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# Achilles numbers

Achilles numbers
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 This page uses content from Wikipedia. The original article was at Achilles number. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

An Achilles number is a number that is powerful but imperfect. Named after Achilles, a hero of the Trojan war, who was also powerful but imperfect.

A positive integer n is a powerful number if, for every prime factor p of n, p2 is also a divisor.

In other words, every prime factor appears at least squared in the factorization.

All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as mk, where m and k are positive integers greater than 1.

A strong Achilles number is an Achilles number whose Euler totient (𝜑) is also an Achilles number.

E.G.

108 is a powerful number. Its prime factorization is 22 × 33, and thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number.

360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by 52 = 25.

Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power; its square root is an even integer, so it is not an Achilles number.

500 = 22 × 53 is a strong Achilles number as its Euler totient, 𝜑(500), is 200 = 23 × 52 which is also an Achilles number.

• Find and show the first 50 Achilles numbers.
• Find and show at least the first 20 strong Achilles numbers.
• For at least 2 through 5, show the count of Achilles numbers with that many digits.

## AArch64 Assembly

Works with: as version Raspberry Pi 3B version Buster 64 bits
or android 64 bits with application Termux
` /* ARM assembly AARCH64 Raspberry PI 3B *//*  program achilleNumber.s   */ /************************************//* Constantes                       *//************************************/.include "../includeConstantesARM64.inc" .equ NBFACT,    33.equ MAXI,      50.equ MAXI1,     20.equ MAXI2,     1000000  /*********************************//* Initialized data              *//*********************************/.dataszMessNumber:       .asciz " @ "szCarriageReturn:   .asciz "\n"szErrorGen:         .asciz "Program error !!!\n"szMessPrime:        .asciz "This number is prime.\n"szMessErrGen:       .asciz "Error end program.\n"szMessNbPrem:       .asciz "This number is prime !!!.\n"szMessOverflow:     .asciz "Overflow function isPrime.\n"szMessError:        .asciz "\033[31mError  !!!\n"szMessTitAchille:   .asciz "First 50 Achilles Numbers:\n"szMessTitStrong:    .asciz "First 20 Strong Achilles Numbers:\n"szMessDigitsCounter: .asciz "Numbers with @ digits : @ \n"/*********************************//* UnInitialized data            *//*********************************/.bss  sZoneConv:           .skip 24tbZoneDecom:         .skip 16 * NBFACT  // factor 8 bytes, number of each factor 8 bytes/*********************************//*  code section                 *//*********************************/.text.global main main:                             // entry of program      ldr x0,qAdrszMessTitAchille    bl affichageMess    mov x4,#1                      // start number    mov x5,#0                      // total counter    mov x6,#0                      // line display counter1:     mov x0,x4    bl controlAchille    cmp x0,#0                      // achille number ?    beq 2f                         // no    mov x0,x4    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 x5,x5,#1                   // increment counter    add x6,x6,#1                   // increment indice line display    cmp x6,#10                     // if = 10  new line    bne 2f    mov x6,#0    ldr x0,qAdrszCarriageReturn    bl affichageMess 2:    add x4,x4,#1                   // increment number    cmp x5,#MAXI    blt 1b                         // and loop     ldr x0,qAdrszMessTitStrong    bl affichageMess    mov x4,#1                      // start number    mov x5,#0                      // total counter    mov x6,#0 3:     mov x0,x4    bl controlAchille    cmp x0,#0    beq 4f    mov x0,x4    bl computeTotient    bl controlAchille    cmp x0,#0    beq 4f    mov x0,x4    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 x5,x5,#1    add x6,x6,#1    cmp x6,#10    bne 4f    mov x6,#0    ldr x0,qAdrszCarriageReturn    bl affichageMess 4:    add x4,x4,#1    cmp x5,#MAXI1    blt 3b     ldr x3,icstMaxi2    mov x4,#1                      // start number    mov x6,#0                      // total counter 2 digits    mov x7,#0                      // total counter 3 digits    mov x8,#0                      // total counter 4 digits    mov x9,#0                      // total counter 5 digits    mov x10,#0                     // total counter 6 digits5:     mov x0,x4    bl controlAchille    cmp x0,#0    beq 10f     mov x0,x4    ldr x1,qAdrsZoneConv    bl conversion10             // call décimal conversion x0 return digit number    cmp x0,#6    bne 6f    add x10,x10,#1    beq 10f 6:    cmp x0,#5    bne 7f    add x9,x9,#1    b 10f 7:    cmp x0,#4    bne 8f    add x8,x8,#1    b 10f 8:    cmp x0,#3    bne 9f    add x7,x7,#1    b 10f 9:    cmp x0,#2    bne 10f    add x6,x6,#110:     add x4,x4,#1    cmp x4,x3    blt 5b    mov x0,#2    mov x1,x6    bl displayCounter    mov x0,#3    mov x1,x7    bl displayCounter    mov x0,#4    mov x1,x8    bl displayCounter    mov x0,#5    mov x1,x9    bl displayCounter    mov x0,#6    mov x1,x10    bl displayCounter    b 100f98:    ldr x0,qAdrszErrorGen    bl affichageMess 100:                              // standard end of the program     mov x0, #0                    // return code    mov x8,EXIT     svc #0                        // perform the system callqAdrszCarriageReturn:    .quad szCarriageReturnqAdrszErrorGen:          .quad szErrorGenqAdrsZoneConv:           .quad sZoneConv  qAdrtbZoneDecom:         .quad tbZoneDecomqAdrszMessNumber:        .quad szMessNumberqAdrszMessTitAchille:    .quad szMessTitAchilleqAdrszMessTitStrong:     .quad szMessTitStrongicstMaxi2:               .quad MAXI2/******************************************************************//*     display digit counter                        */ /******************************************************************//* x0 contains limit  *//* x1 contains counter */displayCounter:    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,qAdrszMessDigitsCounter    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 message100:    ldp x2,x3,[sp],16           // restaur  registers     ldp x1,lr,[sp],16           // restaur  registers    ret qAdrszMessDigitsCounter:   .quad szMessDigitsCounter/******************************************************************//*     control if number is Achille number                        */ /******************************************************************//* x0 contains number  *//* x0 return 0 if not else return 1 */controlAchille:    stp x1,lr,[sp,-16]!          // save  registers     stp x2,x3,[sp,-16]!          // save  registers     stp x4,x5,[sp,-16]!          // save  registers     mov x4,x0    ldr x1,qAdrtbZoneDecom    bl decompFact               // factor decomposition    cmp x0,#-1    beq 99f                     // error ?    cmp x0,#1                   // one only factor or prime ?    ble 98f    mov x1,x0    ldr x0,qAdrtbZoneDecom    mov x2,x4    bl controlDivisor    b 100f98:    mov x0,#0    b 100f99:    ldr x0,qAdrszErrorGen    bl affichageMess 100:    ldp x4,x5,[sp],16        // restaur  registers     ldp x2,x3,[sp],16        // restaur  registers     ldp x1,lr,[sp],16            // restaur  registers    ret /******************************************************************//*     control divisors function                         */ /******************************************************************//* x0 contains address of divisors area *//* x1 contains the number of area items  *//* x2 contains number  */controlDivisor:    stp x1,lr,[sp,-16]!          // save  registers     stp x2,x3,[sp,-16]!          // save  registers     stp x4,x5,[sp,-16]!          // save  registers     stp x6,x7,[sp,-16]!          // save  registers     stp x8,x9,[sp,-16]!          // save  registers     stp x10,x11,[sp,-16]!          // save  registers      mov x6,x1                   // factors number    mov x8,x2                   // save number    mov x9,#0                   // indice    mov x4,x0                   // save area address    add x5,x4,x9,lsl #4         // compute address first factor    ldr x7,[x5,#8]              // load first exposant of factor    add x2,x9,#11:    add x5,x4,x2,lsl #4         // compute address next factor    ldr x3,[x5,#8]              // load exposant of factor    cmp x3,x7                   // factor exposant <> ?    bne 2f                      // yes -> end verif    add x2,x2,#1                // increment indice    cmp x2,x6                   // factor maxi ?    blt 1b                      // no -> loop    mov x0,#0    b 100f                      // all exposants are equals2:    mov x10,x2                  // save indice21:    bge 22f    mov x2,x7                 // if x3 < x7 -> inversion    mov x7,x3    mov x3,x2                 // x7 is the smaller exposant22:    mov x0,x3    mov x1,x7                   // x7 < x3     bl calPGCDmod    cmp x0,#1    beq 24f                     // no commun multiple -> ne peux donc pas etre une puissance23:    add x10,x10,#1              // increment indice    cmp x10,x6                  // factor maxi ?    bge 99f                     // yes -> all exposants are multiples to smaller    add x5,x4,x10,lsl #4    ldr x3,[x5,#8]              // load exposant of next factor    cmp x3,x7    beq 23b                     // for next    b 21b                       // for compare the 2 exposants 24:    mov x9,#0                   // indice3:    add x5,x4,x9,lsl #4    ldr x7,[x5]                 // load factor    mul x1,x7,x7                // factor square    udiv x2,x8,x1    msub x3,x1,x2,x8            // compute remainder    cmp x3,#0                   // remainder null ?    bne 99f     add x9,x9,#1                // other factor    cmp x9,x6                   // factors maxi ?    blt 3b    mov x0,#1                   // achille number ok    b 100f99:                             // achille not ok    mov x0,0100:    ldp x10,x11,[sp],16            // restaur  registers    ldp x8,x9,[sp],16            // restaur  registers    ldp x6,x7,[sp],16            // restaur  registers    ldp x4,x5,[sp],16            // restaur  registers    ldp x2,x3,[sp],16            // restaur  registers    ldp x1,lr,[sp],16            // restaur  registers    ret  /***************************************************//*   Compute pgcd  modulo use                     *//***************************************************//* x0 contains first number *//* x1 contains second number *//* x0 return  PGCD            *//* if error carry set to 1    */calPGCDmod:    stp x1,lr,[sp,-16]!        // save  registres    stp x2,x3,[sp,-16]!        // save  registres    cbz x0,99f                 // if = 0 error    cbz x1,99f    cmp x0,0    bgt 1f    neg x0,x0                  // if negative inversion number 11:    cmp x1,0    bgt 2f    neg x1,x1                  // if negative inversion number 22:    cmp x0,x1                  // compare two numbers    bgt 3f    mov x2,x0                  // inversion    mov x0,x1    mov x1,x23:    udiv x2,x0,x1              // division    msub x0,x2,x1,x0           // compute remainder    cmp x0,0    bgt 2b                     // loop    mov x0,x1    cmn x0,0                   // clear carry    b 100f99:                            // error    mov x0,0    cmp x0,0                   // set carry100:    ldp x2,x3,[sp],16          // restaur des  2 registres    ldp x1,lr,[sp],16          // restaur des  2 registres    ret                        // retour adresse lr x30/******************************************************************//*     compute totient of number                                  */ /******************************************************************//* x0 contains number  */computeTotient:    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 divisor1:                            // 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 3f2:                            // 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 totient3:    cmp x1,#2                 // first divisor ?    mov x0,1    csel x1,x0,x1,eq          // divisor = 1    b 1b                      // and loop4:    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 totient5:     mov x0,x4100:    ldp x4,x5,[sp],16         // restaur  registers     ldp x2,x3,[sp],16         // restaur  registers     ldp x1,lr,[sp],16         // restaur  registers    ret /******************************************************************//*     factor decomposition                                               */ /******************************************************************//* x0 contains number *//* x1 contains address of divisors area *//* x0 return divisors items in table */decompFact:    stp x1,lr,[sp,-16]!          // save  registers     stp x2,x3,[sp,-16]!          // save  registers     stp x4,x5,[sp,-16]!          // save  registers     stp x6,x7,[sp,-16]!          // save  registers     stp x8,x9,[sp,-16]!          // save  registers     mov x5,x1    mov x8,x0                  // save number    bl isPrime                 // prime ?    cmp x0,#1    beq 98f                    // yes is prime    mov x4,#0                  // raz indice    mov x1,#2                  // first divisor    mov x6,#0                  // previous divisor    mov x7,#0                  // number of same divisors2:    udiv x2,x8,x1              // divide number or other result    msub x3,x2,x1,x8           // compute remainder    cmp x3,#0    bne 5f                     // if remainder <> zero  -> no divisor    mov x8,x2                  // else quotient -> new dividende    cmp x1,x6                  // same divisor ?    beq 4f                     // yes    cmp x6,#0                  // no but is the first divisor ?    beq 3f                     // yes     str x6,[x5,x4,lsl #3]      // else store in the table    add x4,x4,#1               // and increment counter    str x7,[x5,x4,lsl #3]      // store counter    add x4,x4,#1               // next item    mov x7,#0                  // and raz counter3:    mov x6,x1                  // new divisor4:    add x7,x7,#1               // increment counter    b 7f                       // and loop     /* not divisor -> increment next divisor */5:    cmp x1,#2                  // if divisor = 2 -> add 1     mov x0,#1    mov x3,#2                  // else add 2    csel x3,x0,x3,eq    add x1,x1,x3    b 2b     /* divisor -> test if new dividende is prime */7:     mov x3,x1                  // save divisor    cmp x8,#1                  // dividende = 1 ? -> end    beq 10f    mov x0,x8                  // new dividende is prime ?    mov x1,#0    bl isPrime                 // the new dividende is prime ?    cmp x0,#1    bne 10f                    // the new dividende is not prime     cmp x8,x6                  // else dividende is same divisor ?    beq 9f                     // yes    cmp x6,#0                  // no but is the first divisor ?    beq 8f                     // yes it is a first    str x6,[x5,x4,lsl #3]      // else store in table    add x4,x4,#1               // and increment counter    str x7,[x5,x4,lsl #3]      // and store counter     add x4,x4,#1               // next item8:    mov x6,x8                  // new dividende -> divisor prec    mov x7,#0                  // and raz counter9:    add x7,x7,#1               // increment counter    b 11f 10:    mov x1,x3                  // current divisor = new divisor    cmp x1,x8                  // current divisor  > new dividende ?    ble 2b                     // no -> loop     /* end decomposition */ 11:    str x6,[x5,x4,lsl #3]      // store last divisor    add x4,x4,#1    str x7,[x5,x4,lsl #3]      // and store last number of same divisors    add x4,x4,#1    lsr x0,x4,#1               // return number of table items    mov x3,#0    str x3,[x5,x4,lsl #3]      // store zéro in last table item    add x4,x4,#1    str x3,[x5,x4,lsl #3]      // and zero in counter same divisor    b 100f  98:     //ldr x0,qAdrszMessPrime    //bl   affichageMess    mov x0,#0                  // return code 0 = number is prime    b 100f99:    ldr x0,qAdrszMessErrGen    bl   affichageMess    mov x0,#-1                  // error code    b 100f100:    ldp x8,x9,[sp],16        // restaur  registers     ldp x6,x7,[sp],16        // restaur  registers     ldp x4,x5,[sp],16        // restaur  registers     ldp x2,x3,[sp],16        // restaur  registers     ldp x1,lr,[sp],16            // restaur  registers    ret qAdrszMessErrGen:          .quad szMessErrGen /***************************************************//*   Verification si un nombre est premier         *//***************************************************//* x0 contient le nombre à verifier *//* x0 retourne 1 si premier  0 sinon */isPrime:    stp x1,lr,[sp,-16]!        // save  registres    stp x2,x3,[sp,-16]!        // save  registres    mov x2,x0    sub x1,x0,#1    cmp x2,0    beq 99f                    // retourne zéro    cmp x2,2                   // pour 1 et 2 retourne 1    ble 2f    mov x0,#2    bl moduloPur64    bcs 100f                   // erreur overflow    cmp x0,#1    bne 99f                    // Pas premier    cmp x2,3    beq 2f    mov x0,#3    bl moduloPur64    blt 100f                   // erreur overflow    cmp x0,#1    bne 99f     cmp x2,5    beq 2f    mov x0,#5    bl moduloPur64    bcs 100f                   // erreur overflow    cmp x0,#1    bne 99f                    // Pas premier     cmp x2,7    beq 2f    mov x0,#7    bl moduloPur64    bcs 100f                   // erreur overflow    cmp x0,#1    bne 99f                    // Pas premier     cmp x2,11    beq 2f    mov x0,#11    bl moduloPur64    bcs 100f                   // erreur overflow    cmp x0,#1    bne 99f                    // Pas premier     cmp x2,13    beq 2f    mov x0,#13    bl moduloPur64    bcs 100f                   // erreur overflow    cmp x0,#1    bne 99f                    // Pas premier     cmp x2,17    beq 2f    mov x0,#17    bl moduloPur64    bcs 100f                   // erreur overflow    cmp x0,#1    bne 99f                    // Pas premier2:    cmn x0,0                   // carry à zero pas d'erreur    mov x0,1                   // premier    b 100f99:    cmn x0,0                   // carry à zero pas d'erreur    mov x0,#0                  // Pas premier100:    ldp x2,x3,[sp],16          // restaur des  2 registres    ldp x1,lr,[sp],16          // restaur des  2 registres    ret                        // retour adresse lr x30 /**************************************************************//********************************************************//*   Calcul modulo de b puissance e modulo m  *//*    Exemple 4 puissance 13 modulo 497 = 445         *//********************************************************//* x0  nombre  *//* x1 exposant *//* x2 modulo   */moduloPur64:    stp x1,lr,[sp,-16]!        // save  registres    stp x3,x4,[sp,-16]!        // save  registres    stp x5,x6,[sp,-16]!        // save  registres    stp x7,x8,[sp,-16]!        // save  registres    stp x9,x10,[sp,-16]!        // save  registres    cbz x0,100f    cbz x1,100f    mov x8,x0    mov x7,x1    mov x6,1                   // resultat    udiv x4,x8,x2    msub x9,x4,x2,x8           // contient le reste1:    tst x7,1    beq 2f    mul x4,x9,x6    umulh x5,x9,x6    //cbnz x5,99f    mov x6,x4    mov x0,x6    mov x1,x5    bl divisionReg128U    cbnz x1,99f                // overflow    mov x6,x32:    mul x8,x9,x9    umulh x5,x9,x9    mov x0,x8    mov x1,x5    bl divisionReg128U    cbnz x1,99f                // overflow    mov x9,x3    lsr x7,x7,1    cbnz x7,1b    mov x0,x6                  // result    cmn x0,0                   // carry à zero pas d'erreur    b 100f99:    ldr x0,qAdrszMessOverflow    bl  affichageMess    cmp x0,0                   // carry à un car erreur    mov x0,-1                  // code erreur 100:    ldp x9,x10,[sp],16          // restaur des  2 registres    ldp x7,x8,[sp],16          // restaur des  2 registres    ldp x5,x6,[sp],16          // restaur des  2 registres    ldp x3,x4,[sp],16          // restaur des  2 registres    ldp x1,lr,[sp],16          // restaur des  2 registres    ret                        // retour adresse lr x30qAdrszMessOverflow:         .quad  szMessOverflow/***************************************************//*   division d un nombre de 128 bits par un nombre de 64 bits *//***************************************************//* x0 contient partie basse dividende *//* x1 contient partie haute dividente *//* x2 contient le diviseur *//* x0 retourne partie basse quotient *//* x1 retourne partie haute quotient *//* x3 retourne le reste */divisionReg128U:    stp x6,lr,[sp,-16]!        // save  registres    stp x4,x5,[sp,-16]!        // save  registres    mov x5,#0                  // raz du reste R    mov x3,#128                // compteur de boucle    mov x4,#0                  // dernier bit1:        lsl x5,x5,#1               // on decale le reste de 1    tst x1,1<<63               // test du bit le plus à gauche    lsl x1,x1,#1               // on decale la partie haute du quotient de 1    beq 2f    orr  x5,x5,#1              // et on le pousse dans le reste R2:    tst x0,1<<63    lsl x0,x0,#1               // puis on decale la partie basse     beq 3f    orr x1,x1,#1               // et on pousse le bit de gauche dans la partie haute3:    orr x0,x0,x4               // position du dernier bit du quotient    mov x4,#0                  // raz du bit    cmp x5,x2    blt 4f    sub x5,x5,x2                // on enleve le diviseur du reste    mov x4,#1                   // dernier bit à 14:                               // et boucle    subs x3,x3,#1    bgt 1b        lsl x1,x1,#1               // on decale le quotient de 1    tst x0,1<<63    lsl x0,x0,#1              // puis on decale la partie basse     beq 5f    orr x1,x1,#15:    orr x0,x0,x4                  // position du dernier bit du quotient    mov x3,x5100:    ldp x4,x5,[sp],16          // restaur des  2 registres    ldp x6,lr,[sp],16          // restaur des  2 registres    ret                        // retour adresse lr x30/***************************************************//*      ROUTINES INCLUDE                           *//***************************************************/.include "../includeARM64.inc"  `
```First 50 Achilles Numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972  1125  1152  1323  1352  1372  1568  1800
1944  2000  2312  2592  2700  2888  3087  3200  3267  3456
3528  3872  3888  4000  4232  4500  4563  4608  5000  5292
5324  5400  5408  5488  6075  6125  6272  6728  6912  7200
First 20 Strong Achilles Numbers:
500  864  1944  2000  2592  3456  5000  10125  10368  12348
12500  16875  19652  19773  30375  31104  32000  33275  37044  40500
Numbers with 2 digits : 1
Numbers with 3 digits : 12
Numbers with 4 digits : 47
Numbers with 5 digits : 192
Numbers with 6 digits : 664
```

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
`BEGIN # find Achilles Numbers: numbers whose prime factors p appear at least  #      # twice (i.e. if p is a prime factor, so is p^2) and cannot be          #      # expressed as m^k for any integer m, k > 1                             #      # also find strong Achilles Numbers: Achilles Numbers where the Euler's #      # totient of the number is also Achilles                                #    # 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               #        in the Totient Function task #        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 numbers                                                        #    INT max number = 1 000 000;                 # max number we will consider #    PR read "primes.incl.a68" PR                #     include prime utilities #    []BOOL prime = PRIMESIEVE max number;       # construct a sieve of primes #    # table of numbers, will be set to TRUE for the Achilles Numbers          #    [ 1 : max number ]BOOL achiles;    FOR a TO UPB achiles DO        achiles[ a ] := TRUE    OD;    # remove the numbers that don't have squared primes as factors            #    achiles[ 1 ] := FALSE;    FOR a TO UPB achiles DO        IF prime[ a ] THEN            # have a prime, remove it and every multiple of it that isn't a   #            # multiple of a squared                                           #            INT a count := 0;            FOR j FROM a BY a TO UPB achiles DO                a count +:= 1;                IF a count = a THEN # have a multiple of i^2, keep the number #                    a count := 0                ELSE               # not a multiple of i^2, remove the number #                    achiles[ j ] := FALSE                FI            OD        FI    OD;    # achiles now has TRUE for the powerful numbers, remove all m^k (m,k > 1) #    FOR m FROM 2 TO UPB achiles DO        INT mk    := m;        INT max mk = UPB achiles OVER m;    # avoid overflow if INT is 32 bit #        WHILE mk <= max mk DO            mk           *:= m;            achiles[ mk ] := FALSE        OD    OD;    # achiles now has TRUE for imperfect powerful numbers                     #    # show the first 50 Achilles Numbers                                      #    BEGIN        print( ( "First 50 Achilles Numbers:", newline ) );        INT a count := 0;        FOR a WHILE a count < 50 DO            IF achiles[ a ] THEN                a count +:= 1;                print( ( " ", whole( a, -6 ) ) );                IF a count MOD 10 = 0 THEN                    print( ( newline ) )                FI            FI        OD    END;    # show the first 50 Strong Achilles numbers                               #    BEGIN        print( ( "First 20 Strong Achilles Numbers:", newline ) );        INT s count := 0;        FOR s WHILE s count < 20 DO            IF achiles[ s ] THEN                IF achiles[ totient( s ) ] THEN                    s count +:= 1;                    print( ( " ", whole( s, -6 ) ) );                    IF s count MOD 10 = 0 THEN                        print( ( newline ) )                    FI                FI            FI        OD    END;    # count the number of Achilles Numbers by their digit counts              #    BEGIN        INT a count     :=   0;        INT power of 10 := 100;        INT digit count :=   2;        FOR a TO UPB achiles DO            IF achiles[ a ] THEN                # have an Achilles Number                                     #                a count +:= 1            FI;            IF a = power of 10 THEN                # have reached a power of 10                                  #                print( ( "Achilles Numbers with ", whole( digit count, 0 )                       , " digits: ",             whole( a count,    -6 )                       , newline                       )                     );                digit count +:=  1;                power of 10 *:= 10;                a count      :=  0            FI        OD    ENDEND`
Output:
```First 50 Achilles Numbers:
72    108    200    288    392    432    500    648    675    800
864    968    972   1125   1152   1323   1352   1372   1568   1800
1944   2000   2312   2592   2700   2888   3087   3200   3267   3456
3528   3872   3888   4000   4232   4500   4563   4608   5000   5292
5324   5400   5408   5488   6075   6125   6272   6728   6912   7200
First 20 Strong Achilles Numbers:
500    864   1944   2000   2592   3456   5000  10125  10368  12348
12500  16875  19652  19773  30375  31104  32000  33275  37044  40500
Achilles Numbers with 2 digits:      1
Achilles Numbers with 3 digits:     12
Achilles Numbers with 4 digits:     47
Achilles Numbers with 5 digits:    192
Achilles Numbers with 6 digits:    664```

## ARM Assembly

Works with: as version Raspberry Pi
or android 32 bits with application Termux
` /* ARM assembly Raspberry PI  *//*  program achilleNumber.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 NBFACT,    33.equ MAXI,      50.equ MAXI1,     20.equ MAXI2,     1000000 /*********************************//* Initialized data              *//*********************************/.dataszMessNumber:       .asciz " @ "szCarriageReturn:   .asciz "\n"szErrorGen:         .asciz "Program error !!!\n"szMessPrime:        .asciz "This number is prime.\n"szMessTitAchille:   .asciz "First 50 Achilles Numbers:\n"szMessTitStrong:    .asciz "First 20 Strong Achilles Numbers:\n"szMessDigitsCounter: .asciz "Numbers with @ digits : @ \n"/*********************************//* UnInitialized data            *//*********************************/.bss  sZoneConv:           .skip 24tbZoneDecom:         .skip 8 * NBFACT          // factor 4 bytes, number of each factor 4 bytes/*********************************//*  code section                 *//*********************************/.text.global main main:                             @ entry of program     ldr r0,iAdrszMessTitAchille    bl affichageMess    mov r4,#1                      @ start number    mov r5,#0                      @ total counter    mov r6,#0                      @ line display counter1:     mov r0,r4    bl controlAchille    cmp r0,#0                      @ achille number ?    beq 2f                         @ no    mov r0,r4    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 r5,r5,#1                   @ increment counter    add r6,r6,#1                   @ increment indice line display    cmp r6,#10                     @ if = 10  new line    bne 2f    mov r6,#0    ldr r0,iAdrszCarriageReturn    bl affichageMess 2:    add r4,r4,#1                   @ increment number    cmp r5,#MAXI    blt 1b                         @ and loop     ldr r0,iAdrszMessTitStrong    bl affichageMess    mov r4,#1                      @ start number    mov r5,#0                      @ total counter    mov r6,#0 3:     mov r0,r4    bl controlAchille    cmp r0,#0    beq 4f    mov r0,r4    bl computeTotient    bl controlAchille    cmp r0,#0    beq 4f    mov r0,r4    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 r5,r5,#1    add r6,r6,#1    cmp r6,#10    bne 4f    mov r6,#0    ldr r0,iAdrszCarriageReturn    bl affichageMess 4:    add r4,r4,#1    cmp r5,#MAXI1    blt 3b     ldr r3,icstMaxi2    mov r4,#1                      @ start number    mov r6,#0                      @ total counter 2 digits    mov r7,#0                      @ total counter 3 digits    mov r8,#0                      @ total counter 4 digits    mov r9,#0                      @ total counter 5 digits    mov r10,#0                     @ total counter 6 digits5:     mov r0,r4    bl controlAchille    cmp r0,#0    beq 6f     mov r0,r4    ldr r1,iAdrsZoneConv    bl conversion10             @ call décimal conversion r0 return digit number    cmp r0,#6    addeq r10,r10,#1    beq 6f    cmp r0,#5    addeq r9,r9,#1    beq 6f    cmp r0,#4    addeq r8,r8,#1    beq 6f    cmp r0,#3    addeq r7,r7,#1    beq 6f    cmp r0,#2    addeq r6,r6,#1    beq 6f6:     add r4,r4,#1    cmp r4,r3    blt 5b    mov r0,#2    mov r1,r6    bl displayCounter    mov r0,#3    mov r1,r7    bl displayCounter    mov r0,#4    mov r1,r8    bl displayCounter    mov r0,#5    mov r1,r9    bl displayCounter    mov r0,#6    mov r1,r10    bl displayCounter    b 100f98:    ldr r0,iAdrszErrorGen    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 calliAdrszCarriageReturn:    .int szCarriageReturniAdrszErrorGen:          .int szErrorGeniAdrsZoneConv:           .int sZoneConv  iAdrtbZoneDecom:         .int tbZoneDecomiAdrszMessNumber:        .int szMessNumberiAdrszMessTitAchille:    .int szMessTitAchilleiAdrszMessTitStrong:     .int szMessTitStrongicstMaxi2:               .int MAXI2/******************************************************************//*     display digit counter                        */ /******************************************************************//* r0 contains limit  *//* r1 contains counter */displayCounter:    push {r1-r3,lr}            @ save  registers     mov r2,r1    ldr r1,iAdrsZoneConv    bl conversion10             @ call décimal conversion    ldr r0,iAdrszMessDigitsCounter    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 message100:    pop {r1-r3,pc}             @ restaur registersiAdrszMessDigitsCounter:   .int szMessDigitsCounter/******************************************************************//*     control if number is Achille number                        */ /******************************************************************//* r0 contains number  *//* r0 return 0 if not else return 1 */controlAchille:    push {r1-r4,lr}            @ save  registers     mov r4,r0    ldr r1,iAdrtbZoneDecom    bl decompFact               @ factor decomposition    cmp r0,#-1    beq 98f                     @ error ?    cmp r0,#1                   @ one only factor ?    moveq r0,#0    beq 100f    mov r1,r0    ldr r0,iAdrtbZoneDecom    mov r2,r4    bl controlDivisor    b 100f98:    ldr r0,iAdrszErrorGen    bl affichageMess 100:    pop {r1-r4,pc}             @ restaur registers/******************************************************************//*     control divisors function                         */ /******************************************************************//* r0 contains address of divisors area *//* r1 contains the number of area items  *//* r2 contains number  */controlDivisor:    push {r1-r10,lr}            @ save  registers     cmp r1,#0    moveq r0,#0    beq 100f    mov r6,r1                   @ factors number    mov r8,r2                   @ save number    mov r9,#0                   @ indice    mov r4,r0                   @ save area address    add r5,r4,r9,lsl #3         @ compute address first factor    ldr r7,[r5,#4]              @ load first exposant of factor    add r2,r9,#11:    add r5,r4,r2,lsl #3         @ compute address next factor    ldr r3,[r5,#4]              @ load exposant of factor    cmp r3,r7                   @ factor exposant <> ?    bne 2f                      @ yes -> end verif    add r2,r2,#1                @ increment indice    cmp r2,r6                   @ factor maxi ?    blt 1b                      @ no -> loop    mov r0,#0    b 100f                      @ all exposants are equals2:    mov r10,r2                  @ save indice21:    movlt r2,r7                 @ if r3 < r7 -> inversion    movlt r7,r3    movlt r3,r2                 @ r7 is the smaller exposant    mov r0,r3    mov r1,r7                   @ r7 < r3     bl computePgcd    cmp r0,#1    beq 23f                     @ no commun multiple -> ne peux donc pas etre une puissance22:    add r10,r10,#1              @ increment indice    cmp r10,r6                  @ factor maxi ?    movge r0,#0    bge 100f                    @ yes -> all exposants are multiples to smaller    add r5,r4,r10,lsl #3    ldr r3,[r5,#4]              @ load exposant of next factor    cmp r3,r7    beq 22b                     @ for next    b 21b                       @ for compare the 2 exposants 23:    mov r9,#0                   @ indice3:    add r5,r4,r9,lsl #3    ldr r7,[r5]                 @ load factor    mul r1,r7,r7                @ factor square    mov r0,r8                   @ number    bl division    cmp r3,#0                   @ remainder null ?    movne r0,#0    bne 100f     add r9,#1                   @ other factor    cmp r9,r6                   @ factors maxi ?    blt 3b    mov r0,#1                   @ achille number ok100:    pop {r1-r10,lr}             @ restaur registers    bx lr                       @ return /******************************************//* calcul du pgcd                         *//*****************************************//* r0 number one  *//* r1 number two  *//* r0 result return */computePgcd:    push {r2,lr}       @ save registers1:    cmp r0,#0    ble 2f    cmp r1,r0    movgt r2,r0    movgt r0,r1    movgt r1,r2    sub r0,r1    b 1b2:        mov r0,r1             pop {r2,pc}       @ restaur registers/******************************************************************//*     compute totient of number                                  */ /******************************************************************//* r0 contains number  */computeTotient:    push {r1-r5,lr}           @ save  registers     mov r4,r0                 @ totient    mov r5,r0                 @ save number    mov r1,#0                 @ for first divisor1:                            @ 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 3f2:                            @ 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 totient3:    cmp r1,#2                 @ first divisor ?    moveq r1,#1               @ divisor = 1    b 1b                      @ and loop4:    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 totient5:     mov r0,r4100:    pop {r1-r5,pc}             @ restaur registers /******************************************************************//*     factor decomposition                                               */ /******************************************************************//* r0 contains number *//* r1 contains address of divisors area *//* r0 return divisors items in table */decompFact:    push {r1-r8,lr}            @ save  registers    mov r5,r1    mov r8,r0                  @ save number    bl isPrime                 @ prime ?    cmp r0,#1    beq 98f                    @ yes is prime    mov r4,#0                  @ raz indice    mov r1,#2                  @ first divisor    mov r6,#0                  @ previous divisor    mov r7,#0                  @ number of same divisors2:    mov r0,r8                  @ dividende    bl division                @  r1 divisor r2 quotient r3 remainder    cmp r3,#0    bne 5f                     @ if remainder <> zero  -> no divisor    mov r8,r2                  @ else quotient -> new dividende    cmp r1,r6                  @ same divisor ?    beq 4f                     @ yes    cmp r6,#0                  @ no but is the first divisor ?    beq 3f                     @ yes     str r6,[r5,r4,lsl #2]      @ else store in the table    add r4,r4,#1               @ and increment counter    str r7,[r5,r4,lsl #2]      @ store counter    add r4,r4,#1               @ next item    mov r7,#0                  @ and raz counter3:    mov r6,r1                  @ new divisor4:    add r7,r7,#1               @ increment counter    b 7f                       @ and loop     /* not divisor -> increment next divisor */5:    cmp r1,#2                  @ if divisor = 2 -> add 1     addeq r1,#1    addne r1,#2                @ else add 2    b 2b     /* divisor -> test if new dividende is prime */7:     mov r3,r1                  @ save divisor    cmp r8,#1                  @ dividende = 1 ? -> end    beq 10f    mov r0,r8                  @ new dividende is prime ?    mov r1,#0    bl isPrime                 @ the new dividende is prime ?    cmp r0,#1    bne 10f                    @ the new dividende is not prime     cmp r8,r6                  @ else dividende is same divisor ?    beq 9f                     @ yes    cmp r6,#0                  @ no but is the first divisor ?    beq 8f                     @ yes it is a first    str r6,[r5,r4,lsl #2]      @ else store in table    add r4,r4,#1               @ and increment counter    str r7,[r5,r4,lsl #2]      @ and store counter     add r4,r4,#1               @ next item8:    mov r6,r8                  @ new dividende -> divisor prec    mov r7,#0                  @ and raz counter9:    add r7,r7,#1               @ increment counter    b 11f 10:    mov r1,r3                  @ current divisor = new divisor    cmp r1,r8                  @ current divisor  > new dividende ?    ble 2b                     @ no -> loop     /* end decomposition */ 11:    str r6,[r5,r4,lsl #2]      @ store last divisor    add r4,r4,#1    str r7,[r5,r4,lsl #2]      @ and store last number of same divisors    add r4,r4,#1    lsr r0,r4,#1               @ return number of table items    mov r3,#0    str r3,[r5,r4,lsl #2]      @ store zéro in last table item    add r4,r4,#1    str r3,[r5,r4,lsl #2]      @ and zero in counter same divisor    b 100f  98:     //ldr r0,iAdrszMessPrime    //bl   affichageMess    mov r0,#1                   @ return code    b 100f99:    ldr r0,iAdrszErrorGen    bl   affichageMess    mov r0,#-1                  @ error code    b 100f100:    pop {r1-r8,lr}              @ restaur registers    bx lriAdrszMessPrime:           .int szMessPrime /***************************************************//*   check if a number is prime              *//***************************************************//* r0 contains the number            *//* r0 return 1 if prime  0 else */@2147483647@4294967297@131071isPrime:    push {r1-r6,lr}    @ save registers     cmp r0,#0    beq 90f    cmp r0,#17    bhi 1f    cmp r0,#3    bls 80f            @ for 1,2,3 return prime    cmp r0,#5    beq 80f            @ for 5 return prime    cmp r0,#7    beq 80f            @ for 7 return prime    cmp r0,#11    beq 80f            @ for 11 return prime    cmp r0,#13    beq 80f            @ for 13 return prime    cmp r0,#17    beq 80f            @ for 17 return prime1:    tst r0,#1          @ even ?    beq 90f            @ yes -> not prime    mov r2,r0          @ save number    sub r1,r0,#1       @ exposant n - 1    mov r0,#3          @ base    bl moduloPuR32     @ compute base power n - 1 modulo n    cmp r0,#1    bne 90f            @ if <> 1  -> not prime     mov r0,#5    bl moduloPuR32    cmp r0,#1    bne 90f     mov r0,#7    bl moduloPuR32    cmp r0,#1    bne 90f     mov r0,#11    bl moduloPuR32    cmp r0,#1    bne 90f     mov r0,#13    bl moduloPuR32    cmp r0,#1    bne 90f     mov r0,#17    bl moduloPuR32    cmp r0,#1    bne 90f80:    mov r0,#1        @ is prime    b 100f90:    mov r0,#0        @ no prime100:                 @ fin standard de la fonction     pop {r1-r6,lr}   @ restaur des registres    bx lr            @ retour de la fonction en utilisant lr /********************************************************//*   Calcul modulo de b puissance e modulo m  *//*    Exemple 4 puissance 13 modulo 497 = 445         *//*                                             *//********************************************************//* r0  nombre  *//* r1 exposant *//* r2 modulo   *//* r0 return result  */moduloPuR32:    push {r1-r7,lr}    @ save registers      cmp r0,#0          @ verif <> zero     beq 100f    cmp r2,#0          @ verif <> zero     beq 100f           @ TODO: vérifier les cas erreur1:    mov r4,r2          @ save modulo    mov r5,r1          @ save exposant     mov r6,r0          @ save base    mov r3,#1          @ start result     mov r1,#0          @ division de r0,r1 par r2    bl division32R    mov r6,r2          @ base <- remainder2:    tst r5,#1          @  exposant even or odd    beq 3f    umull r0,r1,r6,r3    mov r2,r4    bl division32R    mov r3,r2          @ result <- remainder3:    umull r0,r1,r6,r6    mov r2,r4    bl division32R    mov r6,r2          @ base <- remainder     lsr r5,#1          @ left shift 1 bit    cmp r5,#0          @ end ?    bne 2b    mov r0,r3100:                   @ fin standard de la fonction    pop {r1-r7,lr}     @ restaur des registres    bx lr              @ retour de la fonction en utilisant lr     /***************************************************//*   division number 64 bits in 2 registers by number 32 bits *//***************************************************//* r0 contains lower part dividende   *//* r1 contains upper part dividende   *//* r2 contains divisor   *//* r0 return lower part quotient    *//* r1 return upper part quotient    *//* r2 return remainder               */division32R:    push {r3-r9,lr}    @ save registers    mov r6,#0          @ init upper upper part remainder  !!    mov r7,r1          @ init upper part remainder with upper part dividende    mov r8,r0          @ init lower part remainder with lower part dividende    mov r9,#0          @ upper part quotient     mov r4,#0          @ lower part quotient    mov r5,#32         @ bits number1:                     @ begin loop    lsl r6,#1          @ shift upper upper part remainder    lsls r7,#1         @ shift upper  part remainder    orrcs r6,#1            lsls r8,#1         @ shift lower  part remainder    orrcs r7,#1    lsls r4,#1         @ shift lower part quotient    lsl r9,#1          @ shift upper part quotient    orrcs r9,#1                       @ divisor sustract  upper  part remainder    subs r7,r2    sbcs  r6,#0        @ and substract carry    bmi 2f             @ négative ?                        @ positive or equal    orr r4,#1          @ 1 -> right bit quotient    b 3f2:                     @ negative     orr r4,#0          @ 0 -> right bit quotient    adds r7,r2         @ and restaur remainder    adc  r6,#0 3:    subs r5,#1         @ decrement bit size     bgt 1b             @ end ?    mov r0,r4          @ lower part quotient    mov r1,r9          @ upper part quotient    mov r2,r7          @ remainder100:                   @ function end    pop {r3-r9,lr}     @ restaur registers    bx lr    /***************************************************//*      ROUTINES INCLUDE                           *//***************************************************/.include "../affichage.inc" `
```First 50 Achilles Numbers:
72           108          200          288          392          432          500          648          675          800
864          968          972          1125         1152         1323         1352         1372         1568         1800
1944         2000         2312         2592         2700         2888         3087         3200         3267         3456
3528         3872         3888         4000         4232         4500         4563         4608         5000         5292
5324         5400         5408         5488         6075         6125         6272         6728         6912         7200
First 20 Strong Achilles Numbers:
500          864          1944         2000         2592         3456         5000         10125        10368        12348
12500        16875        19652        19773        30375        31104        32000        33275        37044        40500
Numbers with 2           digits : 1
Numbers with 3           digits : 12
Numbers with 4           digits : 47
Numbers with 5           digits : 192
Numbers with 6           digits : 664
```

## C++

Translation of: Wren
Library: Boost
`#include <algorithm>#include <chrono>#include <cmath>#include <cstdint>#include <iomanip>#include <iostream>#include <vector> #include <boost/multiprecision/cpp_int.hpp> using boost::multiprecision::uint128_t; template <typename T> void unique_sort(std::vector<T>& vector) {    std::sort(vector.begin(), vector.end());    vector.erase(std::unique(vector.begin(), vector.end()), vector.end());} auto perfect_powers(uint128_t n) {    std::vector<uint128_t> result;    for (uint128_t i = 2, s = sqrt(n); i <= s; ++i)        for (uint128_t p = i * i; p < n; p *= i)            result.push_back(p);    unique_sort(result);    return result;} auto achilles(uint128_t from, uint128_t to, const std::vector<uint128_t>& pps) {    std::vector<uint128_t> result;    auto c = static_cast<uint128_t>(std::cbrt(static_cast<double>(to / 4)));    auto s = sqrt(to / 8);    for (uint128_t b = 2; b <= c; ++b) {        uint128_t b3 = b * b * b;        for (uint128_t a = 2; a <= s; ++a) {            uint128_t p = b3 * a * a;            if (p >= to)                break;            if (p >= from && !binary_search(pps.begin(), pps.end(), p))                result.push_back(p);        }    }    unique_sort(result);    return result;} uint128_t totient(uint128_t n) {    uint128_t tot = n;    if ((n & 1) == 0) {        while ((n & 1) == 0)            n >>= 1;        tot -= tot >> 1;    }    for (uint128_t p = 3; p * p <= n; p += 2) {        if (n % p == 0) {            while (n % p == 0)                n /= p;            tot -= tot / p;        }    }    if (n > 1)        tot -= tot / n;    return tot;} int main() {    auto start = std::chrono::high_resolution_clock::now();     const uint128_t limit = 1000000000000000;     auto pps = perfect_powers(limit);    auto ach = achilles(1, 1000000, pps);     std::cout << "First 50 Achilles numbers:\n";    for (size_t i = 0; i < 50 && i < ach.size(); ++i)        std::cout << std::setw(4) << ach[i] << ((i + 1) % 10 == 0 ? '\n' : ' ');     std::cout << "\nFirst 50 strong Achilles numbers:\n";    for (size_t i = 0, count = 0; count < 50 && i < ach.size(); ++i)        if (binary_search(ach.begin(), ach.end(), totient(ach[i])))            std::cout << std::setw(6) << ach[i]                      << (++count % 10 == 0 ? '\n' : ' ');     int digits = 2;    std::cout << "\nNumber of Achilles numbers with:\n";    for (uint128_t from = 1, to = 100; to <= limit; to *= 10, ++digits) {        size_t count = achilles(from, to, pps).size();        std::cout << std::setw(2) << digits << " digits: " << count << '\n';        from = to;    }     auto end = std::chrono::high_resolution_clock::now();    std::chrono::duration<double> duration(end - start);    std::cout << "\nElapsed time: " << duration.count() << " seconds\n";}`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 50 strong Achilles numbers:
500    864   1944   2000   2592   3456   5000  10125  10368  12348
12500  16875  19652  19773  30375  31104  32000  33275  37044  40500
49392  50000  52488  55296  61731  64827  67500  69984  78608  80000
81000  83349  84375  93312 108000 111132 124416 128000 135000 148176
151875 158184 162000 165888 172872 177957 197568 200000 202612 209952

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008
10 digits: 77330
11 digits: 247449
12 digits: 788855
13 digits: 2508051
14 digits: 7960336
15 digits: 25235383

Elapsed time: 13.2644 seconds
```

## FreeBASIC

`Function GCD(n As Uinteger, d As Uinteger) As Uinteger    Return Iif(d = 0, n, GCD(d, n Mod d))End Function Function Totient(n As Integer) As Integer    Dim As Integer m, tot = 0    For m = 1 To n        If GCD(m, n) = 1 Then tot += 1    Next m    Return totEnd Function Function isPowerful(m As Integer) As Boolean    Dim As Integer n = m, f = 2, q, l = Sqr(m)     If m <= 1 Then Return false    Do        q = n/f        If (n Mod f) = 0 Then            If (m Mod(f*f)) Then Return false            n = q            If f > n Then Exit Do        Else                f += 1            If f > l Then                If (m Mod (n*n)) Then Return false                Exit Do            End If        End If    Loop    Return trueEnd Function Function isAchilles(n As Integer) As Boolean    If Not isPowerful(n) Then Return false    Dim As Integer m = 2, a = m*m    Do        Do            If a = n Then Return false            If a > n Then Exit Do            a *= m        Loop        m += 1        a = m*m    Loop Until a > n    Return trueEnd Function Dim As Integer num, n, iDim As Single inicioDim As Double t0 = Timer Print "First 50 Achilles numbers:"num = 0n = 1Do    If isAchilles(n) Then        Print Using "#####"; n;        num += 1        If num >= 50 Then Exit Do        If (num Mod 10) Then Print Space(3); Else Print    End If    n += 1Loop Print !"\n\nFirst 20 strong Achilles numbers:"num = 0n = 1Do    If isAchilles(n) And isAchilles(Totient(n)) Then        Print Using "#####"; n;        num += 1        If num >= 20 Then Exit Do        If (num Mod 10) Then Print Space(3); Else Print    End If    n += 1Loop Print !"\n\nNumber of Achilles numbers with:"For i = 2 To 6    inicio = Fix(10.0 ^ (i-1))    num = 0    For n = inicio To inicio*10-1        If isAchilles(n) Then num += 1    Next n    Print i; " digits:"; numNext iSleep`
Output:
```First 50 Achilles numbers:
72     108     200     288     392     432     500     648     675     800
864     968     972    1125    1152    1323    1352    1372    1568    1800
1944    2000    2312    2592    2700    2888    3087    3200    3267    3456
3528    3872    3888    4000    4232    4500    4563    4608    5000    5292
5324    5400    5408    5488    6075    6125    6272    6728    6912    7200

First 20 strong Achilles numbers:
500     864    1944    2000    2592    3456    5000   10125   10368   12348
12500   16875   19652   19773   30375   31104   32000   33275   37044   40500

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664```

## Go

Translation of: Wren

Based on Version 2, takes around 19 seconds.

`package main import (    "fmt"    "math"    "sort") func totient(n int) int {    tot := n    i := 2    for i*i <= n {        if n%i == 0 {            for n%i == 0 {                n /= i            }            tot -= tot / i        }        if i == 2 {            i = 1        }        i += 2    }    if n > 1 {        tot -= tot / n    }    return tot} var pps = make(map[int]bool) func getPerfectPowers(maxExp int) {    upper := math.Pow(10, float64(maxExp))    for i := 2; i <= int(math.Sqrt(upper)); i++ {        fi := float64(i)        p := fi        for {            p *= fi            if p >= upper {                break            }            pps[int(p)] = true        }    }} func getAchilles(minExp, maxExp int) map[int]bool {    lower := math.Pow(10, float64(minExp))    upper := math.Pow(10, float64(maxExp))    achilles := make(map[int]bool)    for b := 1; b <= int(math.Cbrt(upper)); b++ {        b3 := b * b * b        for a := 1; a <= int(math.Sqrt(upper)); a++ {            p := b3 * a * a            if p >= int(upper) {                break            }            if p >= int(lower) {                if _, ok := pps[p]; !ok {                    achilles[p] = true                }            }        }    }    return achilles} func main() {    maxDigits := 15    getPerfectPowers(maxDigits)    achillesSet := getAchilles(1, 5) // enough for first 2 parts    achilles := make([]int, len(achillesSet))    i := 0    for k := range achillesSet {        achilles[i] = k        i++    }    sort.Ints(achilles)     fmt.Println("First 50 Achilles numbers:")    for i = 0; i < 50; i++ {        fmt.Printf("%4d ", achilles[i])        if (i+1)%10 == 0 {            fmt.Println()        }    }     fmt.Println("\nFirst 30 strong Achilles numbers:")    var strongAchilles []int    count := 0    for n := 0; count < 30; n++ {        tot := totient(achilles[n])        if _, ok := achillesSet[tot]; ok {            strongAchilles = append(strongAchilles, achilles[n])            count++        }    }    for i = 0; i < 30; i++ {        fmt.Printf("%5d ", strongAchilles[i])        if (i+1)%10 == 0 {            fmt.Println()        }    }     fmt.Println("\nNumber of Achilles numbers with:")    for d := 2; d <= maxDigits; d++ {        ac := len(getAchilles(d-1, d))        fmt.Printf("%2d digits: %d\n", d, ac)    }}`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008
10 digits: 77330
11 digits: 247449
12 digits: 788855
13 digits: 2508051
14 digits: 7960336
15 digits: 25235383
```

## J

Implementation:

`achilles=: (*/ .>&1 * 1 = +./)@(1{__&q:)"0strong=: [email protected](5&p:)`

`   5 10\$(#~ achilles) 1+i.10000  NB. first 50 achilles numbers  72  108  200  288  392  432  500  648  675  800 864  968  972 1125 1152 1323 1352 1372 1568 18001944 2000 2312 2592 2700 2888 3087 3200 3267 34563528 3872 3888 4000 4232 4500 4563 4608 5000 52925324 5400 5408 5488 6075 6125 6272 6728 6912 7200    20{.(#~ strong * achilles) 1+i.100000 NB. first twenty strong achilles numbers500 864 1944 2000 2592 3456 5000 10125 10368 12348 12500 16875 19652 19773 30375 31104 32000 33275 37044 40500    +/achilles (+i.)/1 9*10^<:2  NB. count of two digit achilles numbers1   +/achilles (+i.)/1 9*10^<:312   +/achilles (+i.)/1 9*10^<:447   +/achilles (+i.)/1 9*10^<:5192   +/achilles (+i.)/1 9*10^<:6664`

Explanation of the code:

(1{__&q:) is a function which returns the non-zero powers of the prime factors of a positive integer. (__&q: returns both the primes and their factors, but here we do not care about the primes themselves.)

+./ returns the greatest common divisor of a list, and 1=+./ is true if that gcd is 1 (0 if it's false).

*/ .>&1 is true if all the values in a list are greater than 1 (0 if not).

"0 maps a function onto the individual (rank 0) items of a list or array (we use this to avoid complexities: for example if we padded our lists of prime factor powers with zeros, we could still find the gcd, but our test that the powers are greater than 1 would fail). (Actually... we could change */ .>&1 to (0 = 1 e. ]) but padding would still be a bad idea here, for performance reasons. Perhaps we ought to have an option for q: to return a sparse array...)

5&p: is euler's totient function.

(#~ predicate) list selects the elements of list where predicate is true.

## Julia

`using Primes isAchilles(n) = (p = [x[2] for x in factor(n).pe]; all(>(1), p) && gcd(p) == 1) isstrongAchilles(n) = isAchilles(n) && isAchilles(totient(n)) function teststrongachilles(nachilles = 50, nstrongachilles = 100)    # task 1    println("First \$nachilles Achilles numbers:")    n, found = 0, 0    while found < nachilles        if isAchilles(n)            found += 1            print(rpad(n, 5), found % 10 == 0 ? "\n" : "")        end        n += 1    end    # task 2    println("\nFirst \$nstrongachilles strong Achilles numbers:")    n, found = 0, 0    while found < nstrongachilles        if isstrongAchilles(n)            found += 1            print(rpad(n, 7), found % 10 == 0 ? "\n" : "")        end        n += 1    end    # task 3    println("\nCount of Achilles numbers for various intervals:")    intervals = [10:99, 100:999, 1000:9999, 10000:99999, 100000:999999]    for interval in intervals        println(lpad(interval, 15), " ", count(isAchilles, interval))    endend teststrongachilles() `
Output:
```First 50 Achilles numbers:
72   108  200  288  392  432  500  648  675  800
864  968  972  1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 100 strong Achilles numbers:
500    864    1944   2000   2592   3456   5000   10125  10368  12348
12500  16875  19652  19773  30375  31104  32000  33275  37044  40500
49392  50000  52488  55296  61731  64827  67500  69984  78608  80000
81000  83349  84375  93312  108000 111132 124416 128000 135000 148176
151875 158184 162000 165888 172872 177957 197568 200000 202612 209952
219488 221184 237276 243000 246924 253125 266200 270000 273375 296352
320000 324000 333396 364500 397953 405000 432000 444528 453789 455877
493848 497664 500000 518616 533871 540000 555579 583443 605052 607500
629856 632736 648000 663552 665500 666792 675000 691488 740772 750141
790272 800000 810448 820125 831875 877952 949104 972000 987696 1000188

Count of Achilles numbers for various intervals:
10:99 1
100:999 12
1000:9999 47
10000:99999 192
100000:999999 664
```

## Mathematica/Wolfram Language

`ClearAll[PowerfulNumberQ, StrongAchillesNumberQ]PowerfulNumberQ[n_Integer] := AllTrue[FactorInteger[n][[All, 2]], GreaterEqualThan[2]]AchillesNumberQ[n_Integer] := Module[{divs},  If[PowerfulNumberQ[n],   divs = Divisors[n];   If[Length[divs] > 2,    divs = divs[[2 ;; -2]];    !AnyTrue[Log[#, n] & /@ divs, IntegerQ]    ,    True    ]   ,   False   ]  ]StrongAchillesNumberQ[n_] := AchillesNumberQ[n] \[And] AchillesNumberQ[EulerPhi[n]] n = 0;i = 0;Reap[While[n < 50,   i++;   If[AchillesNumberQ[i], n++; Sow[i]]   ]][[2, 1]] n = 0;i = 0;Reap[While[n < 20,   i++;   If[StrongAchillesNumberQ[i], n++; Sow[i]]   ]][[2, 1]] Tally[IntegerLength /@ Select[Range[9999999], AchillesNumberQ]] // Grid`
Output:
```{72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075, 6125, 6272, 6728, 6912, 7200}

{500, 864, 1944, 2000, 2592, 3456, 5000, 10125, 10368, 12348, 12500, 16875, 19652, 19773, 30375, 31104, 32000, 33275, 37044, 40500}

2	1
3	12
4	47
5	192
6	664
7	2242```

## Perl

Borrowed, and lightly modified, code from Powerful_numbers

Library: ntheory
`use strict;use warnings;use feature <say current_sub>;use experimental 'signatures';use List::AllUtils <max head uniqint>;use ntheory <is_square_free is_power euler_phi>;use Math::AnyNum <:overload idiv iroot ipow is_coprime>; sub table { my \$t = shift() * (my \$c = 1 + length max @_); ( sprintf( ('%'.\$c.'d')x@_, @_) ) =~ s/.{1,\$t}\K/\n/gr } sub powerful_numbers (\$n, \$k = 2) {    my @powerful;    sub (\$m, \$r) {        \$r < \$k and push @powerful, \$m and return;        for my \$v (1 .. iroot(idiv(\$n, \$m), \$r)) {            if (\$r > \$k) { next unless is_square_free(\$v) and is_coprime(\$m, \$v) }            __SUB__->(\$m * ipow(\$v, \$r), \$r - 1);        }    }->(1, 2*\$k - 1);    sort { \$a <=> \$b } @powerful;} my(@P, @achilles, %Ahash, @strong);@P = uniqint @P, powerful_numbers(10**9, \$_) for 2..9; shift @P;!is_power(\$_) and push @achilles, \$_ and \$Ahash{\$_}++ for @P;\$Ahash{euler_phi \$_} and push @strong, \$_ for @achilles; say "First 50 Achilles numbers:\n"        . table 10, head 50, @achilles;say "First 30 strong Achilles numbers:\n" . table 10, head 30, @strong;say "Number of Achilles numbers with:\n";for my \$l (2..9) {    my \$c; \$l == length and \$c++ for @achilles;    say "\$l digits: \$c";}`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008```

Here is a translation from Wren version 2, as an alternative.

`use strict;use warnings; my %pps;my \$maxDigits = 9; sub totient {    my \$tot = my \$n = shift;    my \$i   = 2;   while (\$i*\$i <= \$n) {      unless (\$n % \$i) {         until(\$n % \$i) { \$n = int(\$n/\$i) }         \$tot -= int(\$tot/\$i)      }      if (\$i == 2) { \$i = 1 }      \$i += 2;    }   if (\$n > 1) { \$tot -= int(\$tot/\$n) }   return \$tot} sub getPerfectPowers {   for my \$i (2..int(sqrt(my \$upper = 10**( shift )))) {      my \$p = \$i;      while ((\$p *= \$i) < \$upper) { \$pps{\$p}++ }   }} sub getAchilles {    my (\$lower, \$upper) = map { 10** \$_ } @_ ;   my %achilles = ();    my \$count = 0;   for my \$b (1..int(\$upper**(1/3))) {      my (\$b3,\$p) = \$b * \$b * \$b;      for my \$a (1..int(sqrt(\$upper))) {         last if ((\$p = \$b3 * \$a * \$a) >= \$upper);         \$achilles{\$p}++ if (\$p >= \$lower and !\$pps{\$p})        }   }   return keys %achilles} getPerfectPowers \$maxDigits; my @achilles = sort { \$a <=> \$b } getAchilles(1,5);my %achillesSet;@achillesSet{ @achilles } = undef;  print "First 50 Achilles numbers:\n";for (0..49) { printf "%5d".(\$_%10 == 9 ? "\n" : " "), \$achilles[\$_] } my %strongAchilles;my \$count = my \$n = 0;for (my \$count = my \$n = 0; \$count < 30; \$n++) {   if ( exists(\$achillesSet{ totient( \$achilles[\$n] ) })) {      \$strongAchilles{ \$achilles[\$n] }++;      \$count++   }} my @strongAchilles30 = (sort { \$a <=> \$b } keys %strongAchilles)[0..29]; print "\nFirst 30 strong Achilles numbers:\n";for (0..29) { printf "%5d".(\$_%10 == 9 ? "\n" : " "), \$strongAchilles30[\$_] } print "\nNumber of Achilles numbers with:\n";for my \$d (2..\$maxDigits) {   printf "%2d digits: %d\n", \$d, scalar getAchilles(\$d-1, \$d)}`

Output is the same.

## Phix

Library: Phix/online

You can run this online here, though [slightly outdated and] you should expect a blank screen for about 9s.

Translation of: Wren
```with javascript_semantics
requires("1.0.2") -- [join_by(fmt)]
atom t0 = time()
constant maxDigits = iff(platform()=JS?10:12)
integer pps = new_dict()

procedure getPerfectPowers(integer maxExp)
atom hi = power(10, maxExp)
integer imax = floor(sqrt(hi))
for i=2 to imax do
atom p = i
while true do
p *= i
if p>=hi then exit end if
setd(p,true,pps)
end while
end for
end procedure

function get_achilles(integer minExp, maxExp)
atom lo10 = power(10,minExp),
hi10 = power(10,maxExp)
integer bmax = floor(power(hi10,1/3)),
amax = floor(sqrt(hi10))
sequence achilles = {}
for b=2 to bmax do
atom b3 = b * b * b
for a=2 to amax do
atom p = b3 * a * a
if p>=hi10 then exit end if
if p>=lo10 then
integer node = getd_index(p,pps)
if node=NULL then
achilles &= p
end if
end if
end for
end for
achilles = unique(achilles)
return achilles
end function

getPerfectPowers(maxDigits)
sequence achilles = get_achilles(1,5)

function strong_achilles(integer n)
integer totient = sum(sq_eq(apply(true,gcd,{tagset(n),n}),1))
return find(totient,achilles)
end function

sequence a = join_by(achilles[1..50],1,10," ",fmt:="%4d"),
sa = filter(achilles,strong_achilles)[1..30],
ssa = join_by(sa,1,10," ",fmt:="%5d")

printf(1,"First 50 Achilles numbers:\n%s\n",{a})
printf(1,"First 30 strong Achilles numbers:\n%s\n",{ssa})
for d=2 to maxDigits do
printf(1,"Achilles numbers with %d digits:%d\n",{d,length(get_achilles(d-1,d))})
end for
?elapsed(time()-t0)
```
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Achilles numbers with 2 digits:1
Achilles numbers with 3 digits:12
Achilles numbers with 4 digits:47
Achilles numbers with 5 digits:192
Achilles numbers with 6 digits:664
Achilles numbers with 7 digits:2242
Achilles numbers with 8 digits:7395
Achilles numbers with 9 digits:24008
Achilles numbers with 10 digits:77330
Achilles numbers with 11 digits:247449
Achilles numbers with 12 digits:788855
"30.7s"
```

## Raku

Timing is going to be system / OS dependent.

`use Prime::Factor;use Math::Root; sub is-square-free (Int \n) {    constant @p = ^100 .map: { next unless .is-prime; .² };    for @p -> \p { return False if n %% p }    True} sub powerful (\n, \k = 2) {    my @p;    p(1, 2*k - 1);    sub p (\m, \r) {        @p.push(m) and return if r < k;        for 1 .. (n / m).&root(r) -> \v {            if r > k {                next unless is-square-free(v);                next unless m gcd v == 1;            }            p(m * v ** r, r - 1)        }    }    @p} my \$achilles = powerful(10**9).hyper(:500batch).grep( {    my \$f = .&prime-factors.Bag;    (+\$f.keys > 1) && (1 == [gcd] \$f.values) && (.sqrt.Int² !== \$_)} ).classify: { .chars } my \𝜑 = 0, |(1..*).hyper.map: -> \t { t × [×] t.&prime-factors.squish.map: { 1 - 1/\$_ } } my %as = Set.new: flat \$achilles.values».list; my \$strong = lazy (flat \$achilles.sort».value».list».sort).grep: { ?%as{𝜑[\$_]} }; put "First 50 Achilles numbers:";put (flat \$achilles.sort».value».list».sort)[^50].batch(10)».fmt("%4d").join("\n"); put "\nFirst 30 strong Achilles numbers:";put   \$strong[^30].batch(10)».fmt("%5d").join("\n"); put "\nNumber of Achilles numbers with:";say "\$_ digits: " ~ +\$achilles{\$_} // 0 for 2..9; printf "\n%.1f total elapsed seconds\n", now - INIT now;`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008

6.1 total elapsed seconds```

Could go further but slows to a crawl and starts chewing up memory in short order.

```10 digits: 77330
11 digits: 247449
12 digits: 788855

410.4 total elapsed seconds
```

## Rust

Translation of: Wren
`fn perfect_powers(n: u128) -> Vec<u128> {    let mut powers = Vec::<u128>::new();    let sqrt = (n as f64).sqrt() as u128;    for i in 2..=sqrt {        let mut p = i * i;        while p < n {            powers.push(p);            p *= i;        }    }    powers.sort();    powers.dedup();    powers} fn bsearch<T: Ord>(vector: &Vec<T>, value: &T) -> bool {    match vector.binary_search(value) {        Ok(_) => true,        _ => false,    }} fn achilles(from: u128, to: u128, pps: &Vec<u128>) -> Vec<u128> {    let mut result = Vec::<u128>::new();    let cbrt = ((to / 4) as f64).cbrt() as u128;    let sqrt = ((to / 8) as f64).sqrt() as u128;    for b in 2..=cbrt {        let b3 = b * b * b;        for a in 2..=sqrt {            let p = b3 * a * a;            if p >= to {                break;            }            if p >= from && !bsearch(&pps, &p) {                result.push(p);            }        }    }    result.sort();    result.dedup();    result} fn totient(mut n: u128) -> u128 {    let mut tot = n;    if (n & 1) == 0 {        while (n & 1) == 0 {            n >>= 1;        }        tot -= tot >> 1;    }    let mut p = 3;    while p * p <= n {        if n % p == 0 {            while n % p == 0 {                n /= p;            }            tot -= tot / p;        }        p += 2;    }    if n > 1 {        tot -= tot / n;    }    tot} fn main() {    use std::time::Instant;    let t0 = Instant::now();    let limit = 1000000000000000u128;     let pps = perfect_powers(limit);    let ach = achilles(1, 1000000, &pps);     println!("First 50 Achilles numbers:");    for i in 0..50 {        print!("{:4}{}", ach[i], if (i + 1) % 10 == 0 { "\n" } else { " " });    }     println!("\nFirst 50 strong Achilles numbers:");    for (i, n) in ach        .iter()        .filter(|&x| bsearch(&ach, &totient(*x)))        .take(50)        .enumerate()    {        print!("{:6}{}", n, if (i + 1) % 10 == 0 { "\n" } else { " " });    }    println!();     let mut from = 1u128;    let mut to = 100u128;    let mut digits = 2;    while to <= limit {        let count = achilles(from, to, &pps).len();        println!("{:2} digits: {}", digits, count);        from = to;        to *= 10;        digits += 1;    }     let duration = t0.elapsed();    println!("\nElapsed time: {} milliseconds", duration.as_millis());}`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 50 strong Achilles numbers:
500    864   1944   2000   2592   3456   5000  10125  10368  12348
12500  16875  19652  19773  30375  31104  32000  33275  37044  40500
49392  50000  52488  55296  61731  64827  67500  69984  78608  80000
81000  83349  84375  93312 108000 111132 124416 128000 135000 148176
151875 158184 162000 165888 172872 177957 197568 200000 202612 209952

2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008
10 digits: 77330
11 digits: 247449
12 digits: 788855
13 digits: 2508051
14 digits: 7960336
15 digits: 25235383

Elapsed time: 12608 milliseconds
```

## Wren

Library: Wren-math
Library: Wren-seq
Library: Wren-fmt

### Version 1 (Brute force)

This finds the number of 6 digit Achilles numbers in 2.5 seconds, 7 digits in 51 seconds but 8 digits needs a whopping 21 minutes!

`import "./math" for Intimport "./seq" for Lstimport "./fmt" for Fmt var maxDigits = 8var limit = 10.pow(maxDigits)var c = Int.primeSieve(limit-1, false) var totient = Fn.new { |n|    var tot = n    var i = 2    while (i*i <= n) {        if (n%i == 0) {            while(n%i == 0) n = (n/i).floor            tot = tot - (tot/i).floor        }        if (i == 2) i = 1        i = i + 2    }    if (n > 1) tot = tot - (tot/n).floor    return tot} var isPerfectPower = Fn.new { |n|    if (n == 1) return true    var x = 2    while (x * x <= n) {        var y = 2        var p = x.pow(y)        while (p > 0 && p <= n) {            if (p == n) return true            y = y + 1            p = x.pow(y)        }        x = x + 1    }    return false} var isPowerful = Fn.new { |n|    while (n % 2 == 0) {        var p = 0        while (n % 2 == 0) {            n = (n/2).floor            p = p + 1        }        if (p == 1) return false    }    var f = 3    while (f * f <= n) {        var p = 0        while (n % f == 0) {            n = (n/f).floor            p = p + 1        }        if (p == 1) return false        f = f + 2    }    return n == 1} var isAchilles = Fn.new { |n| c[n] && isPowerful.call(n) && !isPerfectPower.call(n) } var isStrongAchilles = Fn.new { |n|    if (!isAchilles.call(n)) return false    var tot = totient.call(n)    return isAchilles.call(tot)} System.print("First 50 Achilles numbers:")var achilles = []var count = 0var n = 2while (count < 50) {    if (isAchilles.call(n)) {        achilles.add(n)        count = count + 1    }    n = n + 1}for (chunk in Lst.chunks(achilles, 10)) Fmt.print("\$4d", chunk) System.print("\nFirst 30 strong Achilles numbers:")var strongAchilles = []count = 0n = achilles[0]while (count < 30) {    if (isStrongAchilles.call(n)) {        strongAchilles.add(n)        count = count + 1    }    n = n + 1}for (chunk in Lst.chunks(strongAchilles, 10)) Fmt.print("\$5d", chunk) System.print("\nNumber of Achilles numbers with:")var pow = 10for (i in 2..maxDigits) {    var count = 0    for (j in pow..pow*10-1) {        if (isAchilles.call(j)) count = count + 1    }    System.print("%(i) digits: %(count)")    pow = pow * 10}`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
```

### Version 2 (Much faster)

Library: Wren-set

Here we make use of the fact that powerful numbers are always of the form a²b³, where a and b > 0, to generate such numbers up to a given limit. We also generate in advance all perfect powers up to the same limit.

Ridiculously fast compared to the previous method: 12 digits can now be reached in 1.03 seconds, 13 digits in 3.7 seconds, 14 digits in 12.2 seconds and 15 digits in 69 seconds.

`import "./set" for Setimport "./seq" for Lstimport "./fmt" for Fmt var totient = Fn.new { |n|    var tot = n    var i = 2    while (i*i <= n) {        if (n%i == 0) {            while(n%i == 0) n = (n/i).floor            tot = tot - (tot/i).floor        }        if (i == 2) i = 1        i = i + 2    }    if (n > 1) tot = tot - (tot/n).floor    return tot} var pps = Set.new() var getPerfectPowers = Fn.new { |maxExp|    var upper = 10.pow(maxExp)    for (i in 2..upper.sqrt.floor) {        var p = i        while ((p = p * i) < upper) pps.add(p)    }} var getAchilles = Fn.new { |minExp, maxExp|    var lower = 10.pow(minExp)    var upper = 10.pow(maxExp)    var achilles = Set.new() // avoids duplicates    for (b in 1..upper.cbrt.floor) {        var b3 = b * b * b        for (a in 1..upper.sqrt.floor) {            var p = b3 * a * a            if (p >= upper) break            if (p >= lower) {                if (!pps.contains(p)) achilles.add(p)            }        }    }    return achilles} var maxDigits = 15getPerfectPowers.call(maxDigits) var achillesSet = getAchilles.call(1, 5) // enough for first 2 partsvar achilles = achillesSet.toListachilles.sort() System.print("First 50 Achilles numbers:")for (chunk in Lst.chunks(achilles[0..49], 10)) Fmt.print("\$4d", chunk) System.print("\nFirst 30 strong Achilles numbers:")var strongAchilles = []var count = 0var n = 0while (count < 30) {    var tot = totient.call(achilles[n])    if (achillesSet.contains(tot)) {        strongAchilles.add(achilles[n])        count = count + 1    }    n = n + 1}for (chunk in Lst.chunks(strongAchilles, 10)) Fmt.print("\$5d", chunk) System.print("\nNumber of Achilles numbers with:")for (d in 2..maxDigits) {    var ac = getAchilles.call(d-1, d).count    Fmt.print("\$2d digits: \$d", d, ac)}`
Output:
```First 50 Achilles numbers:
72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008
10 digits: 77330
11 digits: 247449
12 digits: 788855
13 digits: 2508051
14 digits: 7960336
15 digits: 25235383
```

## XPL0

`func GCD(N, D);         \Return the greatest common divisor of N and Dint  N, D;              \numerator and denominatorint  R;[if D > N then    [R:= D;  D:= N;  N:= R];    \swap D and Nwhile D > 0 do    [R:= rem(N/D);    N:= D;    D:= R;    ];return N;];      \GCD func Totient(N);        \Return the totient of Nint  N, Phi, M;[Phi:= 0;for M:= 1 to N do    if GCD(M, N) = 1 then Phi:= Phi+1;return Phi;]; func Powerful(N0);      \Return 'true' if N0 is a powerful numberint  N0, N, F, Q, L;[if N0 <= 1 then return false;N:= N0;  F:= 2;L:= sqrt(N0);loop    [Q:= N/F;        if rem(0) = 0 then      \found a factor                [if rem(N0/(F*F)) then return false;                N:= Q;                if F>N then quit;                ]        else    [F:= F+1;                if F > L then                    [if rem(N0/(N*N)) then return false;                    quit;                    ];                ];        ];return true;]; func Achilles(N);       \Return 'true' if N is an Achilles numberint  N, M, A;[if not Powerful(N) then return false;M:= 2;A:= M*M;repeat  loop    [if A = N then return false;                if A > N then quit;                A:= A*M;                ];        M:= M+1;        A:= M*M;until   A > N;return true;]; int Cnt, N, Pwr, Start;[Cnt:= 0;N:= 1;loop    [if Achilles(N) then            [IntOut(0, N);            Cnt:= Cnt+1;            if Cnt >= 50 then quit;            if rem(Cnt/10) then ChOut(0, 9) else CrLf(0);            ];        N:= N+1;        ];CrLf(0);  CrLf(0);Cnt:= 0;N:= 1;loop    [if Achilles(N) then            if Achilles(Totient(N)) then                [IntOut(0, N);                Cnt:= Cnt+1;                if Cnt >= 20 then quit;                if rem(Cnt/10) then ChOut(0, 9) else CrLf(0);                ];        N:= N+1;        ];CrLf(0);  CrLf(0);for Pwr:= 1 to 6 do    [IntOut(0, Pwr);  Text(0, ": ");    Start:= fix(Pow(10.0, float(Pwr-1)));    Cnt:= 0;    for N:= Start to Start*10-1 do        if Achilles(N) then Cnt:= Cnt+1;    IntOut(0, Cnt);  CrLf(0);    ];]`
Output:
```72      108     200     288     392     432     500     648     675     800
864     968     972     1125    1152    1323    1352    1372    1568    1800
1944    2000    2312    2592    2700    2888    3087    3200    3267    3456
3528    3872    3888    4000    4232    4500    4563    4608    5000    5292
5324    5400    5408    5488    6075    6125    6272    6728    6912    7200

500     864     1944    2000    2592    3456    5000    10125   10368   12348
12500   16875   19652   19773   30375   31104   32000   33275   37044   40500

1: 0
2: 1
3: 12
4: 47
5: 192
6: 664
```