Perfect totient numbers
You are encouraged to solve this task according to the task description, using any language you may know.
Generate and show here, the first twenty Perfect totient numbers.
- Related task
- Also see
-
- the OEIS entry for perfect totient numbers.
- mrob list of the first 54
11l
F φ(n)
R sum((1..n).filter(k -> gcd(@n, k) == 1).map(k -> 1))
F perfect_totient(cnt)
[Int] r
L(n0) 1..
V parts = 0
V n = n0
L n != 1
n = φ(n)
parts += n
I parts == n0
r [+]= n0
I r.len == cnt
R r
print(perfect_totient(20))- Output:
[3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
AArch64 Assembly
/* ARM assembly AARCH64 Raspberry PI 3B */
/* program totientPerfect64.s */
/************************************/
/* Constantes */
/************************************/
.include "../includeConstantesARM64.inc"
.equ MAXI, 20
/*********************************/
/* Initialized data */
/*********************************/
.data
szMessNumber: .asciz " @ "
szCarriageReturn: .asciz "\n"
/*********************************/
/* UnInitialized data */
/*********************************/
.bss
sZoneConv: .skip 24
/*********************************/
/* code section */
/*********************************/
.text
.global main
main:
mov x4,#2 // start number
mov x6,#0 // line counter
mov x7,#0 // result counter
1:
mov x0,x4
mov x5,#0 // totient sum
2:
bl totient // compute totient
add x5,x5,x0 // add totient
cmp x0,#1
beq 3f
b 2b
3:
cmp x5,x4 // compare number and totient sum
bne 4f
mov x0,x4 // display result if equals
ldr x1,qAdrsZoneConv
bl conversion10 // call décimal conversion
ldr x0,qAdrszMessNumber
ldr x1,qAdrsZoneConv // insert conversion in message
bl strInsertAtCharInc
bl affichageMess // display message
add x7,x7,#1
add x6,x6,#1 // increment indice line display
cmp x6,#5 // if = 5 new line
bne 4f
mov x6,#0
ldr x0,qAdrszCarriageReturn
bl affichageMess
4:
add x4,x4,#1 // increment number
cmp x7,#MAXI // maxi ?
blt 1b // and loop
ldr x0,qAdrszCarriageReturn
bl affichageMess
100: // standard end of the program
mov x0, #0 // return code
mov x8,EXIT
svc #0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrsZoneConv: .quad sZoneConv
qAdrszMessNumber: .quad szMessNumber
/******************************************************************/
/* compute totient of number */
/******************************************************************/
/* x0 contains number */
totient:
stp x1,lr,[sp,-16]! // save registers
stp x2,x3,[sp,-16]! // save registers
stp x4,x5,[sp,-16]! // save registers
mov x4,x0 // totient
mov x5,x0 // save number
mov x1,#0 // for first divisor
1: // begin loop
mul x3,x1,x1 // compute square
cmp x3,x5 // compare number
bgt 4f // end
add x1,x1,#2 // next divisor
udiv x2,x5,x1
msub x3,x1,x2,x5 // compute remainder
cmp x3,#0 // remainder null ?
bne 3f
2: // begin loop 2
udiv x2,x5,x1
msub x3,x1,x2,x5 // compute remainder
cmp x3,#0
csel x5,x2,x5,eq // new value = quotient
beq 2b
udiv x2,x4,x1 // divide totient
sub x4,x4,x2 // compute new totient
3:
cmp x1,#2 // first divisor ?
mov x0,1
csel x1,x0,x1,eq // divisor = 1
b 1b // and loop
4:
cmp x5,#1 // final value > 1
ble 5f
mov x0,x4 // totient
mov x1,x5 // divide by value
udiv x2,x4,x5 // totient divide by value
sub x4,x4,x2 // compute new totient
5:
mov x0,x4
100:
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
.include "../includeARM64.inc"3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
ALGOL 68
BEGIN # find the first 20 perfect totient numbers #
# returns the number of integers k where 1 <= k <= n that are mutually prime to n #
PROC totient = ( INT n )INT: # algorithm from the second Go sample #
IF n < 3 THEN 1
ELIF n = 3 THEN 2
ELSE
INT result := n;
INT v := n;
INT i := 2;
WHILE i * i <= v DO
IF v MOD i = 0 THEN
WHILE v MOD i = 0 DO v OVERAB i OD;
result -:= result OVER i
FI;
IF i = 2 THEN
i := 1
FI;
i +:= 2
OD;
IF v > 1 THEN result -:= result OVER v FI;
result
FI # totient # ;
# find the first 20 perfect totient numbers #
INT p count := 0;
FOR i FROM 2 WHILE p count < 20 DO
INT t := totient( i );
INT sum := t;
WHILE t /= 1 DO
t := totient( t );
sum +:= t
OD;
IF sum = i THEN
# have a perfect totient #
p count +:= 1;
print( ( " ", whole( i, 0 ) ) )
FI
OD;
print( ( newline ) )
END- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
APL
(⊢(/⍨)((+/((1+.=⍳∨⊢)∘⊃,⊢)⍣(1=(⊃⊣)))=2∘×)¨)1↓⍳6000
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
ARM Assembly
/* ARM assembly Raspberry PI or android with termux */
/* program totientPerfect.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 MAXI, 20
/*********************************/
/* Initialized data */
/*********************************/
.data
szMessNumber: .asciz " @ "
szCarriageReturn: .asciz "\n"
/*********************************/
/* UnInitialized data */
/*********************************/
.bss
sZoneConv: .skip 24
/*********************************/
/* code section */
/*********************************/
.text
.global main
main:
mov r4,#2 @ start number
mov r6,#0 @ line counter
mov r7,#0 @ result counter
1:
mov r0,r4
mov r5,#0 @ totient sum
2:
bl totient @ compute totient
add r5,r5,r0 @ add totient
cmp r0,#1
beq 3f
b 2b
3:
cmp r5,r4 @ compare number and totient sum
bne 4f
mov r0,r4 @ display result if equals
ldr r1,iAdrsZoneConv
bl conversion10 @ call décimal conversion
ldr r0,iAdrszMessNumber
ldr r1,iAdrsZoneConv @ insert conversion in message
bl strInsertAtCharInc
bl affichageMess @ display message
add r7,r7,#1
add r6,r6,#1 @ increment indice line display
cmp r6,#5 @ if = 5 new line
bne 4f
mov r6,#0
ldr r0,iAdrszCarriageReturn
bl affichageMess
4:
add r4,r4,#1 @ increment number
cmp r7,#MAXI @ maxi ?
blt 1b @ and loop
ldr r0,iAdrszCarriageReturn
bl affichageMess
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
iAdrsZoneConv: .int sZoneConv
iAdrszMessNumber: .int szMessNumber
/******************************************************************/
/* compute totient of number */
/******************************************************************/
/* r0 contains number */
totient:
push {r1-r5,lr} @ save registers
mov r4,r0 @ totient
mov r5,r0 @ save number
mov r1,#0 @ for first divisor
1: @ begin loop
mul r3,r1,r1 @ compute square
cmp r3,r5 @ compare number
bgt 4f @ end
add r1,r1,#2 @ next divisor
mov r0,r5
bl division
cmp r3,#0 @ remainder null ?
bne 3f
2: @ begin loop 2
mov r0,r5
bl division
cmp r3,#0
moveq r5,r2 @ new value = quotient
beq 2b
mov r0,r4 @ totient
bl division
sub r4,r4,r2 @ compute new totient
3:
cmp r1,#2 @ first divisor ?
moveq r1,#1 @ divisor = 1
b 1b @ and loop
4:
cmp r5,#1 @ final value > 1
ble 5f
mov r0,r4 @ totient
mov r1,r5 @ divide by value
bl division
sub r4,r4,r2 @ compute new totient
5:
mov r0,r4
100:
pop {r1-r5,pc} @ restaur registers
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
.include "../affichage.inc"3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Arturo
totient: function [n][
tt: new n
nn: new n
i: new 2
while [nn >= i ^ 2][
if zero? nn % i [
while [zero? nn % i]->
'nn / i
'tt - tt/i
]
if i = 2 ->
i: new 1
'i + 2
]
if nn > 1 ->
'tt - tt/nn
return tt
]
x: new 1
num: new 0
while [num < 20][
tot: new x
s: new 0
while [tot <> 1][
tot: totient tot
's + tot
]
if s = x [
prints ~"|x| "
inc 'num
]
'x + 2
]
print ""
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
AutoHotkey
MsgBox, 262144, , % result := perfect_totient(20)
perfect_totient(n){
count := sum := tot := 0, str:= "", m := 1
while (count < n) {
tot := m, sum := 0
while (tot != 1) {
tot := totient(tot)
sum += tot
}
if (sum = m) {
str .= m ", "
count++
}
m++
}
return Trim(str, ", ")
}
totient(n) {
tot := n, i := 2
while (i*i <= n) {
if !Mod(n, i) {
while !Mod(n, i)
n /= i
tot -= tot / i
}
if (i = 2)
i := 1
i+=2
}
if (n > 1)
tot -= tot / n
return tot
}
- Output:
3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571
AWK
# syntax: GAWK -f PERFECT_TOTIENT_NUMBERS.AWK
BEGIN {
i = 20
printf("The first %d perfect totient numbers:\n%s\n",i,perfect_totient(i))
exit(0)
}
function perfect_totient(n, count,m,str,sum,tot) {
for (m=1; count<n; m++) {
tot = m
sum = 0
while (tot != 1) {
tot = totient(tot)
sum += tot
}
if (sum == m) {
str = str m " "
count++
}
}
return(str)
}
function totient(n, i,tot) {
tot = n
for (i=2; i*i<=n; i+=2) {
if (n % i == 0) {
while (n % i == 0) {
n /= i
}
tot -= tot / i
}
if (i == 2) {
i = 1
}
}
if (n > 1) {
tot -= tot / n
}
return(tot)
}
- Output:
The first 20 perfect totient numbers: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
BASIC
10 DEFINT A-Z
20 N=3
30 S=N: T=0
40 X=S: S=0
50 FOR I=1 TO X-1
60 A=X: B=I
70 IF B>0 THEN C=A: A=B: B=C MOD B: GOTO 70
80 IF A=1 THEN S=S+1
90 NEXT
100 T=T+S
110 IF S>1 GOTO 40
120 IF T=N THEN PRINT N,: Z=Z+1
130 N=N+2
140 IF Z<20 GOTO 30
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Applesoft BASIC
100 HOME
110 PRINT "The first 20 perfect totient numbers:"
120 n = 3
130 s = n : t = 0
140 x = s : s = 0
150 FOR i = 1 TO x-1
160 a = x : b = i
170 IF B > 0 THEN C = A : A = B : B = C - INT(C / B) * B : GOTO 170
180 IF a = 1 THEN s = s+1
190 NEXT i
200 t = t+s
210 IF s > 1 THEN GOTO 140
220 IF t = n THEN PRINT n;" "; : z = z+1
230 n = n+2
240 IF z < 20 THEN GOTO 130
250 END
BASIC256
found = 0
curr = 3
while found < 20
sum = Totient(curr)
toti = sum
while toti <> 1
toti = Totient(toti)
sum += toti
end while
if sum = curr then
print sum
found += 1
end if
curr += 1
end while
end
function GCD(n, d)
if n = 0 then return d
if d = 0 then return n
if n > d then return GCD(d, (n mod d))
return GCD(n, (d mod n))
end function
function Totient(n)
phi = 0
for m = 1 to n
if GCD(m, n) = 1 then phi += 1
next m
return phi
end function
Chipmunk Basic
100 CLS
110 PRINT "The first 20 perfect totient numbers:"
120 n = 3
130 s = n : t = 0
140 x = s : s = 0
150 FOR i = 1 TO x-1
160 a = x : b = i
170 IF b > 0 THEN c = a : a = b : b = c MOD b : GOTO 170
180 IF a = 1 THEN s = s+1
190 NEXT i
200 t = t+s
210 IF s > 1 THEN GOTO 140
220 IF t = n THEN PRINT n;" "; : z = z+1
230 n = n+2
240 IF z < 20 THEN GOTO 130
250 END
FreeBASIC
Uses the code from the Totient Function example as an include.
#include"totient.bas"
dim as uinteger found = 0, curr = 3, sum, toti
while found < 20
sum = totient(curr)
toti = sum
do
toti = totient(toti)
sum += toti
loop while toti <> 1
if sum = curr then
print sum
found += 1
end if
curr += 1
wend
Gambas
Public Sub Main()
Dim found As Integer = 0, curr As Integer = 3
Dim sum As Integer, toti As Integer
Print "The first 20 perfect totient numbers are:"
While found < 20
sum = totient(curr)
toti = sum
Do
toti = totient(toti)
sum += toti
Loop While toti <> 1
If sum = curr Then
Print sum; ", ";
found += 1
End If
curr += 1
Wend
Print Chr(8); Chr(8); " "
End
Function GCD(n As Integer, d As Integer) As Integer
If d = 0 Then Return n Else Return GCD(d, n Mod d)
End Function
Function Totient(n As Integer) As Integer
Dim m As Integer, phi As Integer = 0
For m = 1 To n
If GCD(m, n) = 1 Then phi += 1
Next
Return phi
End Function
- Output:
Same as FreeBASIC entry.
GW-BASIC
The Chipmunk Basic solution works without any changes.
MSX Basic
The Chipmunk Basic solution works without any changes.
QBasic
The BASIC solution works without any changes. Also The Chipmunk Basic solution works without any changes.
bc
define gcd (i, j) {
while(j != 0) {
k = j
j = i % j
i = k
}
return i
}
define is_perfect_totient (num) {
tot = 0
for (i = 1; i < num; i++) {
if (gcd(num, i) == 1) {
tot += 1
}
}
sum = tot + cache[tot]
cache[num] = sum
return num == sum
}
j = 1
count = 0
# we only go to 15 (not 20) because bc is very slow
while (count <= 15) {
if (is_perfect_totient(j)) {
print j, " "
count += 1
}
j += 1
}
print "\n"
quit
- Output:
$ time bc -q perfect-totient.bc 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 real 0m35,553s user 0m35,437s sys 0m0,030s
BCPL
get "libhdr"
let gcd(a,b) = b=0 -> a, gcd(b, a rem b)
let totient(n) = valof
$( let r = 0
for i=1 to n-1
if gcd(n,i) = 1 then r := r + 1
resultis r
$)
let perfect(n) = valof
$( let sum = 0 and x = n
$( x := totient(x)
sum := sum + x
$) repeatuntil x = 1
resultis sum = n
$)
let start() be
$( let seen = 0 and n = 3
while seen < 20
$( if perfect(n)
$( writef("%N ", n)
seen := seen + 1
$)
n := n + 2
$)
wrch('*N')
$)- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
C
Calculates as many perfect Totient numbers as entered on the command line.
#include<stdlib.h>
#include<stdio.h>
long totient(long n){
long tot = n,i;
for(i=2;i*i<=n;i+=2){
if(n%i==0){
while(n%i==0)
n/=i;
tot-=tot/i;
}
if(i==2)
i=1;
}
if(n>1)
tot-=tot/n;
return tot;
}
long* perfectTotients(long n){
long *ptList = (long*)malloc(n*sizeof(long)), m,count=0,sum,tot;
for(m=1;count<n;m++){
tot = m;
sum = 0;
while(tot != 1){
tot = totient(tot);
sum += tot;
}
if(sum == m)
ptList[count++] = m;
}
return ptList;
}
long main(long argC, char* argV[])
{
long *ptList,i,n;
if(argC!=2)
printf("Usage : %s <number of perfect Totient numbers required>",argV[0]);
else{
n = atoi(argV[1]);
ptList = perfectTotients(n);
printf("The first %d perfect Totient numbers are : \n[",n);
for(i=0;i<n;i++)
printf(" %d,",ptList[i]);
printf("\b]");
}
return 0;
}
Output for multiple runs, a is the default executable file name produced by GCC
C:\rossetaCode>a 10 The first 10 perfect Totient numbers are : [ 3, 9, 15, 27, 39, 81, 111, 183, 243, 255] C:\rossetaCode>a 20 The first 20 perfect Totient numbers are : [ 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571] C:\rossetaCode>a 30 The first 30 perfect Totient numbers are : [ 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147] C:\rossetaCode>a 40 The first 40 perfect Totient numbers are : [ 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, 1594323, 4190263, 4782969, 9056583, 14348907, 43046721, 57395631]
C++
#include <cassert>
#include <iostream>
#include <vector>
class totient_calculator {
public:
explicit totient_calculator(int max) : totient_(max + 1) {
for (int i = 1; i <= max; ++i)
totient_[i] = i;
for (int i = 2; i <= max; ++i) {
if (totient_[i] < i)
continue;
for (int j = i; j <= max; j += i)
totient_[j] -= totient_[j] / i;
}
}
int totient(int n) const {
assert (n >= 1 && n < totient_.size());
return totient_[n];
}
bool is_prime(int n) const {
return totient(n) == n - 1;
}
private:
std::vector<int> totient_;
};
bool perfect_totient_number(const totient_calculator& tc, int n) {
int sum = 0;
for (int m = n; m > 1; ) {
int t = tc.totient(m);
sum += t;
m = t;
}
return sum == n;
}
int main() {
totient_calculator tc(10000);
int count = 0, n = 1;
std::cout << "First 20 perfect totient numbers:\n";
for (; count < 20; ++n) {
if (perfect_totient_number(tc, n)) {
if (count > 0)
std::cout << ' ';
++count;
std::cout << n;
}
}
std::cout << '\n';
return 0;
}
- Output:
First 20 perfect totient numbers: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
CLU
gcd = proc (a, b: int) returns (int)
while b ~= 0 do
a, b := b, a//b
end
return(a)
end gcd
totient = proc (n: int) returns (int)
tot: int := 0
for i: int in int$from_to(1,n-1) do
if gcd(n,i)=1 then tot := tot + 1 end
end
return(tot)
end totient
perfect = proc (n: int) returns (bool)
sum: int := 0
x: int := n
while true do
x := totient(x)
sum := sum + x
if x=1 then break end
end
return(sum = n)
end perfect
start_up = proc ()
po: stream := stream$primary_output()
seen: int := 0
n: int := 3
while seen < 20 do
if perfect(n) then
stream$puts(po, int$unparse(n) || " ")
seen := seen + 1
end
n := n + 2
end
end start_up- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Cowgol
include "cowgol.coh";
sub gcd(a: uint16, b: uint16): (r: uint16) is
while b != 0 loop
r := a;
a := b;
b := r % b;
end loop;
r := a;
end sub;
sub totient(n: uint16): (tot: uint16) is
var i: uint16 := 1;
tot := 0;
while i < n loop
if gcd(n,i) == 1 then
tot := tot + 1;
end if;
i := i + 1;
end loop;
end sub;
sub totientSum(n: uint16): (sum: uint16) is
var x := n;
sum := 0;
while x > 1 loop
x := totient(x);
sum := sum + x;
end loop;
end sub;
var seen: uint8 := 0;
var n: uint16 := 3;
while seen < 20 loop
if totientSum(n) == n then
print_i16(n);
print_char(' ');
seen := seen + 1;
end if;
n := n + 2;
end loop;
print_nl();- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
D
import std.stdio;
import std.numeric;
int[10000] cache;
bool is_perfect_totient(int num) {
int tot = 0;
for (int i = 1; i < num; i++) {
if (gcd(num, i) == 1) {
tot++;
}
}
int sum = tot + cache[tot];
cache[num] = sum;
return num == sum;
}
void main() {
int j = 1;
int count = 0;
while (count < 20) {
if (is_perfect_totient(j)) {
printf("%d ", j);
count++;
}
j++;
}
writeln(" ");
}
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Dart
import "dart:io";
var cache = List<int>.filled(10000, 0, growable: true);
void main() {
cache[0] = 0;
var count = 0;
var i = 1;
while (count < 20) {
if (is_perfect_totient(i)) {
stdout.write("$i ");
count++;
}
i++;
}
print(" ");
}
bool is_perfect_totient(n) {
var tot = 0;
for (int i = 1; i < n; i++ ) {
if (i.gcd(n) == 1) {
tot++;
}
}
int sum = tot + cache[tot];
cache[n] = sum;
return n == sum;
}
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Delphi
program Perfect_totient_numbers;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
function totient(n: Integer): Integer;
begin
var tot := n;
var i := 2;
while i * i <= n do
begin
if (n mod i) = 0 then
begin
while (n mod i) = 0 do
n := n div i;
dec(tot, tot div i);
end;
if i = 2 then
i := 1;
inc(i, 2);
end;
if n > 1 then
dec(tot, tot div n);
Result := tot;
end;
begin
var perfect: TArray<Integer>;
var n := 1;
while length(perfect) < 20 do
begin
var tot := n;
var sum := 0;
while tot <> 1 do
begin
tot := totient(tot);
inc(sum, tot);
end;
if sum = n then
begin
SetLength(perfect, Length(perfect) + 1);
perfect[High(perfect)] := n;
end;
inc(n, 2);
end;
writeln('The first 20 perfect totient numbers are:');
write('[');
for var e in perfect do
write(e, ' ');
writeln(']');
readln;
end.
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Draco
proc nonrec gcd(word a, b) word:
word c;
while b ~= 0 do
c := a;
a := b;
b := c % b
od;
a
corp
proc nonrec totient(word n) word:
word r, i;
r := 0;
for i from 1 upto n-1 do
if gcd(n,i) = 1 then r := r+1 fi
od;
r
corp
proc nonrec perfect(word n) bool:
word sum, x;
sum := 0;
x := n;
while
x := totient(x);
sum := sum + x;
x ~= 1
do od;
sum = n
corp
proc nonrec main() void:
word seen, n;
seen := 0;
n := 3;
while seen < 20 do
if perfect(n) then
write(n, " ");
seen := seen + 1
fi;
n := n + 2
od
corp- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
EasyLang
func totient n .
tot = n
i = 2
while i <= sqrt n
if n mod i = 0
while n mod i = 0
n = n div i
.
tot -= tot div i
.
if i = 2
i = 1
.
i += 2
.
if n > 1
tot -= tot div n
.
return tot
.
n = 1
while cnt < 20
tot = n
sum = 0
while tot <> 1
tot = totient tot
sum += tot
.
if sum = n
write n & " "
cnt += 1
.
n += 2
.
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Factor
USING: formatting kernel lists lists.lazy math
math.primes.factors ;
: perfect? ( n -- ? )
[ 0 ] dip dup [ dup 2 < ] [ totient tuck [ + ] 2dip ] until
drop = ;
20 1 lfrom [ perfect? ] lfilter ltake list>array
"%[%d, %]\n" printf
- Output:
{ 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571 }
Go
package main
import "fmt"
func gcd(n, k int) int {
if n < k || k < 1 {
panic("Need n >= k and k >= 1")
}
s := 1
for n&1 == 0 && k&1 == 0 {
n >>= 1
k >>= 1
s <<= 1
}
t := n
if n&1 != 0 {
t = -k
}
for t != 0 {
for t&1 == 0 {
t >>= 1
}
if t > 0 {
n = t
} else {
k = -t
}
t = n - k
}
return n * s
}
func totient(n int) int {
tot := 0
for k := 1; k <= n; k++ {
if gcd(n, k) == 1 {
tot++
}
}
return tot
}
func main() {
var perfect []int
for n := 1; len(perfect) < 20; n += 2 {
tot := n
sum := 0
for tot != 1 {
tot = totient(tot)
sum += tot
}
if sum == n {
perfect = append(perfect, n)
}
}
fmt.Println("The first 20 perfect totient numbers are:")
fmt.Println(perfect)
}
- Output:
The first 20 perfect totient numbers are: [3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571]
The following much quicker version uses Euler's product formula rather than repeated invocation of the gcd function to calculate the totient:
package main
import "fmt"
func totient(n int) int {
tot := n
for i := 2; i*i <= n; i += 2 {
if n%i == 0 {
for n%i == 0 {
n /= i
}
tot -= tot / i
}
if i == 2 {
i = 1
}
}
if n > 1 {
tot -= tot / n
}
return tot
}
func main() {
var perfect []int
for n := 1; len(perfect) < 20; n += 2 {
tot := n
sum := 0
for tot != 1 {
tot = totient(tot)
sum += tot
}
if sum == n {
perfect = append(perfect, n)
}
}
fmt.Println("The first 20 perfect totient numbers are:")
fmt.Println(perfect)
}
The output is the same as before.
Haskell
perfectTotients :: [Int]
perfectTotients =
filter ((==) <*> (succ . sum . tail . takeWhile (1 /=) . iterate φ)) [2 ..]
φ :: Int -> Int
φ = memoize (\n -> length (filter ((1 ==) . gcd n) [1 .. n]))
memoize :: (Int -> a) -> (Int -> a)
memoize f = (!!) (f <$> [0 ..])
main :: IO ()
main = print $ take 20 perfectTotients
- Output:
[3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571]
J
Until =: conjunction def 'u^:(0 -: v)^:_'
Filter =: (#~`)(`:6)
totient =: 5&p:
totient_chain =: [: }. (, totient@{:)Until(1={:)
ptnQ =: (= ([: +/ totient_chain))&>
With these definitions I've found the first 28 perfect totient numbers
PTN =: ptnQ Filter >: i.99999 #PTN 28 PTN 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571 6561 8751 15723 19683 36759 46791 59049 65535
Java
import java.util.ArrayList;
import java.util.List;
public class PerfectTotientNumbers {
public static void main(String[] args) {
computePhi();
int n = 20;
System.out.printf("The first %d perfect totient numbers:%n%s%n", n, perfectTotient(n));
}
private static final List<Integer> perfectTotient(int n) {
int test = 2;
List<Integer> results = new ArrayList<Integer>();
for ( int i = 0 ; i < n ; test++ ) {
int phiLoop = test;
int sum = 0;
do {
phiLoop = phi[phiLoop];
sum += phiLoop;
} while ( phiLoop > 1);
if ( sum == test ) {
i++;
results.add(test);
}
}
return results;
}
private static final int max = 100000;
private static final int[] phi = new int[max+1];
private static final void computePhi() {
for ( int i = 1 ; i <= max ; i++ ) {
phi[i] = i;
}
for ( int i = 2 ; i <= max ; i++ ) {
if (phi[i] < i) continue;
for ( int j = i ; j <= max ; j += i ) {
phi[j] -= phi[j] / i;
}
}
}
}
- Output:
The first 20 perfect totient numbers: [3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
JavaScript
(() => {
'use strict';
// main :: IO ()
const main = () =>
showLog(
take(20, perfectTotients())
);
// perfectTotients :: Generator [Int]
function* perfectTotients() {
const
phi = memoized(
n => length(
filter(
k => 1 === gcd(n, k),
enumFromTo(1, n)
)
)
),
imperfect = n => n !== sum(
tail(iterateUntil(
x => 1 === x,
phi,
n
))
);
let ys = dropWhileGen(imperfect, enumFrom(1))
while (true) {
yield ys.next().value - 1;
ys = dropWhileGen(imperfect, ys)
}
}
// GENERIC FUNCTIONS ----------------------------
// abs :: Num -> Num
const abs = Math.abs;
// dropWhileGen :: (a -> Bool) -> Gen [a] -> [a]
const dropWhileGen = (p, xs) => {
let
nxt = xs.next(),
v = nxt.value;
while (!nxt.done && p(v)) {
nxt = xs.next();
v = nxt.value;
}
return xs;
};
// enumFrom :: Int -> [Int]
function* enumFrom(x) {
let v = x;
while (true) {
yield v;
v = 1 + v;
}
}
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
m <= n ? iterateUntil(
x => n <= x,
x => 1 + x,
m
) : [];
// filter :: (a -> Bool) -> [a] -> [a]
const filter = (f, xs) => xs.filter(f);
// gcd :: Int -> Int -> Int
const gcd = (x, y) => {
const
_gcd = (a, b) => (0 === b ? a : _gcd(b, a % b)),
abs = Math.abs;
return _gcd(abs(x), abs(y));
};
// iterateUntil :: (a -> Bool) -> (a -> a) -> a -> [a]
const iterateUntil = (p, f, x) => {
const vs = [x];
let h = x;
while (!p(h))(h = f(h), vs.push(h));
return vs;
};
// Returns Infinity over objects without finite length.
// This enables zip and zipWith to choose the shorter
// argument when one is non-finite, like cycle, repeat etc
// length :: [a] -> Int
const length = xs =>
(Array.isArray(xs) || 'string' === typeof xs) ? (
xs.length
) : Infinity;
// memoized :: (a -> b) -> (a -> b)
const memoized = f => {
const dctMemo = {};
return x => {
const v = dctMemo[x];
return undefined !== v ? v : (dctMemo[x] = f(x));
};
};
// showLog :: a -> IO ()
const showLog = (...args) =>
console.log(
args
.map(JSON.stringify)
.join(' -> ')
);
// sum :: [Num] -> Num
const sum = xs => xs.reduce((a, x) => a + x, 0);
// tail :: [a] -> [a]
const tail = xs => 0 < xs.length ? xs.slice(1) : [];
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
'GeneratorFunction' !== xs.constructor.constructor.name ? (
xs.slice(0, n)
) : [].concat.apply([], Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}));
// MAIN ---
main();
})();
- Output:
[3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571]
jq
Adapted from Julia
Works with gojq, the Go implementation of jq
One small point of interest in the following implementation is the way the cacheing of totient values is accomplished using a helper function (`cachephi`).
# jq optimizes the recursive call of _gcd in the following:
def gcd(a;b):
def _gcd:
if .[1] != 0 then [.[1], .[0] % .[1]] | _gcd else .[0] end;
[a,b] | _gcd ;
def count(s): reduce s as $x (0; .+1);
# A perfect totient number is an integer that is equal to the sum of its iterated totients.
# aka Euler's phi function
def totient:
. as $n
| count( range(0; .) | select( gcd($n; .) == 1) );
# input: the cache
# output: the updated cache
def cachephi($n):
($n|tostring) as $s
| if (has($s)|not) then .[$s] = ($n|totient) else . end ;
# Emit the stream of perfect totients
def perfect_totients:
. as $n
| foreach range(1; infinite) as $i ({cache: {}};
.tot = $i
| .tsum = 0
| until( .tot == 1;
.tot as $tot
| .cache |= cachephi($tot)
| .tot = .cache[$tot|tostring]
| .tsum += .tot);
if .tsum == $i then $i else empty end );
"The first 20 perfect totient numbers:",
limit(20; perfect_totients)- Output:
The first 20 perfect totient numbers: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Julia
using Primes
eulerphi(n) = (r = one(n); for (p,k) in factor(abs(n)) r *= p^(k-1)*(p-1) end; r)
const phicache = Dict{Int, Int}()
cachedphi(n) = (if !haskey(phicache, n) phicache[n] = eulerphi(n) end; phicache[n])
function perfecttotientseries(n)
perfect = Vector{Int}()
i = 1
while length(perfect) < n
tot = i
tsum = 0
while tot != 1
tot = cachedphi(tot)
tsum += tot
end
if tsum == i
push!(perfect, i)
end
i += 1
end
perfect
end
println("The first 20 perfect totient numbers are: $(perfecttotientseries(20))")
println("The first 40 perfect totient numbers are: $(perfecttotientseries(40))")
- Output:
The first 20 perfect totient numbers are: [3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571] The first 40 perfect totient numbers are: [3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, 1594323, 4190263, 4782969, 9056583, 14348907, 43046721, 57395631]
Kotlin
// Version 1.3.21
fun totient(n: Int): Int {
var tot = n
var nn = n
var i = 2
while (i * i <= nn) {
if (nn % i == 0) {
while (nn % i == 0) nn /= i
tot -= tot / i
}
if (i == 2) i = 1
i += 2
}
if (nn > 1) tot -= tot / nn
return tot
}
fun main() {
val perfect = mutableListOf<Int>()
var n = 1
while (perfect.size < 20) {
var tot = n
var sum = 0
while (tot != 1) {
tot = totient(tot)
sum += tot
}
if (sum == n) perfect.add(n)
n += 2
}
println("The first 20 perfect totient numbers are:")
println(perfect)
}
- Output:
The first 20 perfect totient numbers are: [3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
Lua
local function phi(n)
assert(type(n) == 'number', 'n must be a number!')
local result, i = n, 2
while i <= n do
if n % i == 0 then
while n % i == 0 do n = n // i end
result = result - (result // i)
end
if i == 2 then i = 1 end
i = i + 2
end
if n > 1 then result = result - (result // n) end
return result
end
local function phi_iter(n)
assert(type(n) == 'number', 'n must be a number!')
if n == 2 then
return phi(n) + 0
else
return phi(n) + phi_iter(phi(n))
end
end
local i, count = 2, 0
while count ~= 20 do
if i == phi_iter(i) then
io.write(i, ' ')
count = count + 1
end
i = i + 1
end
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
MAD
NORMAL MODE IS INTEGER
INTERNAL FUNCTION(Y,Z)
ENTRY TO GCD.
A = Y
B = Z
LOOP WHENEVER A.E.B, FUNCTION RETURN A
WHENEVER A.G.B, A = A-B
WHENEVER A.L.B, B = B-A
TRANSFER TO LOOP
END OF FUNCTION
INTERNAL FUNCTION(C)
ENTRY TO TOTENT.
E = 0
THROUGH LOOP, FOR D=1, 1, D.GE.C
LOOP WHENEVER GCD.(C,D).E.1, E = E+1
FUNCTION RETURN E
END OF FUNCTION
INTERNAL FUNCTION(G)
ENTRY TO PERFCT.
H = G
I = 0
LOOP H = TOTENT.(H)
I = I+H
WHENEVER H.G.1, TRANSFER TO LOOP
FUNCTION RETURN I.E.G
END OF FUNCTION
SEEN = 0
THROUGH LOOP, FOR N=3, 2, SEEN.GE.20
WHENEVER PERFCT.(N)
SEEN = SEEN+1
PRINT FORMAT FMT,N
END OF CONDITIONAL
LOOP CONTINUE
VECTOR VALUES FMT = $I9*$
END OF PROGRAM- Output:
3
9
15
27
39
81
111
183
243
255
327
363
471
729
2187
2199
3063
4359
4375
5571
Maple
iterated_totient := proc(n::posint, total)
if NumberTheory:-Totient(n) = 1 then
return total + 1;
else
return iterated_totient(NumberTheory:-Totient(n), total + NumberTheory:-Totient(n));
end if;
end proc:
isPerfect := n -> evalb(iterated_totient(n, 0) = n):
count := 0:
num_list := []:
for i while count < 20 do
if isPerfectTotient(i) then
num_list := [op(num_list), i];
count := count + 1;
end if;
end do;
num_list;- Output:
[3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
Mathematica / Wolfram Language
ClearAll[PerfectTotientNumberQ]
PerfectTotientNumberQ[n_Integer] := Total[Rest[Most[FixedPointList[EulerPhi, n]]]] == n
res = {};
i = 0;
While[Length[res] < 20,
i++;
If[PerfectTotientNumberQ[i], AppendTo[res, i]]
]
res
- Output:
{3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571}
Maxima
totientseq(n):=block(
[depth:1,L:n],
while totient(L)#1 do (L:totient(L),depth:depth+1),
j:n,
makelist(j:totient(j),depth),
append([n],%%))$
perfect_totient_p(n):=if n=1 then false else is(equal(n,apply("+",rest(totientseq(n)))))$
/* Function that returns a list of the first len perfect totient numbers */
perfect_totient_count(len):=block(
[i:1,count:0,result:[]],
while count<len do (if perfect_totient_p(i) then (result:endcons(i,result),count:count+1),i:i+1),
result)$
/*Test cases */
perfect_totient_count(20);
- Output:
[3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571]
Miranda
main :: [sys_message]
main = [Stdout (show (take 20 perfect) ++ "\n")]
perfect :: [num]
perfect = [k | k<-[1..]; k = totientsum k]
totientsum :: num->num
totientsum = sum . takewhile (>0) . tl . iterate totient
totient :: num->num
totient n = #[k | k<-[1..n-1]; n $gcd k = 1]
gcd :: num->num->num
gcd a 0 = a
gcd a b = gcd b (a mod b)- Output:
[3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571]
Modula-2
MODULE PerfectTotient;
FROM InOut IMPORT WriteCard, WriteLn;
CONST Amount = 20;
VAR n, seen: CARDINAL;
PROCEDURE GCD(a, b: CARDINAL): CARDINAL;
VAR c: CARDINAL;
BEGIN
WHILE b # 0 DO
c := a MOD b;
a := b;
b := c;
END;
RETURN a;
END GCD;
PROCEDURE Totient(n: CARDINAL): CARDINAL;
VAR i, tot: CARDINAL;
BEGIN
tot := 0;
FOR i := 1 TO n/2 DO
IF GCD(n,i) = 1 THEN
tot := tot + 1;
END;
END;
RETURN tot;
END Totient;
PROCEDURE Perfect(n: CARDINAL): BOOLEAN;
VAR sum, x: CARDINAL;
BEGIN
sum := 0;
x := n;
REPEAT
x := Totient(x);
sum := sum + x;
UNTIL x = 1;
RETURN sum = n;
END Perfect;
BEGIN
seen := 0;
n := 3;
WHILE seen < Amount DO
IF Perfect(n) THEN
WriteCard(n,5);
seen := seen + 1;
IF seen MOD 14 = 0 THEN
WriteLn();
END;
END;
n := n + 2;
END;
WriteLn();
END PerfectTotient.
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Nim
import strformat
func totient(n: int): int =
var tot = n
var nn = n
var i = 2
while i * i <= nn:
if nn mod i == 0:
while nn mod i == 0:
nn = nn div i
dec tot, tot div i
if i == 2:
i = 1
inc i, 2
if nn > 1:
dec tot, tot div nn
tot
var n = 1
var num = 0
echo "The first 20 perfect totient numbers are:"
while num < 20:
var tot = n
var sum = 0
while tot != 1:
tot = totient(tot)
inc sum, tot
if sum == n:
write(stdout, fmt"{n} ")
inc num
inc n, 2
write(stdout, "\n")
- Output:
The first 20 perfect totient numbers are: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Pascal
I am using a really big array to calculate the Totient of every number up to 1.162.261.467, the 46.te perfect totient number.
( I can only test up to 1.5e9 before I get - out of memory ( 6.5 GB ) ).
I'm doing this, by using only prime numbers to calculate the Totientnumbers.
After that I sum up the totient numbers Tot[i] := Tot[i]+Tot[Tot[i]];
Tot[Tot[i]] is always < Tot[i], so it is already calculated. So I needn't calculations going trough so whole array ending up in Tot[2].
With limit 57395631 it takes "real 0m2,025s "
The c-program takes "real 3m12,481s"
A test with using floating point/SSE is by 2 seconds faster for 46.th perfect totient number, with the coming new Version of Freepascal 3.2.0
program Perftotient;
{$IFdef FPC}
{$MODE DELPHI} {$CodeAlign proc=32,loop=1}
{$IFEND}
uses
sysutils;
const
cLimit = 57395631;//177147;//4190263;//57395631;//1162261467;//
//global
var
TotientList : array of LongWord;
Sieve : Array of byte;
SolList : array of LongWord;
T1,T0 : INt64;
procedure SieveInit(svLimit:NativeUint);
var
pSieve:pByte;
i,j,pr :NativeUint;
Begin
svlimit := (svLimit+1) DIV 2;
setlength(sieve,svlimit+1);
pSieve := @Sieve[0];
For i := 1 to svlimit do
Begin
IF pSieve[i]= 0 then
Begin
pr := 2*i+1;
j := (sqr(pr)-1) DIV 2;
IF j> svlimit then
BREAK;
repeat
pSieve[j]:= 1;
inc(j,pr);
until j> svlimit;
end;
end;
pr := 0;
j := 0;
For i := 1 to svlimit do
Begin
IF pSieve[i]= 0 then
Begin
pSieve[j] := i-pr;
inc(j);
pr := i;
end;
end;
setlength(sieve,j);
end;
procedure TotientInit(len: NativeUint);
var
pTotLst : pLongWord;
pSieve : pByte;
test : double;
i: NativeInt;
p,j,k,svLimit : NativeUint;
Begin
SieveInit(len);
T0:= GetTickCount64;
setlength(TotientList,len+12);
pTotLst := @TotientList[0];
//Fill totient with simple start values for odd and even numbers
//and multiples of 3
j := 1;
k := 1;// k == j DIV 2
p := 1;// p == j div 3;
repeat
pTotLst[j] := j;//1
pTotLst[j+1] := k;//2 j DIV 2; //2
inc(k);
inc(j,2);
pTotLst[j] := j-p;//3
inc(p);
pTotLst[j+1] := k;//4 j div 2
inc(k);
inc(j,2);
pTotLst[j] := j;//5
pTotLst[j+1] := p;//6 j DIV 3 <= (div 2) * 2 DIV/3
inc(j,2);
inc(p);
inc(k);
until j>len+6;
//correct values of totient by prime factors
svLimit := High(sieve);
p := 3;// starting after 3
pSieve := @Sieve[svLimit+1];
i := -svlimit;
repeat
p := p+2*pSieve[i];
j := p;
// Test := (1-1/p);
while j <= cLimit do
Begin
// pTotLst[j] := trunc(pTotLst[j]*Test);
k:= pTotLst[j];
pTotLst[j]:= k-(k DIV p);
inc(j,p);
end;
inc(i);
until i=0;
T1:= GetTickCount64;
writeln('totient calculated in ',T1-T0,' ms');
setlength(sieve,0);
end;
function GetPerfectTotient(len: NativeUint):NativeUint;
var
pTotLst : pLongWord;
i,sum: NativeUint;
Begin
T0:= GetTickCount64;
pTotLst := @TotientList[0];
setlength(SolList,100);
result := 0;
For i := 3 to Len do
Begin
sum := pTotLst[i];
pTotLst[i] := sum+pTotLst[sum];
end;
//Check for solution ( IF ) in seperate loop ,reduces time consuption ~ 12% for this function
For i := 3 to Len do
IF pTotLst[i] =i then
Begin
SolList[result] := i;
inc(result);
end;
T1:= GetTickCount64;
setlength(SolList,result);
writeln('calculated totientsum in ',T1-T0,' ms');
writeln('found ',result,' perfect totient numbers');
end;
var
j,k : NativeUint;
Begin
TotientInit(climit);
GetPerfectTotient(climit);
k := 0;
For j := 0 to High(Sollist) do
Begin
inc(k);
if k > 4 then
Begin
writeln(Sollist[j]);
k := 0;
end
else
write(Sollist[j],',');
end;
end.
- OutPut
compiled with fpc 3.0.4 -O3 "Perftotient.pas" totient calculated in 32484 ms calculated totientsum in 8244 ms found 46 perfect totient numbers 3,9,15,27,39 81,111,183,243,255 327,363,471,729,2187 2199,3063,4359,4375,5571 6561,8751,15723,19683,36759 46791,59049,65535,140103,177147 208191,441027,531441,1594323,4190263 4782969,9056583,14348907,43046721,57395631 129140163,172186887,236923383,387420489,918330183 1162261467, real 0m47,690s * found 40 perfect totient numbers ... real 0m2,025s
Perl
use ntheory qw(euler_phi);
sub phi_iter {
my($p) = @_;
euler_phi($p) + ($p == 2 ? 0 : phi_iter(euler_phi($p)));
}
my @perfect;
for (my $p = 2; @perfect < 20 ; ++$p) {
push @perfect, $p if $p == phi_iter($p);
}
printf "The first twenty perfect totient numbers:\n%s\n", join ' ', @perfect;
- Output:
The first twenty Perfect totient numbers: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Phix
with javascript_semantics function totient(integer n) integer tot = n, i = 2 while i*i<=n do if mod(n,i)=0 then while true do n /= i if mod(n,i)!=0 then exit end if end while tot -= tot/i end if i += iff(i=2?1:2) end while if n>1 then tot -= tot/n end if return tot end function sequence perfect = {} integer n = 1 while length(perfect)<20 do integer tot = n, tsum = 0 while tot!=1 do tot = totient(tot) tsum += tot end while if tsum=n then perfect &= n end if n += 2 end while printf(1,"The first 20 perfect totient numbers are:\n") ?perfect
- Output:
The first 20 perfect totient numbers are:
{3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571}
PicoLisp
(gc 16)
(de gcd (A B)
(until (=0 B)
(let M (% A B)
(setq A B B M) ) )
(abs A) )
(de totient (N)
(let C 0
(for I N
(and (=1 (gcd N I)) (inc 'C)) )
C ) )
(de totients (NIL)
(let (C 0 N 1)
(while (> 20 C)
(let (Cur N S 0)
(while (> Cur 1)
(inc 'S (setq Cur (totient Cur))) )
(when (= S N)
(inc 'C)
(prin N " ")
(flush) )
(inc 'N 2) ) )
(prinl) ) )
(totients)- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
PILOT
C :z=0
:n=3
*num
C :s=0
:x=n
*perfect
C :t=0
:i=1
*totient
C :a=x
:b=i
*gcd
C :c=a-b*(a/b)
:a=b
:b=c
J (b>0):*gcd
C (a=1):t=t+1
C :i=i+1
J (i<=x-1):*totient
C :x=t
:s=s+x
J (x<>1):*perfect
T (s=n):#n
C (s=n):z=z+1
C :n=n+2
J (z<20):*num
E :- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
PL/I
perfectTotient: procedure options(main);
gcd: procedure(aa, bb) returns(fixed);
declare (aa, bb, a, b, c) fixed;
a = aa;
b = bb;
do while(b ^= 0);
c = a;
a = b;
b = mod(c, b);
end;
return(a);
end gcd;
totient: procedure(n) returns(fixed);
declare (i, n, s) fixed;
s = 0;
do i=1 to n-1;
if gcd(n,i) = 1 then s = s+1;
end;
return(s);
end totient;
perfect: procedure(n) returns(bit);
declare (n, x, sum) fixed;
sum = 0;
x = n;
do while(x > 1);
x = totient(x);
sum = sum + x;
end;
return(sum = n);
end perfect;
declare (n, seen) fixed;
seen = 0;
do n=3 repeat(n+2) while(seen<20);
if perfect(n) then do;
put edit(n) (F(5));
seen = seen+1;
if mod(seen,10) = 0 then put skip;
end;
end;
end perfectTotient;- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
PL/M
100H:
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$NUMBER: PROCEDURE (N);
DECLARE S (7) 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;
GCD: PROCEDURE (A, B) ADDRESS;
DECLARE (A, B, C) ADDRESS;
DO WHILE B <> 0;
C = A;
A = B;
B = C MOD B;
END;
RETURN A;
END GCD;
TOTIENT: PROCEDURE (N) ADDRESS;
DECLARE (I, N, S) ADDRESS;
S = 0;
DO I=1 TO N-1;
IF GCD(N,I) = 1 THEN S = S+1;
END;
RETURN S;
END TOTIENT;
PERFECT: PROCEDURE (N) BYTE;
DECLARE (N, X, SUM) ADDRESS;
X = N;
SUM = 0;
DO WHILE X > 1;
X = TOTIENT(X);
SUM = SUM + X;
END;
RETURN SUM = N;
END PERFECT;
DECLARE N ADDRESS, SEEN BYTE;
SEEN = 0;
N = 3;
DO WHILE SEEN < 20;
IF PERFECT(N) THEN DO;
CALL PRINT$NUMBER(N);
SEEN = SEEN+1;
END;
N = N+2;
END;
CALL EXIT;
EOF- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Python
from math import gcd
from functools import lru_cache
from itertools import islice, count
@lru_cache(maxsize=None)
def φ(n):
return sum(1 for k in range(1, n + 1) if gcd(n, k) == 1)
def perfect_totient():
for n0 in count(1):
parts, n = 0, n0
while n != 1:
n = φ(n)
parts += n
if parts == n0:
yield n0
if __name__ == '__main__':
print(list(islice(perfect_totient(), 20)))
- Output:
[3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
Alternatively, by composition of generic functions:
'''Perfect totient numbers'''
from functools import lru_cache
from itertools import count, islice
from math import gcd
import operator
# perfectTotients :: () -> [Int]
def perfectTotients():
'''An unbounded sequence of perfect totients.
OEIS A082897
'''
def p(x):
return x == 1 + sum(
iterateUntil(eq(1))(
phi
)(x)[1:]
)
return filter(p, count(2))
@lru_cache(maxsize=None)
def phi(n):
'''Euler's totient function.
The count of integers up to n which
are relatively prime to n.
'''
return len([
x for x in enumFromTo(1)(n)
if 1 == gcd(n, x)
])
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''First twenty perfect totient numbers'''
print(
take(20)(
perfectTotients()
)
)
# GENERIC -------------------------------------------------
# curry :: ((a, b) -> c) -> a -> b -> c
def curry(f):
'''A curried function derived
from an uncurried function.
'''
return lambda x: lambda y: f(x, y)
# enumFromTo :: Int -> Int -> [Int]
def enumFromTo(m):
'''Enumeration of integer values [m..n]'''
return lambda n: range(m, 1 + n)
# eq (==) :: Eq a => a -> a -> Bool
eq = curry(operator.eq)
'''True if a and b are comparable and a equals b.'''
# iterateUntil :: (a -> Bool) -> (a -> a) -> a -> [a]
def iterateUntil(p):
'''A list of the results of repeated
applications of f, until p matches.
'''
def go(f, x):
vs = []
v = x
while True:
if p(v):
break
vs.append(v)
v = f(v)
return vs
return lambda f: lambda x: go(f, x)
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.
'''
return lambda xs: (
xs[0:n]
if isinstance(xs, (list, tuple))
else list(islice(xs, n))
)
# MAIN ---
if __name__ == '__main__':
main()
- Output:
[3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
Quackery
totient is defined at Totient function#Quackery.
[ 0 over
[ dup 1 != while
totient
dup dip +
again ]
drop = ] is perfecttotient ( n --> b )
[ [] 1
[ dup perfecttotient if
[ dup dip join ]
2 +
over size 20 =
until ] drop ] is task ( --> )- Output:
[ 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571 ]
Racket
#lang racket
(require math/number-theory)
(define (tot n)
(match n
[1 0]
[n (define t (totient n))
(+ t (tot t))]))
(define (perfect? n)
(= n (tot n)))
(define-values (ns i)
(for/fold ([ns '()] [i 0])
([n (in-naturals 1)]
#:break (= i 20)
#:when (perfect? n))
(values (cons n ns) (+ i 1))))
(reverse ns)
Raku
(formerly Perl 6)
use Prime::Factor;
my \𝜑 = lazy 0, |(1..*).hyper.map: -> \t { t * [*] t.&prime-factors.squish.map: 1 - 1/* }
my \𝜑𝜑 = Nil, |(3, *+2 … *).grep: -> \p { p == sum 𝜑[p], { 𝜑[$_] } … 1 };
put "The first twenty Perfect totient numbers:\n", 𝜑𝜑[1..20];
- Output:
The first twenty Perfect totient numbers: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
REXX
unoptimized
/*REXX program calculates and displays the first N perfect totient numbers. */
parse arg N . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 20 /*Not specified? Then use the default.*/
@.= . /*memoization array of totient numbers.*/
p= 0 /*the count of perfect " " */
$= /*list of the " " " */
do j=3 by 2 until p==N; s= phi(j) /*obtain totient number for a number. */
a= s /* [↓] search for a perfect totient #.*/
do until a==1; a= phi(a); s= s + a
end /*until*/
if s\==j then iterate /*Is J not a perfect totient number? */
p= p + 1 /*bump count of perfect totient numbers*/
$= $ j /*add to perfect totient numbers list. */
end /*j*/
say 'The first ' N " perfect totient numbers:" /*display the header to the terminal. */
say strip($) /* " " list. " " " */
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: parse arg x,y; do until y==0; parse value x//y y with y x; end; return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
phi: procedure expose @.; parse arg z; if @.z\==. then return @.z /*was found before?*/
#= z==1; do m=1 for z-1; if gcd(m, z)==1 then #= # + 1; end /*m*/
@.z= #; return # /*use memoization. */
- output when using the default input of : 20
The first 20 perfect totient numbers: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
optimized
This REXX version is over twice as fast as the unoptimized version.
It takes advantage of the fact that all known perfect totient numbers less than 322 have at least one of these factors: 3, 5, or 7
(322 = 31,381,059,609).
/*REXX program calculates and displays the first N perfect totient numbers. */
parse arg N . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 20 /*Not specified? Then use the default.*/
@.= . /*memoization array of totient numbers.*/
p= 0 /*the count of perfect " " */
$= /*list of the " " " */
do j=3 by 2 until p==N /*obtain the totient number for index J*/
if j//3\==0 then if j//5\==0 then if j//7\==0 then iterate
s= phi(j); a= s /* [↑] J must have 1 of these factors*/
do until a==1; if @.a==. then a= phi(a); else a= @.a
s= s + a
end /*until*/
if s\==j then iterate /*Is J not a perfect totient number? */
p= p + 1 /*bump count of perfect totient numbers*/
$= $ j /*add to perfect totient numbers list. */
end /*j*/
say 'The first ' N " perfect totient numbers:" /*display the header to the terminal. */
say strip($) /* " " list. " " " */
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: parse arg x,y; do until y==0; parse value x//y y with y x; end; return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
phi: procedure expose @.; parse arg z; if @.z\==. then return @.z /*was found before?*/
#= z==1; do m=1 for z-1; if gcd(m, z)==1 then #= # + 1; end /*m*/
@.z= #; return # /*use memoization. */
- output is identical to the 1st REXX version.
Ring
perfect = []
n = 1
while len(perfect)<20
totnt = n
tsum = 0
while totnt!=1
totnt = totient(totnt)
tsum = tsum + totnt
end
if tsum=n
add(perfect,n)
ok
n = n + 2
end
see "The first 20 perfect totient numbers are:" + nl
showarray(perfect)
func totient n
totnt = n
i = 2
while i*i <= n
if n%i=0
while true
n = n/i
if n%i!=0
exit
ok
end
totnt = totnt - totnt/i
ok
if i=2
i = i + 1
else
i = i + 2
ok
end
if n>1
totnt = totnt - totnt/n
ok
return totnt
func showArray array
txt = ""
see "["
for n = 1 to len(array)
txt = txt + array[n] + ","
next
txt = left(txt,len(txt)-1)
txt = txt + "]"
see txtThe first 20 perfect totient numbers are: [3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571]
RPL
« 0 OVER
WHILE DUP 1 ≠ REPEAT
EULER SWAP OVER + SWAP
END DROP ==
» 'PTOT?' STO
« → n
« { } 1
WHILE n REPEAT
IF DUP PTOT? THEN
SWAP OVER + SWAP
'n' 1 STO-
END
2 + @ Perfect totient numbers are odd
END DROP
» » 'TASK' STO
20 TASK
- Output:
1: { 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571 }
Ruby
require "prime"
class Integer
def φ
prime_division.inject(1) {|res, (pr, exp)| res *= (pr-1) * pr**(exp-1) }
end
def perfect_totient?
f, sum = self, 0
until f == 1 do
f = f.φ
sum += f
end
self == sum
end
end
puts (1..).lazy.select(&:perfect_totient?).first(20).join(", ")
- Output:
3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571
Rust
use num::integer::gcd;
static mut CACHE:[i32;10000] = [0; 10000];
fn is_perfect_totient(n: i32) -> bool {
let mut tot = 0;
for i in 1..n {
if gcd(n, i) == 1 {
tot += 1
}
}
unsafe {
let sum = tot + CACHE[tot as usize];
CACHE[n as usize] = sum;
return n == sum;
}
}
fn main() {
let mut i = 1;
let mut count = 0;
while count < 20 {
if is_perfect_totient(i) {
print!("{} ", i);
count += 1;
}
i += 1;
}
println!("{}", " ")
}
- Output:
3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
Scala
In this example we define a function which determines whether or not a number is a perfect totient number, then use it to construct a lazily evaluated list which contains all perfect totient numbers. Calculating the first n perfect totient numbers only requires taking the first n elements from the list.
//List of perfect totients
def isPerfectTotient(num: Int): Boolean = LazyList.iterate(totient(num))(totient).takeWhile(_ != 1).foldLeft(0L)(_+_) + 1 == num
def perfectTotients: LazyList[Int] = LazyList.from(3).filter(isPerfectTotient)
//Totient Function
@tailrec def scrub(f: Long, num: Long): Long = if(num%f == 0) scrub(f, num/f) else num
def totient(num: Long): Long = LazyList.iterate((num, 2: Long, num)){case (ac, i, n) => if(n%i == 0) (ac*(i - 1)/i, i + 1, scrub(i, n)) else (ac, i + 1, n)}.dropWhile(_._3 != 1).head._1
- Output:
scala> perfectTotients.take(20).mkString(", ")
res1: String = 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571
Sidef
func perfect_totient({.<=1}, sum=0) { sum }
func perfect_totient( n, sum=0) { __FUNC__(var(t = n.euler_phi), sum + t) }
say (1..Inf -> lazy.grep {|n| perfect_totient(n) == n }.first(20))
- Output:
[3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
Swift
public func totient(n: Int) -> Int {
var n = n
var i = 2
var tot = n
while i * i <= n {
if n % i == 0 {
while n % i == 0 {
n /= i
}
tot -= tot / i
}
if i == 2 {
i = 1
}
i += 2
}
if n > 1 {
tot -= tot / n
}
return tot
}
public struct PerfectTotients: Sequence, IteratorProtocol {
private var m = 1
public init() { }
public mutating func next() -> Int? {
while true {
defer {
m += 1
}
var tot = m
var sum = 0
while tot != 1 {
tot = totient(n: tot)
sum += tot
}
if sum == m {
return m
}
}
}
}
print("The first 20 perfect totient numbers are:")
print(Array(PerfectTotients().prefix(20)))
- Output:
The first 20 perfect totient numbers are: [3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
Tcl
array set cache {}
set cache(0) 0
proc gcd {i j} {
while {$j != 0} {
set t [expr {$i % $j}]
set i $j
set j $t
}
return $i
}
proc is_perfect_totient {n} {
global cache
set tot 0
for {set i 1} {$i < $n} {incr i} {
if [ expr [gcd $i $n] == 1 ] {
incr tot
}
}
set sum [expr $tot + $cache($tot)]
set cache($n) $sum
return [ expr $n == $sum ? 1 : 0]
}
set i 1
set count 0
while { $count < 20 } {
if [ is_perfect_totient $i ] {
puts -nonewline "${i} "
incr count
}
incr i
}
puts ""
- Output:
$ time tclsh perfect-totient.tcl 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571 real 1m18,058s user 1m17,593s sys 0m0,046s
Wren
The version using Euler's product formula.
import "./math" for Int
var perfect = []
var n = 1
while (perfect.count < 20) {
var tot = n
var sum = 0
while (tot != 1) {
tot = Int.totient(tot)
sum = sum + tot
}
if (sum == n) perfect.add(n)
n = n + 2
}
System.print("The first 20 perfect totient numbers are:")
System.print(perfect)
- Output:
The first 20 perfect totient numbers are: [3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571]
XPL0
func Totient(N);
int N, Tot, I;
[Tot:= N; I:= 2;
while I*I <= N do
[if rem(N/I) = 0 then
[while rem(N/I) = 0 do N:= N/I;
Tot:= Tot - Tot/I;
];
if I = 2 then I:= 1;
I:= I+2;
];
if N > 1 then Tot:= Tot - Tot/N;
return Tot;
];
proc ShowPerfect;
int N, Count, Tot, Sum;
[N:= 1; Count:= 0;
while Count < 20 do
[Tot:= N; Sum:= 0;
while Tot # 1 do
[Tot:= Totient(Tot);
Sum:= Sum + Tot;
];
if Sum = N then
[IntOut(0, N); ChOut(0, ^ );
Count:= Count+1;
];
N:= N+2;
];
];
[Text(0, "The first 20 perfect totient numbers are:^m^j");
ShowPerfect;
]- Output:
The first 20 perfect totient numbers are: 3 9 15 27 39 81 111 183 243 255 327 363 471 729 2187 2199 3063 4359 4375 5571
zkl
var totients=List.createLong(10_000,0); // cache
fcn totient(n){ if(phi:=totients[n]) return(phi);
totients[n]=[1..n].reduce('wrap(p,k){ p + (n.gcd(k)==1) })
}
fcn perfectTotientW{ // -->iterator
(1).walker(*).tweak(fcn(z){
parts,n := 0,z;
while(n!=1){ parts+=( n=totient(n) ) }
if(parts==z) z else Void.Skip;
})
}perfectTotientW().walk(20).println();- Output:
L(3,9,15,27,39,81,111,183,243,255,327,363,471,729,2187,2199,3063,4359,4375,5571)
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