end of the program
mov r0, #0 @ return code mov r7, #EXIT @ request to exit program svc #0 @ perform the system call
iAdrszCarriageReturn: .int szCarriageReturn iNMaxi: .int NMAXI /***************************************************/ /* display message number */ /***************************************************/ /* r0 contains number 1 */ /* r1 contains number 2 */ displayResult:
push {r1-r3,lr} @ save registers
mov r2,r1
ldr r1,iAdrsZoneConv
bl conversion10 @ call décimal conversion
ldr r0,iAdrsMessResult
ldr r1,iAdrsZoneConv @ insert conversion in message
bl strInsertAtCharInc
mov r3,r0
mov r0,r2
ldr r1,iAdrsZoneConv
bl conversion10 @ call décimal conversion
mov r0,r3
ldr r1,iAdrsZoneConv @ insert conversion in message
bl strInsertAtCharInc
bl affichageMess @ display message
pop {r1-r3,pc} @ restaur des registres
iAdrsMessResult: .int sMessResult iAdrsZoneConv: .int sZoneConv /***************************************************/ /* compute factors sum */ /***************************************************/ /* r0 contains the number */ decFactor:
push {r1-r5,lr} @ save registers
mov r5,#1 @ init sum
mov r4,r0 @ save number
mov r1,#2 @ start factor -> divisor
1:
mov r0,r4 @ dividende bl division cmp r1,r2 @ divisor > quotient ? bgt 3f cmp r3,#0 @ remainder = 0 ? bne 2f add r5,r5,r1 @ add divisor to sum cmp r1,r2 @ divisor = quotient ? beq 3f @ yes -> end add r5,r5,r2 @ no -> add quotient to sum
2:
add r1,r1,#1 @ increment factor b 1b @ and loop
3:
mov r0,r5 @ return sum
pop {r1-r5,pc} @ restaur registers
/***************************************************/ /* search and store result in array */ /***************************************************/ /* r0 contains the number */ /* r1 contains factors sum */ /* r0 return 1 if find 0 else -1 if error */ searchRes:
push {r1-r4,lr} @ save registers
ldr r4,iAdrtResult @ array address
mov r2,#0 @ indice begin
1:
ldr r3,[r4,r2,lsl #2] @ load one result array cmp r3,#0 @ if 0 store new result beq 2f cmp r3,r0 @ equal ? moveq r0,#1 @ find -> return 1 beq 100f add r2,r2,#1 @ increment indice cmp r2,#TABMAXI @ maxi array ? blt 1b ldr r0,iAdrszMessErr1 @ error bl affichageMess mov r0,#-1 b 100f
2:
str r1,[r4,r2,lsl #2] mov r0,#0 @ not find -> store and retun 0
100:
pop {r1-r4,pc} @ restaur registers
iAdrtResult: .int tResult iAdrszMessErr1: .int szMessErr1 /***************************************************/ /* ROUTINES INCLUDE */ /***************************************************/ .include "../affichage.inc" </lang>
220 : 284 1184 : 1210 2620 : 2924 5020 : 5564 6232 : 6368 10744 : 10856 12285 : 14595 17296 : 18416
Arturo
<lang rebol>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</lang>
- Output:
[220 284] [1184 1210] [2620 2924] [5020 5564] [6232 6368] [10744 10856] [12285 14595] [17296 18416]
ATS
<lang ATS> (* ****** ****** *) //
- include
"share/atspre_staload.hats"
- include
"share/HATS/atspre_staload_libats_ML.hats" // (* ****** ****** *) // fun sum_list_vt
(xs: List_vt(int)): int =
(
case+ xs of | ~list_vt_nil() => 0 | ~list_vt_cons(x, xs) => x + sum_list_vt(xs)
) // (* ****** ****** *)
fun propDivs (
x0: int
) : List0_vt(int) =
loop(x0, 2, list_vt_sing(1)) where
{ // fun loop ( x0: int, i: int, res: List0_vt(int) ) : List0_vt(int) = ( if (i * i) > x0 then 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]
(* ****** ****** *)
fun sum_propDivs(x: int): int = sum_list_vt(propDivs(x))
(* ****** ****** *)
val theNat2 = auxmain(2) where { fun auxmain (
n: int
) : stream_vt(int) = $ldelay(stream_vt_cons(n, auxmain(n+1))) }
(* ****** ****** *) // val theAmicable = ( 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 () = ()
(* ****** ****** *) </lang>
- Output:
(220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)
AutoHotkey
<lang d>SetBatchLines -1 Loop, 20000 { m := A_index
; Getting factors loop % floor(sqrt(m)) { if ( mod(m, A_index) = 0 ) { if ( A_index ** 2 == m ) { sum += A_index continue } else if ( A_index != 1 ) { sum += A_index + m//A_index } else if ( A_index = 1 ) { sum += A_index } } } ; Factors obtained
; Checking factors of sum if ( sum > 1 ) { loop % floor(sqrt(sum)) { if ( mod(sum, A_index) = 0 ) { if ( A_index ** 2 == sum ) { sum2 += A_index continue } else if ( A_index != 1 ) { sum2 += A_index + sum//A_index } else if ( A_index = 1 ) { sum2 += A_index } } } if ( m = sum2 ) && ( m != sum ) && ( m < sum ) final .= m . ":" . sum . "`n" } ; Checked
sum := 0 sum2 := 0 } MsgBox % final ExitApp</lang>
- Output:
220:284 1184:1210 2620:2924 5020:5564 6232:6368 10744:10856 12285:14595 17296:18416
AWK
<lang awk>
- !/bin/awk -f
function sumprop(num, i,sum,root) { if (num < 2) return 0 sum=1 root=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=20000 print "Amicable pairs < ",limit for (n=1; n < limit+1; n++)
{
m=sumprop(n)
if (n == sumprop(m) && n < m) print n,m
}
} }</lang>
- Output:
# ./amicable Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
BASIC256
<lang BASIC256>function SumProperDivisors(number) if number < 2 then return 0 sum = 0 for i = 1 to number \ 2 if number mod i = 0 then sum += i next i return sum end function
dim sum(20000) for n = 1 to 19999 sum[n] = SumProperDivisors(n) next n
print "The pairs of amicable numbers below 20,000 are :" print
for n = 1 to 19998 f = sum[n] if f <= n or f < 1 or f > 19999 then continue for if f = sum[n] and n = sum[f] then print rjust(string(n), 5); " and "; sum[n] end if next n end</lang>
- 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
BCPL
<lang bcpl>get "libhdr"
manifest $(
MAXIMUM = 20000
$)
// Calculate proper divisors for 1..N let 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)
$)</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
Befunge
<lang befunge>v_@#-*8*:"2":$_:#!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*:*:*%:*\`!^</lang>
- 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) <lang c>#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;
}</lang>
- 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#
<lang csharp>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);
}
}
}</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
C++
<lang cpp>
- 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;
} </lang>
- Output:
[220, 284] [1184, 1210] [2620, 2924] [5020, 5564] [6232, 6368] [10744, 10856] [12285, 14595] [17296, 18416] 8 amicable pairs discovered
Clojure
<lang lisp> (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))
</lang>
- Output:
#{220 284}
#{6232 6368}
#{1184 1210}
#{5020 5564}
#{2620 2924}
#{12285 14595}
#{17296 18416}
#{10744 10856}
CLU
<lang clu>% Generate proper divisors from 1 to max proper_divisors = proc (max: int) returns (array[int])
divs: array[int] := array[int]$fill(1, max, 0)
for i: int in int$from_to(1, max/2) do
for j: int in int$from_to_by(i*2, max, i) do
divs[j] := divs[j] + i
end
end
return(divs)
end proper_divisors
% Are A and B and amicable pair, given the proper divisors? amicable = proc (divs: array[int], a, b: int) returns (bool)
return(divs[a] = b & divs[b] = a)
end amicable
% Find all amicable pairs up to 20 000 start_up = proc ()
max = 20000
po: stream := stream$primary_output()
divs: array[int] := proper_divisors(max)
for a: int in int$from_to(1, max) do
for b: int in int$from_to(a+1, max) do
if amicable(divs, a, b) then
stream$putl(po, int$unparse(a) || ", " || int$unparse(b))
end
end
end
end start_up</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
Common Lisp
<lang 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)</lang>
- Output:
((220 284) (1184 1210) (2620 2924) (5020 5564) (6232 6368) (10744 10856) (12285 14595) (17296 18416))
Cowgol
<lang cowgol>include "cowgol.coh";
const LIMIT := 20000;
- Calculate sums of proper divisors
var 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 pair
i := 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;</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
Crystal
<lang Crystal> MX = 524_000_000 N = Math.sqrt(MX).to_u32 x = 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
} </lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ...... ...... ....... ....... 426191535 514780497 475838415 514823985 509379344 523679536
D
<lang d>void main() @safe /*@nogc*/ {
import std.stdio, std.algorithm, std.range, std.typecons, std.array;
immutable properDivs = (in uint n) pure nothrow @safe /*@nogc*/ =>
iota(1, (n + 1) / 2 + 1).filter!(x => n % x == 0);
enum rangeMax = 20_000; auto n2d = iota(1, rangeMax + 1).map!(n => properDivs(n).sum);
foreach (immutable n, immutable divSum; n2d.enumerate(1))
if (n < divSum && divSum <= rangeMax && n2d[divSum - 1] == n)
writefln("Amicable pair: %d and %d with proper divisors:\n %s\n %s",
n, divSum, properDivs(n), properDivs(divSum));
}</lang>
- Output:
Amicable pair: 220 and 284 with proper divisors:
[1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110]
[1, 2, 4, 71, 142]
Amicable pair: 1184 and 1210 with proper divisors:
[1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592]
[1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605]
Amicable pair: 2620 and 2924 with proper divisors:
[1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310]
[1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462]
Amicable pair: 5020 and 5564 with proper divisors:
[1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510]
[1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782]
Amicable pair: 6232 and 6368 with proper divisors:
[1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116]
[1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184]
Amicable pair: 10744 and 10856 with proper divisors:
[1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372]
[1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428]
Amicable pair: 12285 and 14595 with proper divisors:
[1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095]
[1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865]
Amicable pair: 17296 and 18416 with proper divisors:
[1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648]
[1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]
Delphi
See Pascal.
Draco
<lang draco>/* Fill a given array such that for each N,
* P[n] is the sum of proper divisors of N */
proc nonrec propdivs([*] word p) void:
word i, j, max;
max := dim(p,1)-1;
for i from 0 upto max do p[i] := 0 od;
for i from 1 upto max/2 do
for j from i*2 by i upto max do
p[j] := p[j] + i
od
od
corp
/* Find all amicable pairs between 0 and 20,000 */ proc nonrec main() void:
word MAX = 20000;
word i, j;
[MAX] word p;
propdivs(p);
for i from 1 upto MAX-1 do
for j from i+1 upto MAX-1 do
if p[i]=j and p[j]=i then
writeln(i:5, ", ", j:5)
fi
od
od
corp</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
EchoLisp
<lang scheme>
- 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))
</lang>
Ela
<lang ela>open monad io number list
divisors n = filter ((0 ==) << (n `mod`)) [1..(n `div` 2)] range = [1 .. 20000] divs = zip range $ map (sum << divisors) range pairs = [(n, m) \\ (n, nd) <- divs, (m, md) <- divs | n < m && nd == m && md == n]
do putLn pairs ::: IO</lang>
- Output:
[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]
Elena
ELENA 5.0 : <lang elena>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)
}
}</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Alternative variant using strong-typed closures
<lang elena>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)
}
}</lang>
Alternative variant using yield enumerator
<lang elena>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)
}
}</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Elixir
With proper_divisors#Elixir in place: <lang elixir>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)</lang>
- Output:
220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416
Erlang
Erlang, slow
Very slow solution. Same functions by and large as in proper divisors and co.
<lang erlang> -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)). </lang>
- 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.]
<lang erlang> 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]).
</lang>
- 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: <lang erlang> 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.
</lang>
- 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
<lang 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=SUM
END 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 FOR
END PROGRAM</lang>
- Output:
Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
F#
<lang fsharp> [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 List.groupBy id |> List.map snd |> List.filter (List.length >> ((=) 2)) |> List.map List.head |> List.iter (printfn "%A") </lang>
- 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.
<lang Factor> 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 </lang>
- Output:
{ 220 284 }
{ 1184 1210 }
{ 2620 2924 }
{ 5020 5564 }
{ 6232 6368 }
{ 10744 10856 }
{ 12285 14595 }
{ 17296 18416 }
Forth
Direct approach
Calculate many times the divisors.
<lang forth>: proper-divisors ( n -- 1..n )
dup 2 / 1+ 1 ?do dup i mod 0= if i swap then loop drop ;
- divisors-sum ( 1..n -- n )
dup 1 = if exit then begin over + swap 1 = until ;
- pair ( n -- n )
dup 1 = if exit then proper-divisors divisors-sum ;
- ?paired ( n -- t | f )
dup pair 2dup pair = >r < r> and ;
- amicable-list
1+ 1 do i ?paired if cr i . i pair . then loop ;
20000 amicable-list</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ok
Storage approach
Use memory to store sum of divisors, a little quicker.
<lang forth>variable amicable-table
- proper-divisors ( n -- 1..n )
dup 1 = if exit then ( not really but useful ) dup 2 / 1+ 1 ?do dup i mod 0= if i swap then loop drop ;
- divisors-sum ( 1..n -- n )
dup 1 = if exit then begin over + swap 1 = until ;
- build-amicable-table
here amicable-table ! 1+ dup , 1 do i proper-divisors divisors-sum , loop ;
- paired cells amicable-table @ + @ ;
- .amicables
amicable-table @ @ 1 do i paired paired i = i paired i > and if cr i . i paired . then loop ;
- amicable-list
build-amicable-table .amicables ;
20000 amicable-list</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ok
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
<lang FORTRAN>
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.
</lang>
FreeBASIC
using Mod
<lang freebasic> ' 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 sum
End Function
Dim As Integer n, f Dim 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 If
Next
Print Print "Press any key to exit the program" Sleep End </lang>
- 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
<lang freebasic>' 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, f Dim Shared As UInteger sum(max_1)
For n = 2 To max_1
sum(n) = 1
Next
For n = 2 To max_1 \ 2
For f = n * 2 To max_1 Step n sum(f) += n Next
Next
Print Print Using " The pairs of amicable numbers below" & u_str & ", are :"; max Print
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 If
Next
' empty keyboard buffer While Inkey <> "" : Wend Print : Print : Print " Hit any key to end program" Sleep End</lang>
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. <lang frink> n = 1 seen = 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 </lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
Futhark
This program is much too parallel and manifests all the pairs, which requires a giant amount of memory.
<lang> 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
</lang>
GFA Basic
<lang> OPENW 1 CLEARW 1 ' DIM f%(20001) ! sum of proper factors for each n FOR i%=1 TO 20000
f%(i%)=@sum_proper_divisors(i%)
NEXT i% ' look for pairs FOR 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% ' PRINT PRINT "-- 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 </lang>
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
<lang 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)
}
}
}</lang>
- 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
<lang Haskell>divisors :: (Integral a) => a -> [a] divisors n = filter ((0 ==) . (n `mod`)) [1 .. (n `div` 2)]
main :: IO () main = do
let range = [1 .. 20000 :: Int]
divs = zip range $ map (sum . divisors) range
pairs = [(n, m) | (n, nd) <- divs, (m, md) <- divs,
n < m, nd == m, md == n]
print pairs</lang>
- Output:
[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]
Or, deriving proper divisors above the square root as cofactors (for better performance)
<lang Haskell>import Data.Bool (bool)
amicablePairsUpTo :: Int -> [(Int, Int)] amicablePairsUpTo n =
let sigma = sum . properDivisors
in [1 .. n] >>=
(\x ->
let y = sigma x
in bool [] [(x, y)] (x < y && x == sigma y))
properDivisors
:: Integral a => a -> [a]
properDivisors n =
let root = (floor . sqrt) (fromIntegral n :: Double)
lows = filter ((0 ==) . rem n) [1 .. root]
in init $
lows ++ drop (bool 0 1 (root * root == n)) (reverse (quot n <$> lows))
main :: IO () main = mapM_ print $ amicablePairsUpTo 20000</lang>
- Output:
(220,284) (1184,1210) (2620,2924) (5020,5564) (6232,6368) (10744,10856) (12285,14595) (17296,18416)
J
Proper Divisor implementation:
<lang J>factors=: [: /:~@, */&>@{@((^ i.@>:)&.>/)@q:~&__ properDivisors=: factors -. -.&1</lang>
Amicable pairs:
<lang J> 1 + 0 20000 #: I. ,(</~@i.@# * (* |:))(=/ +/@properDivisors@>) 1 + i.20000
220 284 1184 1210 2620 2924 5020 5564 6232 6368
10744 10856 12285 14595 17296 18416</lang>
Java
<lang java>import java.util.Map; import java.util.function.Function; import java.util.stream.Collectors; import java.util.stream.LongStream;
public class AmicablePairs {
public static void main(String[] args) {
int limit = 20_000;
Map<Long, Long> map = LongStream.rangeClosed(1, limit)
.parallel()
.boxed()
.collect(Collectors.toMap(Function.identity(), AmicablePairs::properDivsSum));
LongStream.rangeClosed(1, limit)
.forEach(n -> {
long m = map.get(n);
if (m > n && m <= limit && map.get(m) == n)
System.out.printf("%s %s %n", n, m);
});
}
public static Long properDivsSum(long n) {
return LongStream.rangeClosed(1, (n + 1) / 2).filter(i -> n % i == 0).sum();
}
}</lang>
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
JavaScript
ES5
<lang JavaScript>(function (max) {
// Proper divisors
function properDivisors(n) {
if (n < 2) return [];
else {
var rRoot = Math.sqrt(n),
intRoot = Math.floor(rRoot),
lows = range(1, intRoot).filter(function (x) {
return (n % x) === 0;
});
return lows.concat(lows.slice(1).map(function (x) {
return n / x;
}).reverse().slice((rRoot === intRoot) | 0));
}
}
// [m..n]
function range(m, n) {
var a = Array(n - m + 1),
i = n + 1;
while (i--) a[i - 1] = i;
return a;
}
// Filter an array of proper divisor sums,
// reading the array index as a function of N (N-1)
// and the sum of proper divisors as a potential M
var pairs = range(1, max).map(function (x) {
return properDivisors(x).reduce(function (a, d) {
return a + d;
}, 0)
}).reduce(function (a, m, i, lst) {
var n = i + 1;
return (m > n) && lst[m - 1] === n ? a.concat(n, m) : a;
}, []);
// a -> bool -> s -> s
function wikiTable(lstRows, blnHeaderRow, strStyle) {
return '{| class="wikitable" ' + (
strStyle ? 'style="' + strStyle + '"' :
) + lstRows.map(function (lstRow, iRow) {
var strDelim = ((blnHeaderRow && !iRow) ? '!' : '|');
return '\n|-\n' + strDelim + ' ' + lstRow.map(function (v) {
return typeof v === 'undefined' ? ' ' : v;
}).join(' ' + strDelim + strDelim + ' ');
}).join() + '\n|}';
}
return wikiTable(
'N', 'M'.concat(pairs),
true,
'text-align:center'
) + '\n\n' + JSON.stringify(pairs);
})(20000);</lang>
- Output:
| N | M |
|---|---|
| 220 | 284 |
| 1184 | 1210 |
| 2620 | 2924 |
| 5020 | 5564 |
| 6232 | 6368 |
| 10744 | 10856 |
| 12285 | 14595 |
| 17296 | 18416 |
<lang JavaScript>[[220,284],[1184,1210],[2620,2924],[5020,5564],
[6232,6368],[10744,10856],[12285,14595],[17296,18416]]</lang>
ES6
<lang JavaScript>(() => {
'use strict';
// amicablePairsUpTo :: Int -> [(Int, Int)]
const amicablePairsUpTo = n => {
const sigma = compose(sum, properDivisors);
return enumFromTo(1)(n).flatMap(x => {
const y = sigma(x);
return x < y && x === sigma(y) ? ([
[x, y]
]) : [];
});
};
// properDivisors :: Int -> [Int]
const properDivisors = n => {
const
rRoot = Math.sqrt(n),
intRoot = Math.floor(rRoot),
lows = enumFromTo(1)(intRoot)
.filter(x => 0 === (n % x));
return lows.concat(lows.map(x => n / x)
.reverse()
.slice((rRoot === intRoot) | 0, -1));
};
// TEST -----------------------------------------------
// main :: IO ()
const main = () =>
console.log(unlines(
amicablePairsUpTo(20000).map(JSON.stringify)
));
// GENERIC FUNCTIONS ----------------------------------
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (...fs) =>
x => fs.reduceRight((a, f) => f(a), x);
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = m => n =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i);
// sum :: [Num] -> Num const sum = xs => xs.reduce((a, x) => a + x, 0);
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// MAIN --- return main();
})();</lang>
- Output:
[220,284] [1184,1210] [2620,2924] [5020,5564] [6232,6368] [10744,10856] [12285,14595] [17296,18416]
jq
<lang jq># unordered def proper_divisors:
. as $n
| if $n > 1 then 1,
(sqrt|floor as $s
| range(2; $s+1) as $i
| if ($n % $i) == 0 then $i,
(if $i * $i == $n then empty else ($n / $i) end)
else empty end)
else empty end;
def addup(stream): reduce stream as $i (0; . + $i);
def task(n):
(reduce range(0; n+1) as $n ( []; . + [$n | addup(proper_divisors)] )) as $listing | range(1;n+1) as $j | range(1;$j) as $k | if $listing[$j] == $k and $listing[$k] == $j then "\($k) and \($j) are amicable" else empty end ;
task(20000)</lang>
- Output:
<lang sh>$ jq -c -n -f amicable_pairs.jq 220 and 284 are amicable 1184 and 1210 are amicable 2620 and 2924 are amicable 5020 and 5564 are amicable 6232 and 6368 are amicable 10744 and 10856 are amicable 12285 and 14595 are amicable 17296 and 18416 are amicable</lang>
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.
<lang Julia>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 pcon
end
function divisorsum(n::Int64)
dsum = one(n)
for (p, a) in factor(n)
dsum *= pcontrib(p, a)
end
dsum -= n
end
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)
end
end
amicables()
</lang>
- 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
<lang k>
propdivs:{1+&0=x!'1+!x%2}
(8,2)#v@&{(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)
</lang>
Kotlin
<lang scala>// 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))
}
}
}</lang>
- 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)! <lang lua>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 sum
end
for n = 2, 20000 do
m = sumDivs(n)
if m > n then
if sumDivs(m) == n then print(n, m) end
end
end</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
MAD
<lang 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.N
SUMMAT 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.20000
MEMO 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.I
CHECK WHENEVER DIVS(I).E.J .AND. DIVS(J).E.I,
0 PRINT FORMAT AMI,I,J
VECTOR VALUES AMI = $I6,I6*$
END OF PROGRAM</lang>
- Output:
AMICABLE PAIRS 284 220 1210 1184 2924 2620 5564 5020 6368 6232 10856 10744 14595 12285 18416 17296
Maple
<lang Maple> 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; </lang>
Mathematica / Wolfram Language
<lang Mathematica>amicableQ[n_] :=
Module[{sum = Total[Most@Divisors@n]},
sum != n && n == Total[Most@Divisors@sum]]
Grid@Partition[Cases[Range[4, 20000], _?(amicableQ@# &)], 2]</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
MATLAB
<lang 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'}))
toc
end
function D=pd(x)
K=1:ceil(x/2); D=K(~(rem(x, K)));
end</lang>
- 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. <lang Nim> 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
</lang>
- 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
<lang Nim>
from math import sqrt
const N = 524_000_000.int32 var 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
</lang>
- 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
<lang Oberon2> 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 sum
END 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
END
END AmicablePairs. </lang>
- 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) :
<lang Oforth>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
]
- </lang>
- Output:
amicables . [[220, 284], [1184, 1210], [2620, 2924], [5020, 5564], [6232, 6368], [10744, 10856], [12285, 14595], [17296, 18416]]
OCaml
<lang ocaml>let rec isqrt n =
if n = 1 then 1 else let _n = isqrt (n - 1) in (_n + (n / _n)) / 2
let sum_divs n =
let sum = ref 1 in for d = 2 to isqrt n do if (n mod d) = 0 then sum := !sum + (n / d + d); done; !sum
let () =
for n = 2 to 20000 do
let m = sum_divs n in
if (m > n) then
if (sum_divs m) = n then Printf.printf "%d %d\n" n m;
done
</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
PARI/GP
<lang parigp>for(x=1,20000,my(y=sigma(x)-x); if(y>x && x == sigma(y)-y,print(x" "y)))</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Pascal
Direct approach
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:<lang pascal>
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);
{$ENDIF}
if i = sf then {A loop?}
begin {Yes. Reveal its members.}
trail[0]:=i; {The first.}
if l = 0 then write('Perfect!! ')
else if l = 1 then write('Amicable! ')
else write('Sociable: ');
for j:=0 to l do Write(Trail[j],',');
WriteLn;
if l = 0 then write(outf,'Perfect!! ')
else if l = 1 then write(outf,'Amicable! ')
else write(outf,'Sociable: ');
for j:=0 to l do write(outf,Trail[j],',');
WriteLn(outf);
end;
end;
Close (outf);
END.</lang>
More expansive.
a "normal" Version. Nearly fast as perl using nTheory. <lang pascal>program AmicablePairs; {$IFDEF FPC}
{$MODE DELPHI}
{$H+}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF} uses
sysutils;
const
MAX = 20000;
//MAX = 20*1000*1000; type
tValue = LongWord;
tpValue = ^tValue;
tPower = array[0..31] of tValue;
tIndex = record
idxI,
idxS : Uint64;
end;
var
Indices : array[0..511] of tIndex; //primes up to 65536 enough until 2^32 primes : array[0..6542] of tValue;
procedure InitPrimes; // sieve of erathosthenes without multiples of 2 type
tSieve = array[0..(65536-1) div 2] of 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 factorization var
su : string; primNo : tValue; p:tValue;
begin
str(n:8,su); su:= su +' ['; primNo := 0; p := primes[0]; repeat Su_Append(n,p,su); inc(primNo); p := primes[primNo]; until (p=0) OR (p*p >= n); p := n; Su_Append(n,p,su); su[length(su)] := ']'; writeln(su);
end;
procedure AmPairOutput(cnt:tValue); var
i : tValue; r_max,r_min,r : double;
begin
r_max := 1.0;
r_min := 16.0;
For i := 0 to cnt-1 do
with Indices[i] do
begin
r := IdxS/IDxI;
writeln(i+1:4,IdxI:16,IDxS:16,' ratio ',r:10:7);
IF r < 1 then
begin
writeln(i);
readln;
halt;
end;
if r_max < r then
r_max := r
else
if r_min > r then
r_min := r;
IF cnt < 20 then
begin
ProperDivs(IdxI);
ProperDivs(IdxS);
end;
end;
writeln(' min ratio ',r_min:12:10); writeln(' max ratio ',r_max:12:10);
end;
procedure SumOFProperDiv(n: tValue;var SumOfProperDivs:tValue); // calculated by prime factorization var
i,q, primNo, Prime,pot : tValue; SumOfDivs: tValue;
begin
i := N;
SumOfDivs := 1;
primNo := 0;
Prime := Primes[0];
q := i DIV Prime;
repeat
if q*Prime = i then
Begin
pot := 1;
repeat
i := q;
q := i div Prime;
Pot := Pot * Prime+1;
until q*Prime <> i;
SumOfDivs := SumOfDivs * pot;
end;
Inc(primNo);
Prime := Primes[primNo];
q := i DIV Prime;
{check if i already prime}
if Prime > q then
begin
prime := i;
q := 1;
end;
until i = 1;
SumOfProperDivs := SumOfDivs - N;
end;
function Check:tValue; const
//going backwards
DIV23 : array[0..5] of byte =
//== 5,4,3,2,1,0
(1,0,0,0,1,0);
var
i,s,k,n : tValue; idx : nativeInt;
begin
n := 0;
idx := 3;
For i := 2 to MAX do
begin
//must be divisble by 2 or 3 ( n < High(tValue) < 1e14 )
IF DIV23[idx] = 0 then
begin
SumOFProperDiv(i,s);
//only 24.7...%
IF s>i then
Begin
SumOFProperDiv(s,k);
IF k = i then
begin
With indices[n] do
begin
idxI := i;
idxS := s;
end;
inc(n);
end;
end;
end;
dec(idx);
IF idx < 0 then
idx := high(DIV23);
end;
result := n;
end;
var
T2,T1: TDatetime; APcnt: tValue;
begin
InitPrimes;
T1:= time;
APCnt:= Check;
T2:= time;
AmPairOutput(APCnt);
writeln('Time to find amicable pairs ',FormatDateTime('HH:NN:SS.ZZZ' ,T2-T1));
{$IFNDEF UNIX} readln;{$ENDIF}
end.</lang> Output
1 220 284 ratio 1.2909091
220 [2^2*5*11*220]
284 [2^2*284]
2 1184 1210 ratio 1.0219595
1184 [2^5*1184]
1210 [2*5*11^2*1210]
3 2620 2924 ratio 1.1160305
2620 [2^2*5*2620]
2924 [2^2*17*43*2924]
4 5020 5564 ratio 1.1083665
5020 [2^2*5*5020]
5564 [2^2*13*5564]
5 6232 6368 ratio 1.0218228
6232 [2^3*19*41*6232]
6368 [2^5*6368]
6 10744 10856 ratio 1.0104244
10744 [2^3*17*79*10744]
10856 [2^3*23*59*10856]
7 12285 14595 ratio 1.1880342
12285 [3^3*5*7*13*12285]
14595 [3*5*7*14595]
8 17296 18416 ratio 1.0647549
17296 [2^4*23*47*17296]
18416 [2^4*18416]Alternative
about 25-times faster. This will not output the amicable number unless both! numbers are under the given limit.
So there will be differences to "Table of n, a(n) for n=1..39374" https://oeis.org/A002025/b002025.txt Up to 524'000'000 the pairs found are only correct by number up to no. 437 460122410 and only 442 out of 455 are found, because some pairs exceed the limit. The limits of the ratio between the numbers of the amicable pair up to 1E14 are, based on b002025.txt:
No. lower upper 31447 52326552030976 52326637800704 ratio 1.0000016 52326552030976 [2^8*563*6079*59723] 52326637800704 [2^8*797*1439*178223] 38336 92371445691525 154378742017851 ratio 1.6712821 92371445691525 [3^2*5^2*7^2*11*13^2*23*29^2*233] 154378742017851 [3^2*13^2*53*337*5682671]
The distance check is being corrected, the lower number is now not limited.
The used method is not useful for very high limits.
n = p[1]^a[1]*p[2]^a[2]*...p[l]^a[l]
sum of divisors(n) = s(n) = (p[1]^(a[1]+1) -1) / (p[1] -1) * ... * (p[l]^(a[l]+1) -1) / (p[l] -1) with
p[k]^(a[k]+1) -1) / (p[k] -1) = sum (i= [1..a[k]])(p[k]^i)
Using "Sieve of Erathosthenes"-style
<lang pascal>program AmicPair; {find amicable pairs in a limited region 2..MAX beware that >both< numbers must be smaller than MAX there are 455 amicable pairs up to 524*1000*1000 correct up to
- 437 460122410
} //optimized for freepascal 2.6.4 32-Bit {$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 do var
su,so : string; i,q : tValue;
begin
su:= '1';
so:= ;
i := 2;
while i*i <= n do
begin
q := n div i;
IF q*i -n = 0 then
begin
su:= su+','+IntToStr(i);
IF q <> i then
so:= ','+IntToStr(q)+so;
end;
inc(i);
end;
writeln(' [',su+so,']');
end;
procedure AmPairOutput(cnt:tValue); var
i : tValue; r : double;
begin
r := 1.0;
For i := 0 to cnt-1 do
with Indices[i] do
begin
writeln(i+1:4,IdxI:12,IDxS:12,' ratio ',IdxS/IDxI:10:7);
if r < IdxS/IDxI then
r := IdxS/IDxI;
IF cnt < 20 then
begin
ProperDivs(IdxI);
ProperDivs(IdxS);
end;
end;
writeln(' max ratio ',r:10:4);
end;
function Check:tValue; var
i,s,n : tValue;
begin
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^i var
k: tValue; Pot, //== prim^k PFac : Int64;
begin
Pot := prim;
PFac := 1;
For k := 0 to High(PowerFac) do
begin
PFac := PFac+Pot;
IF (POT > MAX) then
BREAK;
PowerFac[k] := PFac;
Pot := Pot*prim;
end;
end;
procedure InitPW(prim:tValue); begin
fillchar(power,SizeOf(power),#0); CalcPotfactor(prim);
end;
function NextPotCnt(p: tValue):tValue; //return the first power <> 0 //power == n to base prim var
i : tValue;
begin
result := 0;
repeat
i := power[result];
Inc(i);
IF i < p then
BREAK
else
begin
i := 0;
power[result] := 0;
inc(result);
end;
until false;
power[result] := i;
end;
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.</lang> output
220 284
[1,2,4,5,10,11,20,22,44,55,110]
[1,2,4,71,142]
1184 1210
[1,2,4,8,16,32,37,74,148,296,592]
[1,2,5,10,11,22,55,110,121,242,605]
2620 2924
[1,2,4,5,10,20,131,262,524,655,1310]
[1,2,4,17,34,43,68,86,172,731,1462]
5020 5564
[1,2,4,5,10,20,251,502,1004,1255,2510]
[1,2,4,13,26,52,107,214,428,1391,2782]
6232 6368
[1,2,4,8,19,38,41,76,82,152,164,328,779,1558,3116]
[1,2,4,8,16,32,199,398,796,1592,3184]
10744 10856
[1,2,4,8,17,34,68,79,136,158,316,632,1343,2686,5372]
[1,2,4,8,23,46,59,92,118,184,236,472,1357,2714,5428]
12285 14595
[1,3,5,7,9,13,15,21,27,35,39,45,63,65,91,105,117,135,189,195,273,315,351,455,585,819,945,1365,1755,2457,4095]
[1,3,5,7,15,21,35,105,139,417,695,973,2085,2919,4865]
17296 18416
[1,2,4,8,16,23,46,47,92,94,184,188,368,376,752,1081,2162,4324,8648]
[1,2,4,8,16,1151,2302,4604,9208]
8 amicable numbers up to 20000
00:00:00.000
.... 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.
<lang perl>use ntheory qw/divisor_sum/; for my $x (1..20000) {
my $y = divisor_sum($x)-$x; say "$x $y" if $y > $x && $x == divisor_sum($y)-$y;
}</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Phix
integer n for m=1 to 20000 do n = sum(factors(m,-1)) if m<n and m=sum(factors(n,-1)) then ?{m,n} end if end for
- Output:
{220,284}
{1184,1210}
{2620,2924}
{5020,5564}
{6232,6368}
{10744,10856}
{12285,14595}
{17296,18416}
Phixmonti
<lang Phixmonti>def sumDivs
var n
1 var sum n sqrt
2 swap 2 tolist
for
var d
n d mod not if
sum d + n d / + var sum
endif
endfor
sum
enddef
2 20000 2 tolist for
var i
i sumDivs var m
m i > if
m sumDivs i == if i print "\t" print m print nl endif
endif
endfor
nl msec print " s" print</lang>
PHP
<lang php><?php
function sumDivs ($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 $n." ".$m."
";
}
}
?></lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Picat
Different approaches to solve this task:
* foreach loop (two variants) * list comprehension * while loop.
Also, the calculation of the sum of divisors is tabled (the table is cleared between each run). <lang Picat>go =>
N = 20000,
println(amicable1), time(amicable1(N)),
% initialize_table is needed to clear the table cache % of sum_divisors/1 between each run. initialize_table, println(amicable2), time(amicable2(N)),
initialize_table, println(amicable3), time(amicable3(N)),
initialize_table, println(amicable4), time(amicable4(N)), nl.
% Foreach loop and a map (hash table)
amicable1(N) =>
Pairs = new_map(),
foreach(A in 1..N)
B = sum_divisors(A),
C = sum_divisors(B),
if A != B, A == C then
Pairs.put([A,B].sort(),1)
end
end,
println(Pairs.keys().sort()).
% List comprehension
amicable2(N) =>
println([[A,B].sort() : A in 1..N,
B = sum_divisors(A),
C = sum_divisors(B),
A != B, A == C].remove_dups()).
% While loop
amicable3(N) =>
A = 1,
while(A <= N)
B = sum_divisors(A),
if A < B, A == sum_divisors(B) then
print([A,B]), print(" ")
end,
A := A + 1
end,
nl.
% Foreach loop, everything in the condition amicable4(N) =>
foreach(A in 1..N, B = sum_divisors(A), A < B, A == sum_divisors(B))
print([A,B]), print(" ")
end,
nl.
% % Sum of divisors of N % table sum_divisors(N) = Sum =>
sum_divisors(2,N,1,Sum).
% Base case: exceeding the limit sum_divisors(I,N,Sum0,Sum), I > floor(sqrt(N)) =>
Sum = Sum0.
% I is a divisor of N sum_divisors(I,N,Sum0,Sum), N mod I == 0 =>
Sum1 = Sum0 + I, (I != N div I -> Sum2 = Sum1 + N div I ; Sum2 = Sum1 ), sum_divisors(I+1,N,Sum2,Sum).
% I is not a divisor of N. sum_divisors(I,N,Sum0,Sum) =>
sum_divisors(I+1,N,Sum0,Sum).
</lang>
- Output:
amicable1 [[220,284],[1184,1210],[2620,2924],[5020,5564],[6232,6368],[10744,10856],[12285,14595],[17296,18416]] CPU time 0.114 seconds. amicable2 [[220,284],[1184,1210],[2620,2924],[5020,5564],[6232,6368],[10744,10856],[12285,14595],[17296,18416]] CPU time 0.106 seconds. amicable3 [220,284] [1184,1210] [2620,2924] [5020,5564] [6232,6368] [10744,10856] [12285,14595] [17296,18416] CPU time 0.111 seconds. amicable4 [220,284] [1184,1210] [2620,2924] [5020,5564] [6232,6368] [10744,10856] [12285,14595] [17296,18416] CPU time 0.107 seconds.
PicoLisp
<lang 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) ) ) ) )</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 0.101 sec
PL/I
<lang pli>*process source xref;
ami: Proc Options(main); p9a=time(); Dcl (p9a,p9b,p9c) Pic'(9)9'; Dcl sumpd(20000) Bin Fixed(31); Dcl pd(300) Bin Fixed(31); Dcl npd Bin Fixed(31); Dcl (x,y) Bin Fixed(31);
Do x=1 To 20000;
Call proper_divisors(x,pd,npd);
sumpd(x)=sum(pd,npd);
End;
p9b=time();
Put Edit('sum(pd) computed in',(p9b-p9a)/1000,' seconds elapsed')
(Skip,col(7),a,f(6,3),a);
Do x=1 To 20000;
Do y=x+1 To 20000;
If y=sumpd(x) &
x=sumpd(y) Then
Put Edit(x,y,' found after ',elapsed(),' seconds')
(Skip,2(f(6)),a,f(6,3),a);
End;
End;
Put Edit(elapsed(),' seconds total search time')(Skip,f(6,3),a);
proper_divisors: Proc(n,pd,npd);
Dcl (n,pd(300),npd) Bin Fixed(31);
Dcl (d,delta) Bin Fixed(31);
npd=0;
If n>1 Then Do;
If mod(n,2)=1 Then /* odd number */
delta=2;
Else /* even number */
delta=1;
Do d=1 To n/2 By delta;
If mod(n,d)=0 Then Do;
npd+=1;
pd(npd)=d;
End;
End;
End;
End;
sum: Proc(pd,npd) Returns(Bin Fixed(31)); Dcl (pd(300),npd) Bin Fixed(31); Dcl sum Bin Fixed(31) Init(0); Dcl i Bin Fixed(31); Do i=1 To npd; sum+=pd(i); End; Return(sum); End;
elapsed: Proc Returns(Dec Fixed(6,3)); p9c=time(); Return((p9c-p9b)/1000); End; End;</lang>
- Output:
sum(pd) computed in 0.510 seconds elapsed 220 284 found after 0.010 seconds 1184 1210 found after 0.060 seconds 2620 2924 found after 0.110 seconds 5020 5564 found after 0.210 seconds 6232 6368 found after 0.260 seconds 10744 10856 found after 2.110 seconds 12285 14595 found after 2.150 seconds 17296 18416 found after 2.240 seconds 2.250 seconds total search time
PL/I-80
Rather than populating an array with the sum of the proper divisors and then searching, the approach here performs an direct test, saving memory, and at a minimal penalty in execution speed, even though about 25% more calls are made to sumf() than would be required simply to initialize an array. <lang PL/I> amicable: procedure options (main);
%replace
search_limit by 20000;
dcl (a, b, found) fixed bin;
put skip list ('Searching for amicable pairs up to ');
put edit (search_limit) (f(5));
found = 0;
do a = 2 to search_limit;
b = sumf(a);
if (b > a) then
do;
if (sumf(b) = a) then
do;
found = found + 1;
put skip edit (a,b) (f(7));
end;
end;
end;
put skip list (found, ' pairs were found');
stop;
/* return sum of the proper divisors of n */
sumf:
procedure(n) returns (fixed bin); dcl (n, sum, f1, f2) fixed bin;
sum = 1; /* 1 is a proper divisor of every number */
f1 = 2;
do while ((f1 * f1) < n);
if mod(n, f1) = 0 then
do;
sum = sum + f1;
f2 = n / f1;
/* don't double count identical co-factors! */
if f2 > f1 then sum = sum + f2;
end;
f1 = f1 + 1;
end;
return (sum);
end sumf;
end amicable; </lang>
- Output:
Searching for amicable pairs up to 20000
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416
8 pairs were found
PL/M
<lang plm>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,'$'));
END;
END;
END;
CALL EXIT; EOF</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
PowerShell
<lang PowerShell> function Get-ProperDivisorSum ( [int]$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 </lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
Prolog
With some guidance from other solutions here:
<lang prolog>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).</lang>
Output:
<lang prolog>?- amicable_pairs_under_20000(R). R = [220-284, 1184-1210, 2620-2924, 5020-5564, 6232-6368, 10744-10856, 12285-14595, 17296-18416].</lang>
PureBasic
<lang 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 sum
EndProcedure
Define n, f Define 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 </lang>
- 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: <lang python>from proper_divisors import proper_divs
def amicable(rangemax=20000):
n2divsum = {n: sum(proper_divs(n)) for n in range(1, rangemax + 1)}
for num, divsum in n2divsum.items():
if num < divsum and divsum <= rangemax and n2divsum[divsum] == num:
yield num, divsum
if __name__ == '__main__':
for num, divsum in amicable():
print('Amicable pair: %i and %i With proper divisors:\n %r\n %r'
% (num, divsum, sorted(proper_divs(num)), sorted(proper_divs(divsum))))</lang>
- Output:
Amicable pair: 220 and 284 With proper divisors:
[1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110]
[1, 2, 4, 71, 142]
Amicable pair: 1184 and 1210 With proper divisors:
[1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592]
[1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605]
Amicable pair: 2620 and 2924 With proper divisors:
[1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310]
[1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462]
Amicable pair: 5020 and 5564 With proper divisors:
[1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510]
[1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782]
Amicable pair: 6232 and 6368 With proper divisors:
[1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116]
[1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184]
Amicable pair: 10744 and 10856 With proper divisors:
[1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372]
[1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428]
Amicable pair: 12285 and 14595 With proper divisors:
[1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095]
[1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865]
Amicable pair: 17296 and 18416 With proper divisors:
[1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648]
[1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]
Or, supplying our own properDivisors function, and defining the harvest in terms of a generic concatMap:
<lang python>Amicable pairs
from itertools import chain from 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 -> c
def 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()</lang>
- 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.
<lang 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 ]</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
R
<lang 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")
} </lang>
- 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>#lang racket (require "proper-divisors.rkt") (define SCOPE 20000)
(define P
(let ((P-v (vector)))
(λ (n)
(set! P-v (fold-divisors P-v n 0 +))
(vector-ref P-v n))))
- returns #f if not an amicable number, amicable pairing otherwise
(define (amicable? n)
(define m (P n))
(define m-sod (P m))
(and (= m-sod n)
(< m n) ; each pair exactly once, also eliminates perfect numbers
m))
(void (amicable? SCOPE)) ; prime the memoisation
(for* ((n (in-range 1 (add1 SCOPE)))
(m (in-value (amicable? n)))
#:when m)
(printf #<<EOS
amicable pair: ~a, ~a
~a: divisors: ~a ~a: divisors: ~a
EOS
n m n (proper-divisors n) m (proper-divisors m)))
</lang>
- Output:
amicable pair: 284, 220 284: divisors: (1 2 4 71 142) 220: divisors: (1 2 4 5 10 11 20 22 44 55 110) amicable pair: 1210, 1184 1210: divisors: (1 2 5 10 11 22 55 110 121 242 605) 1184: divisors: (1 2 4 8 16 32 37 74 148 296 592) amicable pair: 2924, 2620 2924: divisors: (1 2 4 17 34 43 68 86 172 731 1462) 2620: divisors: (1 2 4 5 10 20 131 262 524 655 1310) amicable pair: 5564, 5020 5564: divisors: (1 2 4 13 26 52 107 214 428 1391 2782) 5020: divisors: (1 2 4 5 10 20 251 502 1004 1255 2510) amicable pair: 6368, 6232 6368: divisors: (1 2 4 8 16 32 199 398 796 1592 3184) 6232: divisors: (1 2 4 8 19 38 41 76 82 152 164 328 779 1558 3116) amicable pair: 10856, 10744 10856: divisors: (1 2 4 8 23 46 59 92 118 184 236 472 1357 2714 5428) 10744: divisors: (1 2 4 8 17 34 68 79 136 158 316 632 1343 2686 5372) amicable pair: 14595, 12285 14595: divisors: (1 3 5 7 15 21 35 105 139 417 695 973 2085 2919 4865) 12285: divisors: (1 3 5 7 9 13 15 21 27 35 39 45 63 65 91 105 117 135 189 195 273 315 351 455 585 819 945 1365 1755 2457 4095) amicable pair: 18416, 17296 18416: divisors: (1 2 4 8 16 1151 2302 4604 9208) 17296: divisors: (1 2 4 8 16 23 46 47 92 94 184 188 368 376 752 1081 2162 4324 8648)
Raku
(formerly Perl 6)
<lang perl6>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);
}</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
REBOL
<lang 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] ]
]
]</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
ReScript
<lang rescript>let isqrt = (v) => {
Belt.Float.toInt( sqrt(Belt.Int.toFloat(v)))
}
let sum_divs = (n) => {
let sum = ref(1)
for d in 2 to isqrt(n) {
if mod(n, d) == 0 {
sum.contents = sum.contents + (n / d + d)
}
}
sum.contents
}
{
for n in 2 to 20000 {
let m = sum_divs(n)
if (m > n) {
if sum_divs(m) == n {
Printf.printf("%d %d\n", n, m)
}
}
}
} </lang>
- Output:
$ bsc ampairs.res > ampairs.bs.js $ node ampairs.bs.js 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
REXX
version 1, with factoring
<lang rexx> /*REXX*/
Call time 'R' Do x=1 To 20000
pd=proper_divisors(x) sumpd.x=sum(pd) End
Say 'sum(pd) computed in' time('E') 'seconds' Call time 'R' Do x=1 To 20000
/* If x//1000=0 Then Say x time() */
Do y=x+1 To 20000
If y=sumpd.x &,
x=sumpd.y Then
Say x y 'found after' time('E') 'seconds'
End
End
Say time('E') 'seconds total search time' Exit
proper_divisors: Procedure Parse Arg n Pd= If n=1 Then Return If n//2=1 Then /* odd number */
delta=2
Else /* even number */
delta=1
Do d=1 To n%2 By delta
If n//d=0 Then pd=pd d End
Return space(pd)
sum: Procedure Parse Arg list sum=0 Do i=1 To words(list)
sum=sum+word(list,i) End
Return sum</lang>
- Output:
sum(pd) computed in 48.502000 seconds 220 284 found after 3.775000 seconds 1184 1210 found after 21.611000 seconds 2620 2924 found after 46.817000 seconds 5020 5564 found after 84.296000 seconds 6232 6368 found after 100.918000 seconds 10744 10856 found after 150.126000 seconds 12285 14595 found after 162.124000 seconds 17296 18416 found after 185.600000 seconds 188.836000 seconds total search time
version 2, 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. <lang rexx>/*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 m==@.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*/
say say # ' 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. */</lang>
- 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. <lang rexx>/*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 k==@._ 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. */
say say # '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. */</lang>
- 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. <lang rexx>/*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 k==@._ 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. */
say say # '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. */</lang>
- 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. <lang rexx>/*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 k==@._ 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. */
say say # '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. */</lang>
- output is identical to the 2nd REXX version.
Ring
<lang ring> size = 18500 for n = 1 to size
m = amicable(n)
if m>n and amicable(m)=n
see "" + n + " and " + m + nl ok
next see "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
</lang>
Ruby
With proper_divisors#Ruby in place: <lang ruby>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 </lang>
- Output:
220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416
Run BASIC
<lang Runbasic>size = 18500 for n = 1 to size
m = amicable(n) if m > n and amicable(m) = n then print n ; " and " ; m
next
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</lang>
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
<lang 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);
}
}
}</lang>
- Output:
284 220 1210 1184 2924 2620 5564 5020 6368 6232 10856 10744 14595 12285 18416 17296
Scala
<lang 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).toMap val result = divisorsSum.filter(v => v._1 < v._2 && divisorsSum.get(v._2) == Some(v._1))
println( result mkString ", " )</lang>
- Output:
5020 -> 5564, 220 -> 284, 6232 -> 6368, 17296 -> 18416, 2620 -> 2924, 10744 -> 10856, 12285 -> 14595, 1184 -> 1210
Scheme
<lang 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))))
</lang>
- 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
<lang ruby>func propdivsum(n) {
n.sigma - n
}
for i in (1..20000) {
var j = propdivsum(i)
say "#{i} #{j}" if (j>i && i==propdivsum(j))
}</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Swift
<lang Swift>import func Darwin.sqrt
func sqrt(x:Int) -> Int { return Int(sqrt(Double(x))) }
func properDivs(n: Int) -> [Int] {
if n == 1 { return [] }
var result = [Int]()
for div in filter (1...sqrt(n), { n % $0 == 0 }) {
result.append(div)
if n/div != div && n/div != n { result.append(n/div) }
}
return sorted(result)
}
func sumDivs(n:Int) -> Int {
struct Cache { static var sum = [Int:Int]() }
if let sum = Cache.sum[n] { return sum }
let sum = properDivs(n).reduce(0) { $0 + $1 }
Cache.sum[n] = sum
return sum
}
func amicable(n:Int, m:Int) -> Bool {
if n == m { return false }
if sumDivs(n) != m || sumDivs(m) != n { return false }
return true
}
var pairs = [(Int, Int)]()
for n in 1 ..< 20_000 {
for m in n+1 ... 20_000 {
if amicable(n, m) {
pairs.append(n, m)
println("\(n, m)")
}
}
}</lang>
Alternative
about 800 times faster.<lang Swift>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)</lang>
- Output:
(220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)
tbas
<lang vb> 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 sum end sub
dim i, j for 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 </lang>
>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
<lang tcl>proc properDivisors {n} {
if {$n == 1} return
set divs 1
set sum 1
for {set i 2} {$i*$i <= $n} {incr i} {
if {!($n % $i)} { lappend divs $i incr sum $i if {$i*$i < $n} { lappend divs [set d [expr {$n / $i}]] incr sum $d } }
} return [list $sum $divs]
}
proc amicablePairs {limit} {
set result {}
set sums [set divs {{}}]
for {set n 1} {$n < $limit} {incr n} {
lassign [properDivisors $n] sum d lappend sums $sum lappend divs [lsort -integer $d]
}
for {set n 1} {$n < $limit} {incr n} {
set nsum [lindex $sums $n] for {set m 1} {$m < $n} {incr m} { if {$n==[lindex $sums $m] && $m==$nsum} { lappend result $m $n [lindex $divs $m] [lindex $divs $n] } }
} return $result
}
foreach {m n md nd} [amicablePairs 20000] {
puts "$m and $n are an amicable pair with these proper divisors" puts "\t$m : $md" puts "\t$n : $nd"
}</lang>
- Output:
220 and 284 are an amicable pair with these proper divisors 220 : 1 2 4 5 10 11 20 22 44 55 110 284 : 1 2 4 71 142 1184 and 1210 are an amicable pair with these proper divisors 1184 : 1 2 4 8 16 32 37 74 148 296 592 1210 : 1 2 5 10 11 22 55 110 121 242 605 2620 and 2924 are an amicable pair with these proper divisors 2620 : 1 2 4 5 10 20 131 262 524 655 1310 2924 : 1 2 4 17 34 43 68 86 172 731 1462 5020 and 5564 are an amicable pair with these proper divisors 5020 : 1 2 4 5 10 20 251 502 1004 1255 2510 5564 : 1 2 4 13 26 52 107 214 428 1391 2782 6232 and 6368 are an amicable pair with these proper divisors 6232 : 1 2 4 8 19 38 41 76 82 152 164 328 779 1558 3116 6368 : 1 2 4 8 16 32 199 398 796 1592 3184 10744 and 10856 are an amicable pair with these proper divisors 10744 : 1 2 4 8 17 34 68 79 136 158 316 632 1343 2686 5372 10856 : 1 2 4 8 23 46 59 92 118 184 236 472 1357 2714 5428 12285 and 14595 are an amicable pair with these proper divisors 12285 : 1 3 5 7 9 13 15 21 27 35 39 45 63 65 91 105 117 135 189 195 273 315 351 455 585 819 945 1365 1755 2457 4095 14595 : 1 3 5 7 15 21 35 105 139 417 695 973 2085 2919 4865 17296 and 18416 are an amicable pair with these proper divisors 17296 : 1 2 4 8 16 23 46 47 92 94 184 188 368 376 752 1081 2162 4324 8648 18416 : 1 2 4 8 16 1151 2302 4604 9208
Transd
<lang scheme>
- 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")
)))))
}</lang>
- Output:
284, 220 1210, 1184 2924, 2620 5564, 5020 6368, 6232 10856, 10744 14595, 12285 18416, 17296
uBasic/4tH
<lang>Input "Limit: ";l Print "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 ";m
Next
End
_LeastPower Param(2)
Local(1)
c@ = a@ Do While (b@ % c@) = 0 c@ = c@ * a@ Loop
Return (c@)
' Return the sum of the proper divisors of a@
_SumDivisors Param(1)
Local(4)
b@ = a@ c@ = 1
' Handle two specially
d@ = FUNC(_LeastPower (2,b@)) c@ = c@ * (d@ - 1) b@ = b@ / (d@ / 2)
' Handle odd factors
For e@ = 3 Step 2 While (e@*e@) < (b@+1) d@ = FUNC(_LeastPower (e@,b@)) c@ = c@ * ((d@ - 1) / (e@ - 1)) b@ = b@ / (d@ / e@) Loop
' At this point, t must be one or prime
If (b@ > 1) c@ = c@ * (b@+1)
Return (c@)</lang>
- 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
<lang 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
</lang>
VBA
<lang vb>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 i
End Sub
Private Function S(n As Long) As Long 'returns the sum of the proper divisors of n Dim j As Long
For j = 1 To n \ 2
If n Mod j = 0 Then S = j + S
Next
End Function</lang>
- 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. :-( <lang VBScript>start = Now Set nlookup = CreateObject("Scripting.Dictionary") Set uniquepair = CreateObject("Scripting.Dictionary")
For i = 1 To 20000 sum = 0 For n = 1 To 20000 If n < i Then If i Mod n = 0 Then sum = sum + n End If End If Next nlookup.Add i,sum Next
For j = 1 To 20000 sum = 0 For m = 1 To 20000 If m < j Then If j Mod m = 0 Then sum = sum + m End If End If Next If nlookup.Exists(sum) And nlookup.Item(sum) = j And j <> sum _ And uniquepair.Exists(sum) = False Then uniquepair.Add j,sum End If Next
For Each key In uniquepair.Keys WScript.Echo key & ":" & uniquepair.Item(key) Next
WScript.Echo "Execution Time: " & DateDiff("s",Start,Now) & " seconds"</lang>
- Output:
220:284 1184:1210 2620:2924 5020:5564 6232:6368 10744:10856 12285:14595 17296:18416 Execution Time: 162 seconds
Vlang
<lang Go>fn pfac_sum(i int) int {
mut sum := 0
for p := 1;p <= i/2;p++{
if i%p == 0 {
sum += p
}
}
return sum
}
fn main(){
a := []int{len: 20000, init:pfac_sum(it)}
println('The amicable pairs below 20,000 are:')
for n in 2 .. a.len {
m := a[n]
if m > n && m < 20000 && n == a[m] {
println('${n:5} and ${m:5}')
}
}
}</lang>
- 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
Wren
<lang ecmascript>import "/fmt" for Fmt import "/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))")
}
}</lang>
- 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
XPL0
<lang XPL0>func SumDiv(Num); \Return sum of proper divisors of Num int Num, Div, Sum, Quot; [Div:= 2; Sum:= 0; loop [Quot:= Num/Div;
if Div > Quot then quit;
if rem(0) = 0 then
[Sum:= Sum + Div;
if Div # Quot then Sum:= Sum + Quot;
];
Div:= Div+1;
];
return Sum+1; ];
def Limit = 20000; int Tbl(Limit), N, M; [for N:= 0 to Limit-1 do
Tbl(N):= SumDiv(N);
for N:= 1 to Limit-1 do
[M:= Tbl(N);
if M<Limit & N=Tbl(M) & M>N then
[IntOut(0, N); ChOut(0, 9\tab\);
IntOut(0, M); CrLf(0);
];
];
]</lang>
- Output:
220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
Yabasic
<lang Yabasic>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 sum
end sub
for n = 2 to 20000
m = sumDivs(n)
if m > n then
if sumDivs(m) = n print n, "\t", m
end if
next
print : print peek("millisrunning"), " ms"</lang>
zkl
Slooooow <lang zkl>fcn properDivs(n){ [1.. (n + 1)/2 + 1].filter('wrap(x){ n%x==0 and n!=x }) } const N=20000; sums:=[1..N].pump(T(-1),fcn(n){ properDivs(n).sum(0) }); [0..].zip(sums).filter('wrap([(n,s)]){ (n<s<=N) and sums[s]==n }).println();</lang>
- Output:
L(L(220,284),L(1184,1210),L(2620,2924),L(5020,5564),L(6232,6368),L(10744,10856),L(12285,14595),L(17296,18416))
Zig
<lang 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});
}
}
}</lang>
- Output:
220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416
ZX Spectrum Basic
<lang zxbasic>10 LET limit=20000 20 PRINT "Amicable pairs < ";limit 30 FOR n=1 TO limit 40 LET num=n: GO SUB 1000 50 LET m=num 60 GO SUB 1000 70 IF n=num AND n<m THEN PRINT n;" ";m 80 NEXT n 90 STOP 1000 REM sumprop 1010 IF num<2 THEN LET num=0: RETURN 1020 LET sum=1 1030 LET root=SQR num 1040 FOR i=2 TO root-.01 1050 IF num/i=INT (num/i) THEN LET sum=sum+i+num/i 1060 NEXT i 1070 IF num/root=INT (num/root) THEN LET sum=sum+root 1080 LET num=sum 1090 RETURN</lang>
- Output:
Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416
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