# Perfect totient numbers

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.

Also see

## AWK

` # syntax: GAWK -f PERFECT_TOTIENT_NUMBERS.AWKBEGIN {    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
```

## 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]
```

## Factor

`USING: formatting kernel lists lists.lazy mathmath.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.

`import Data.Bool (bool) perfectTotients :: [Int]perfectTotients =  [2 ..] >>=  ((bool [] . return) <*>   ((==) <*> (succ . sum . tail . takeWhile (1 /=) . iterate φ))) φ :: Int -> Intφ = memoize (\n -> length (filter ((1 ==) . gcd n) [1 .. n])) memoize :: (Int -> a) -> (Int -> a)memoize f = (map 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 =: [: }. (, [email protected]{:)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
```

## 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]`

## 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    perfectend 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

Translation of: Go
`// 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]
```

## 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]
```

## 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 = 1var num = 0echo "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, 2write(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;////globalvar  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

Library: ntheory
`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```

## Perl 6

Works with: Rakudo version 2018.11
`my \𝜑  = Nil, |(1..*).hyper.map: -> \$t { +(^\$t).grep: * gcd \$t == 1 };my \𝜑𝜑 = Nil, |(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```

## Phix

Translation of: Go
`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 totend function sequence perfect = {}integer n = 1while 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 += 2end whileprintf(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
```

## Python

`from math import gcdfrom functools import lru_cachefrom 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]`

## 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) `

## 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                                             /*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 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                                             /*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.

## 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
```

## 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 totientsdef isPerfectTotient(num: Int): Boolean = LazyList.iterate(totient(num))(totient).takeWhile(_ != 1).foldLeft(0L)(_+_) + 1 == numdef 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 numdef 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]```

## zkl

`var totients=List.createLong(10_000,0);	// cachefcn 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)
```