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Achilles numbers

From Rosetta Code
Task
Achilles numbers
You are encouraged to solve this task according to the task description, using any language you may know.
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.


Task
  • 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.


See also


AArch64 Assembly[edit]

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 */
/*********************************/
.data
szMessNumber: .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 24
tbZoneDecom: .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 counter
1:
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 digits
5:
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,#1
10:
 
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 100f
98:
ldr x0,qAdrszErrorGen
bl affichageMess
100: // standard end of the program
mov x0, #0 // return code
mov x8,EXIT
svc #0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrszErrorGen: .quad szErrorGen
qAdrsZoneConv: .quad sZoneConv
qAdrtbZoneDecom: .quad tbZoneDecom
qAdrszMessNumber: .quad szMessNumber
qAdrszMessTitAchille: .quad szMessTitAchille
qAdrszMessTitStrong: .quad szMessTitStrong
icstMaxi2: .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 message
100:
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 100f
98:
mov x0,#0
b 100f
99:
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,#1
1:
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 equals
2:
mov x10,x2 // save indice
21:
bge 22f
mov x2,x7 // if x3 < x7 -> inversion
mov x7,x3
mov x3,x2 // x7 is the smaller exposant
22:
mov x0,x3
mov x1,x7 // x7 < x3
bl calPGCDmod
cmp x0,#1
beq 24f // no commun multiple -> ne peux donc pas etre une puissance
23:
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 // indice
3:
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 100f
99: // achille not ok
mov x0,0
100:
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 1
1:
cmp x1,0
bgt 2f
neg x1,x1 // if negative inversion number 2
2:
cmp x0,x1 // compare two numbers
bgt 3f
mov x2,x0 // inversion
mov x0,x1
mov x1,x2
3:
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 100f
99: // error
mov x0,0
cmp x0,0 // set carry
100:
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 divisor
1: // begin loop
mul x3,x1,x1 // compute square
cmp x3,x5 // compare number
bgt 4f // end
add x1,x1,#2 // next divisor
udiv x2,x5,x1
msub x3,x1,x2,x5 // compute remainder
cmp x3,#0 // remainder null ?
bne 3f
2: // begin loop 2
udiv x2,x5,x1
msub x3,x1,x2,x5 // compute remainder
cmp x3,#0
csel x5,x2,x5,eq // new value = quotient
beq 2b
 
udiv x2,x4,x1 // divide totient
sub x4,x4,x2 // compute new totient
3:
cmp x1,#2 // first divisor ?
mov x0,1
csel x1,x0,x1,eq // divisor = 1
b 1b // and loop
4:
cmp x5,#1 // final value > 1
ble 5f
mov x0,x4 // totient
mov x1,x5 // divide by value
udiv x2,x4,x5 // totient divide by value
sub x4,x4,x2 // compute new totient
5:
 
mov x0,x4
100:
ldp x4,x5,[sp],16 // restaur registers
ldp x2,x3,[sp],16 // restaur registers
ldp x1,lr,[sp],16 // restaur registers
ret
/******************************************************************/
/* 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 divisors
2:
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 counter
3:
mov x6,x1 // new divisor
4:
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 item
8:
mov x6,x8 // new dividende -> divisor prec
mov x7,#0 // and raz counter
9:
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 100f
99:
ldr x0,qAdrszMessErrGen
bl affichageMess
mov x0,#-1 // error code
b 100f
100:
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 premier
2:
cmn x0,0 // carry à zero pas d'erreur
mov x0,1 // premier
b 100f
99:
cmn x0,0 // carry à zero pas d'erreur
mov x0,#0 // Pas premier
100:
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 reste
1:
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,x3
2:
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 100f
99:
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 x30
qAdrszMessOverflow: .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 bit
1:
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 R
2:
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 haute
3:
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 1
4:
// 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,#1
5:
orr x0,x0,x4 // position du dernier bit du quotient
mov x3,x5
100:
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[edit]

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
END
END
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[edit]

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 */
/*********************************/
.data
szMessNumber: .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 24
tbZoneDecom: .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 counter
1:
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 digits
5:
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 6f
6:
 
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 100f
98:
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 call
iAdrszCarriageReturn: .int szCarriageReturn
iAdrszErrorGen: .int szErrorGen
iAdrsZoneConv: .int sZoneConv
iAdrtbZoneDecom: .int tbZoneDecom
iAdrszMessNumber: .int szMessNumber
iAdrszMessTitAchille: .int szMessTitAchille
iAdrszMessTitStrong: .int szMessTitStrong
icstMaxi2: .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 message
100:
pop {r1-r3,pc} @ restaur registers
iAdrszMessDigitsCounter: .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 100f
98:
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,#1
1:
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 equals
2:
mov r10,r2 @ save indice
21:
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 puissance
22:
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 @ indice
3:
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 ok
100:
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 registers
1:
cmp r0,#0
ble 2f
cmp r1,r0
movgt r2,r0
movgt r0,r1
movgt r1,r2
sub r0,r1
b 1b
2:
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 divisor
1: @ begin loop
mul r3,r1,r1 @ compute square
cmp r3,r5 @ compare number
bgt 4f @ end
add r1,r1,#2 @ next divisor
mov r0,r5
bl division
cmp r3,#0 @ remainder null ?
bne 3f
2: @ begin loop 2
mov r0,r5
bl division
cmp r3,#0
moveq r5,r2 @ new value = quotient
beq 2b
 
mov r0,r4 @ totient
bl division
sub r4,r4,r2 @ compute new totient
3:
cmp r1,#2 @ first divisor ?
moveq r1,#1 @ divisor = 1
b 1b @ and loop
4:
cmp r5,#1 @ final value > 1
ble 5f
mov r0,r4 @ totient
mov r1,r5 @ divide by value
bl division
sub r4,r4,r2 @ compute new totient
5:
 
mov r0,r4
100:
pop {r1-r5,pc} @ restaur registers
 
/******************************************************************/
/* 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 divisors
2:
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 counter
3:
mov r6,r1 @ new divisor
4:
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 item
8:
mov r6,r8 @ new dividende -> divisor prec
mov r7,#0 @ and raz counter
9:
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 100f
99:
ldr r0,iAdrszErrorGen
bl affichageMess
mov r0,#-1 @ error code
b 100f
100:
pop {r1-r8,lr} @ restaur registers
bx lr
iAdrszMessPrime: .int szMessPrime
 
/***************************************************/
/* check if a number is prime */
/***************************************************/
/* r0 contains the number */
/* r0 return 1 if prime 0 else */
@2147483647
@4294967297
@131071
isPrime:
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 prime
1:
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 90f
80:
mov r0,#1 @ is prime
b 100f
90:
mov r0,#0 @ no prime
100: @ 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 erreur
1:
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 <- remainder
2:
tst r5,#1 @ exposant even or odd
beq 3f
umull r0,r1,r6,r3
mov r2,r4
bl division32R
mov r3,r2 @ result <- remainder
3:
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,r3
100: @ 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 number
1: @ 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 3f
2: @ 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 @ remainder
100: @ 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++[edit]

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[edit]

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 tot
End 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 true
End 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 true
End Function
 
Dim As Integer num, n, i
Dim As Single inicio
Dim As Double t0 = Timer
 
Print "First 50 Achilles numbers:"
num = 0
n = 1
Do
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 += 1
Loop
 
Print !"\n\nFirst 20 strong Achilles numbers:"
num = 0
n = 1
Do
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 += 1
Loop
 
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:"; num
Next i
Sleep
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[edit]

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[edit]

Implementation:

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

Task examples:

   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 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
 
20{.(#~ strong * achilles) 1+i.100000 NB. first twenty strong achilles numbers
500 864 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 numbers
1
+/achilles (+i.)/1 9*10^<:3
12
+/achilles (+i.)/1 9*10^<:4
47
+/achilles (+i.)/1 9*10^<:5
192
+/achilles (+i.)/1 9*10^<:6
664

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[edit]

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))
end
end
 
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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Version 1 (Brute force)[edit]

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 Int
import "./seq" for Lst
import "./fmt" for Fmt
 
var maxDigits = 8
var 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 = 0
var n = 2
while (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 = 0
n = 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 = 10
for (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)[edit]

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 Set
import "./seq" for Lst
import "./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 = 15
getPerfectPowers.call(maxDigits)
 
var achillesSet = getAchilles.call(1, 5) // enough for first 2 parts
var achilles = achillesSet.toList
achilles.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 = 0
var n = 0
while (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[edit]

func GCD(N, D);         \Return the greatest common divisor of N and D
int N, D; \numerator and denominator
int R;
[if D > N then
[R:= D; D:= N; N:= R]; \swap D and N
while D > 0 do
[R:= rem(N/D);
N:= D;
D:= R;
];
return N;
]; \GCD
 
func Totient(N); \Return the totient of N
int 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 number
int 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 number
int 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