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{{task|Prime Numbers}}
The '''totient''' function is also known as: ::* Euler's totient function ::* Euler's phi totient function ::* phi totient function ::* Φ function (uppercase Greek phi) ::* φ function (lowercase Greek phi)
;Definitions (as per number theory): The totient function: ::* counts the integers up to a given positive integer '''n''' that are relatively prime to '''n''' ::* counts the integers '''k''' in the range '''1 ≤ k ≤ n''' for which the greatest common divisor '''gcd(n,k)''' is equal to '''1''' ::* counts numbers '''≤ n''' and prime to '''n'''
If the totient number (for '''N''') is one less than '''N''', then '''N''' is prime.
;Task: Create a '''totient''' function and: ::* Find and display (1 per line) for the 1st '''25''' integers: ::::* the integer (the index) ::::* the totient number for that integer ::::* indicate if that integer is prime ::* Find and display the ''count'' of the primes up to 100 ::* Find and display the ''count'' of the primes up to 1,000 ::* Find and display the ''count'' of the primes up to 10,000 ::* Find and display the ''count'' of the primes up to 100,000 (optional)
Show all output here.
;Related task: ::* [[Perfect totient numbers]]
;Also see:
::* the Wikipedia entry for [http://wikipedia.org/wiki/Euler's_totient_function Euler's totient function].
::* the MathWorld entry for [http://mathworld.wolfram.com/TotientFunction.html totient function].
::* the OEIS entry for [http://oeis.org/A000010 Euler totient function phi(n)].
AWK
# syntax: GAWK -f TOTIENT_FUNCTION.AWK
BEGIN {
print(" N Phi isPrime")
for (n=1; n<=1000000; n++) {
tot = totient(n)
if (n-1 == tot) {
count++
}
if (n <= 25) {
printf("%2d %3d %s\n",n,tot,(n-1==tot)?"true":"false")
if (n == 25) {
printf("\n Limit PrimeCount\n")
printf("%7d %10d\n",n,count)
}
}
else if (n ~ /^100+$/) {
printf("%7d %10d\n",n,count)
}
}
exit(0)
}
function totient(n, i,tot) {
tot = n
for (i=2; i*i<=n; i+=2) {
if (n % i == 0) {
while (n % i == 0) {
n /= i
}
tot -= tot / i
}
if (i == 2) {
i = 1
}
}
if (n > 1) {
tot -= tot / n
}
return(tot)
}
{{out}}
N Phi isPrime
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
Limit PrimeCount
25 9
100 25
1000 168
10000 1229
100000 9592
1000000 78498
C
Translation of the second Go example
/*Abhishek Ghosh, 7th December 2018*/ #include<stdio.h> int totient(int n){ int tot = n,i; for(i=2;i*i<=n;i+=2){ if(n%i==0){ while(n%i==0) n/=i; tot-=tot/i; } if(i==2) i=1; } if(n>1) tot-=tot/n; return tot; } int main() { int count = 0,n,tot; printf(" n %c prime",237); printf("\n---------------\n"); for(n=1;n<=25;n++){ tot = totient(n); if(n-1 == tot) count++; printf("%2d %2d %s\n", n, tot, n-1 == tot?"True":"False"); } printf("\nNumber of primes up to %6d =%4d\n", 25,count); for(n = 26; n <= 100000; n++){ tot = totient(n); if(tot == n-1) count++; if(n == 100 || n == 1000 || n%10000 == 0){ printf("\nNumber of primes up to %6d = %4d\n", n, count); } } return 0; }
Output :
n φ prime
---------------
1 1 False
2 1 True
3 2 True
4 2 False
5 4 True
6 2 False
7 6 True
8 4 False
9 6 False
10 4 False
11 10 True
12 4 False
13 12 True
14 6 False
15 8 False
16 8 False
17 16 True
18 6 False
19 18 True
20 8 False
21 12 False
22 10 False
23 22 True
24 8 False
25 20 False
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 20000 = 2262
Number of primes up to 30000 = 3245
Number of primes up to 40000 = 4203
Number of primes up to 50000 = 5133
Number of primes up to 60000 = 6057
Number of primes up to 70000 = 6935
Number of primes up to 80000 = 7837
Number of primes up to 90000 = 8713
Number of primes up to 100000 = 9592
C#
using static System.Console; using static System.Linq.Enumerable; public class Program { static void Main() { for (int i = 1; i <= 25; i++) { int t = Totient(i); WriteLine(i + "\t" + t + (t == i - 1 ? "\tprime" : "")); } WriteLine(); for (int i = 100; i <= 100_000; i *= 10) { WriteLine($"{Range(1, i).Count(x => Totient(x) + 1 == x):n0} primes below {i:n0}"); } } static int Totient(int n) { if (n < 3) return 1; if (n == 3) return 2; int totient = n; if ((n & 1) == 0) { totient >>= 1; while (((n >>= 1) & 1) == 0) ; } for (int i = 3; i * i <= n; i += 2) { if (n % i == 0) { totient -= totient / i; while ((n /= i) % i == 0) ; } } if (n > 1) totient -= totient / n; return totient; } }
{{out}}
1 1
2 1 prime
3 2 prime
4 2
5 4 prime
6 2
7 6 prime
8 4
9 6
10 4
11 10 prime
12 4
13 12 prime
14 6
15 8
16 8
17 16 prime
18 6
19 18 prime
20 8
21 12
22 10
23 22 prime
24 8
25 20
25 primes below 100
168 primes below 1,000
1,229 primes below 10,000
9,592 primes below 100,000
```
## Dyalect
{{trans|Go}}
```dyalect
func totient(n) {
var tot = n
var i = 2
while i * i <= n {
if n % i == 0 {
while n % i == 0 {
n /= i
}
tot -= tot / i
}
if i == 2 {
i = 1
}
i += 2
}
if n > 1 {
tot -= tot / n
}
return tot
}
print("n\tphi\tprime")
var count = 0
for n in 1..25 {
var tot = totient(n)
var isPrime = n - 1 == tot
if isPrime {
count += 1
}
print("\(n)\t\(tot)\t\(isPrime)")
}
print("\nNumber of primes up to 25 \t= \(count)")
for n in 26..100000 {
var tot = totient(n)
if tot == n - 1 {
count += 1
}
if n == 100 || n == 1000 || n % 10000 == 0 {
print("Number of primes up to \(n) \t= \(count)")
}
}
```
{{out}}
```txt
n phi prime
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 20000 = 2262
Number of primes up to 30000 = 3245
Number of primes up to 40000 = 4203
Number of primes up to 50000 = 5133
Number of primes up to 60000 = 6057
Number of primes up to 70000 = 6935
Number of primes up to 80000 = 7837
Number of primes up to 90000 = 8713
Number of primes up to 100000 = 9592
```
## Factor
```factor
USING: combinators formatting io kernel math math.primes.factors
math.ranges sequences ;
IN: rosetta-code.totient-function
: Φ ( n -- m )
{
{ [ dup 1 < ] [ drop 0 ] }
{ [ dup 1 = ] [ drop 1 ] }
[
dup unique-factors
[ 1 [ 1 - * ] reduce ] [ product ] bi / *
]
} cond ;
: show-info ( n -- )
[ Φ ] [ swap 2dup - 1 = ] bi ", prime" "" ?
"Φ(%2d) = %2d%s\n" printf ;
: totient-demo ( -- )
25 [1,b] [ show-info ] each nl 0 100,000 [1,b] [
[ dup Φ - 1 = [ 1 + ] when ]
[ dup { 100 1,000 10,000 100,000 } member? ] bi
[ dupd "%4d primes <= %d\n" printf ] [ drop ] if
] each drop ;
MAIN: totient-demo
```
{{out}}
```txt
Φ( 1) = 1
Φ( 2) = 1, prime
Φ( 3) = 2, prime
Φ( 4) = 2
Φ( 5) = 4, prime
Φ( 6) = 2
Φ( 7) = 6, prime
Φ( 8) = 4
Φ( 9) = 6
Φ(10) = 4
Φ(11) = 10, prime
Φ(12) = 4
Φ(13) = 12, prime
Φ(14) = 6
Φ(15) = 8
Φ(16) = 8
Φ(17) = 16, prime
Φ(18) = 6
Φ(19) = 18, prime
Φ(20) = 8
Φ(21) = 12
Φ(22) = 10
Φ(23) = 22, prime
Φ(24) = 8
Φ(25) = 20
25 primes <= 100
168 primes <= 1000
1229 primes <= 10000
9592 primes <= 100000
```
## FreeBASIC
{{trans|Pascal}}
```freebasic
#define esPar(n) (((n) And 1) = 0)
#define esImpar(n) (esPar(n) = 0)
Function Totient(n As Integer) As Integer
'delta son números no divisibles por 2,3,5
Dim delta(7) As Integer = {6,4,2,4,2,4,6,2}
Dim As Integer i, quot, idx, result
' div mod por constante es rápido.
'i = 2
result = n
If (2*2 <= n) Then
If Not(esImpar(n)) Then
' eliminar números con factor 2,4,8,16,...
While Not(esImpar(n))
n = n \ 2
Wend
'eliminar los múltiplos de 2
result -= result \ 2
End If
End If
'i = 3
If (3*3 <= n) And (n Mod 3 = 0) Then
Do
quot = n \ 3
If n <> quot*3 Then
Exit Do
Else
n = quot
End If
Loop Until false
result -= result \ 3
End If
'i = 5
If (5*5 <= n) And (n Mod 5 = 0) Then
Do
quot = n \ 5
If n <> quot*5 Then
Exit Do
Else
n = quot
End If
Loop Until false
result -= result \ 5
End If
i = 7
idx = 1
'i = 7,11,13,17,19,23,29,...,49 ..
While i*i <= n
quot = n \ i
If n = quot*i Then
Do
If n <> quot*i Then
Exit Do
Else
n = quot
End If
quot = n \ i
Loop Until false
result -= result \ i
End If
i = i + delta(idx)
idx = (idx+1) And 7
Wend
If n > 1 Then result -= result \ n
Totient = result
End Function
Sub ContandoPrimos(n As Integer)
Dim As Integer i, cnt = 0
For i = 1 To n
If Totient(i) = (i-1) Then cnt += 1
Next i
Print Using " ####### ######"; i-1; cnt
End Sub
Function esPrimo(n As Ulongint) As String
esPrimo = "False"
If n = 1 then Return "False"
If (n=2) Or (n=3) Then Return "True"
If n Mod 2=0 Then Exit Function
If n Mod 3=0 Then Exit Function
Dim As Ulongint limite = Sqr(N)+1
For i As Ulongint = 6 To limite Step 6
If N Mod (i-1)=0 Then Exit Function
If N Mod (i+1)=0 Then Exit Function
Next i
Return "True"
End Function
Sub display(n As Integer)
Dim As Integer idx, phi
If n = 0 Then Exit Sub
Print " n phi(n) esPrimo"
For idx = 1 To n
phi = Totient(idx)
Print Using "### ### \ \"; idx; phi; esPrimo(idx)
Next idx
End Sub
Dim l As Integer
display(25)
Print Chr(10) & " Limite Son primos"
ContandoPrimos(25)
l = 100
Do
ContandoPrimos(l)
l = l*10
Loop Until l > 1000000
End
```
{{out}}
```txt
n phi(n) esPrimo
1 1 False
2 1 True
3 2 True
4 2 False
5 4 True
6 2 False
7 6 True
8 4 False
9 6 False
10 4 False
11 10 True
12 4 False
13 12 True
14 6 False
15 8 False
16 8 False
17 16 True
18 6 False
19 18 True
20 8 False
21 12 False
22 10 False
23 22 True
24 8 False
25 20 False
Limite Son primos
25 9
100 25
1000 168
10000 1229
100000 9592
1000000 78498
```
## Go
Results for the larger values of n are very slow to emerge.
```go
package main
import "fmt"
func gcd(n, k int) int {
if n < k || k < 1 {
panic("Need n >= k and k >= 1")
}
s := 1
for n&1 == 0 && k&1 == 0 {
n >>= 1
k >>= 1
s <<= 1
}
t := n
if n&1 != 0 {
t = -k
}
for t != 0 {
for t&1 == 0 {
t >>= 1
}
if t > 0 {
n = t
} else {
k = -t
}
t = n - k
}
return n * s
}
func totient(n int) int {
tot := 0
for k := 1; k <= n; k++ {
if gcd(n, k) == 1 {
tot++
}
}
return tot
}
func main() {
fmt.Println(" n phi prime")
fmt.Println("---------------")
count := 0
for n := 1; n <= 25; n++ {
tot := totient(n)
isPrime := n-1 == tot
if isPrime {
count++
}
fmt.Printf("%2d %2d %t\n", n, tot, isPrime)
}
fmt.Println("\nNumber of primes up to 25 =", count)
for n := 26; n <= 100000; n++ {
tot := totient(n)
if tot == n-1 {
count++
}
if n == 100 || n == 1000 || n%10000 == 0 {
fmt.Printf("\nNumber of primes up to %-6d = %d\n", n, count)
}
}
}
```
{{out}}
```txt
n phi prime
---------------
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 20000 = 2262
Number of primes up to 30000 = 3245
Number of primes up to 40000 = 4203
Number of primes up to 50000 = 5133
Number of primes up to 60000 = 6057
Number of primes up to 70000 = 6935
Number of primes up to 80000 = 7837
Number of primes up to 90000 = 8713
Number of primes up to 100000 = 9592
```
The following much quicker version (runs in less than 150 ms on my machine) uses Euler's product formula rather than repeated invocation of the gcd function to calculate the totient:
```go
package main
import "fmt"
func totient(n int) int {
tot := n
for i := 2; i*i <= n; i += 2 {
if n%i == 0 {
for n%i == 0 {
n /= i
}
tot -= tot / i
}
if i == 2 {
i = 1
}
}
if n > 1 {
tot -= tot / n
}
return tot
}
func main() {
fmt.Println(" n phi prime")
fmt.Println("---------------")
count := 0
for n := 1; n <= 25; n++ {
tot := totient(n)
isPrime := n-1 == tot
if isPrime {
count++
}
fmt.Printf("%2d %2d %t\n", n, tot, isPrime)
}
fmt.Println("\nNumber of primes up to 25 =", count)
for n := 26; n <= 100000; n++ {
tot := totient(n)
if tot == n-1 {
count++
}
if n == 100 || n == 1000 || n%10000 == 0 {
fmt.Printf("\nNumber of primes up to %-6d = %d\n", n, count)
}
}
}
```
The output is the same as before.
## Haskell
{{trans|C}}
```Haskell
{-# LANGUAGE BangPatterns #-}
import Control.Monad
totient :: (Integral a) => a -> a
totient n
| n == 0 = 1 -- by definition phi(0) = 1
| n < 0 = totient (-n) -- phi(-n) is taken to be equal to phi(n)
| otherwise = loop n n 2 --
where
loop !m !tot !i
| i * i > m = if m > 1 then tot - (tot `div` m) else tot
| m `mod` i == 0 = loop m' tot' i'
| otherwise = loop m tot i'
where i' = if i == 2 then 3 else (i + 2)
m' = nextM m
tot' = tot - (tot `div` i)
nextM !x | x `mod` i == 0 = nextM $ x `div` i
| otherwise = x
main :: IO ()
main = do
putStrLn "n\tphi\tprime\n---------------------"
let loop !i !count
| i >= 10^6 = return ()
| otherwise = do
let i' = i + 1
tot = totient i'
isPrime = tot == i' - 1
count' = if isPrime then count + 1 else count
when (i' <= 25) $ do
putStrLn $ (show i') ++ "\t" ++ (show tot) ++ "\t" ++ (show isPrime)
when (i' `elem` 25 : [ 10^k | k <- [2..6] ]) $ do
putStrLn $ "Number of primes up to " ++ (show i') ++ " = " ++ (show count')
loop (i + 1) count'
loop 0 0
```
{{out}}
```txt
n phi prime
---------------------
1 1 False
2 1 True
3 2 True
4 2 False
5 4 True
6 2 False
7 6 True
8 4 False
9 6 False
10 4 False
11 10 True
12 4 False
13 12 True
14 6 False
15 8 False
16 8 False
17 16 True
18 6 False
19 18 True
20 8 False
21 12 False
22 10 False
23 22 True
24 8 False
25 20 False
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 100000 = 9592
Number of primes up to 1000000 = 78498
```
## J
```J
nth_prime =: p: NB. 2 is the zeroth prime
totient =: 5&p:
primeQ =: 1&p:
NB. first row contains the integer
NB. second row totient
NB. third 1 iff prime
(, totient ,: primeQ) >: i. 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 12 10 22 8 20
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0
NB. primes first exceeding the limits
[&.:(p:inv) 10 ^ 2 + i. 4
101 1009 10007 100003
p:inv 101 1009 10007 100003
25 168 1229 9592
NB. limit and prime count
(,. p:inv) 10 ^ 2 + i. 5
100 25
1000 168
10000 1229
100000 9592
1e6 78498
```
## Julia
{{trans|Python}}
```julia
φ(n) = sum(1 for k in 1:n if gcd(n, k) == 1)
is_prime(n) = φ(n) == n - 1
function runphitests()
for n in 1:25
println(" φ($n) == $(φ(n))", is_prime(n) ? ", is prime" : "")
end
count = 0
for n in 1:100_000
count += is_prime(n)
if n in [100, 1000, 10_000, 100_000]
println("Primes up to $n: $count")
end
end
end
runphitests()
```
{{output}}
```txt
φ(1) == 1
φ(2) == 1, is prime
φ(3) == 2, is prime
φ(4) == 2
φ(5) == 4, is prime
φ(6) == 2
φ(7) == 6, is prime
φ(8) == 4
φ(9) == 6
φ(10) == 4
φ(11) == 10, is prime
φ(12) == 4
φ(13) == 12, is prime
φ(14) == 6
φ(15) == 8
φ(16) == 8
φ(17) == 16, is prime
φ(18) == 6
φ(19) == 18, is prime
φ(20) == 8
φ(21) == 12
φ(22) == 10
φ(23) == 22, is prime
φ(24) == 8
φ(25) == 20
Primes up to 100: 25
Primes up to 1000: 168
Primes up to 10000: 1229
Primes up to 100000: 9592
```
## Kotlin
{{trans|Go}}
```scala
// Version 1.3.21
fun totient(n: Int): Int {
var tot = n
var nn = n
var i = 2
while (i * i <= nn) {
if (nn % i == 0) {
while (nn % i == 0) nn /= i
tot -= tot / i
}
if (i == 2) i = 1
i += 2
}
if (nn > 1) tot -= tot / nn
return tot
}
fun main() {
println(" n phi prime")
println("---------------")
var count = 0
for (n in 1..25) {
val tot = totient(n)
val isPrime = n - 1 == tot
if (isPrime) count++
System.out.printf("%2d %2d %b\n", n, tot, isPrime)
}
println("\nNumber of primes up to 25 = $count")
for (n in 26..100_000) {
val tot = totient(n)
if (tot == n-1) count++
if (n == 100 || n == 1000 || n % 10_000 == 0) {
System.out.printf("\nNumber of primes up to %-6d = %d\n", n, count)
}
}
}
```
{{output}}
```txt
Same as Go example.
```
## Lua
Averages about 7 seconds under LuaJIT
```lua
-- Return the greatest common denominator of x and y
function gcd (x, y)
return y == 0 and math.abs(x) or gcd(y, x % y)
end
-- Return the totient number for n
function totient (n)
local count = 0
for k = 1, n do
if gcd(n, k) == 1 then count = count + 1 end
end
return count
end
-- Determine (inefficiently) whether p is prime
function isPrime (p)
return totient(p) == p - 1
end
-- Output totient and primality for the first 25 integers
print("n", string.char(237), "prime")
print(string.rep("-", 21))
for i = 1, 25 do
print(i, totient(i), isPrime(i))
end
-- Count the primes up to 100, 1000 and 10000
local pCount, i, limit = 0, 1
for power = 2, 4 do
limit = 10 ^ power
repeat
i = i + 1
if isPrime(i) then pCount = pCount + 1 end
until i == limit
print("\nThere are " .. pCount .. " primes below " .. limit)
end
```
{{out}}
```txt
n φ prime
---------------------
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
There are 25 primes below 100
There are 168 primes below 1000
There are 1229 primes below 10000
```
## Nim
```Nim
import strformat
func totient(n: int): int =
var tot = n
var nn = n
var i = 2
while i * i <= nn:
if nn mod i == 0:
while nn mod i == 0:
nn = nn div i
dec tot, tot div i
if i == 2:
i = 1
inc i, 2
if nn > 1:
dec tot, tot div nn
tot
echo " n φ prime"
echo "---------------"
var count = 0
for n in 1..25:
let tot = totient(n)
let isPrime = n - 1 == tot
if isPrime:
inc count
echo fmt"{n:2} {tot:2} {isPrime}"
echo ""
echo fmt"Number of primes up to {25:>6} = {count:>4}"
for n in 26..100_000:
let tot = totient(n)
if tot == n - 1:
inc count
if n == 100 or n == 1000 or n mod 10_000 == 0:
echo fmt"Number of primes up to {n:>6} = {count:>4}"
```
{{out}}
```txt
n φ prime
---------------
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 20000 = 2262
Number of primes up to 30000 = 3245
Number of primes up to 40000 = 4203
Number of primes up to 50000 = 5133
Number of primes up to 60000 = 6057
Number of primes up to 70000 = 6935
Number of primes up to 80000 = 7837
Number of primes up to 90000 = 8713
Number of primes up to 100000 = 9592
```
## Pascal
Yes, a very slow possibility to check prime
```pascal
{$IFDEF FPC}
{$MODE DELPHI}
{$IFEND}
function gcd_mod(u, v: NativeUint): NativeUint;inline;
//prerequisites u > v and u,v > 0
var
t: NativeUInt;
begin
repeat
t := u;
u := v;
v := t mod v;
until v = 0;
gcd_mod := u;
end;
function Totient(n:NativeUint):NativeUint;
var
i : NativeUint;
Begin
result := 1;
For i := 2 to n do
inc(result,ORD(GCD_mod(n,i)=1));
end;
function CheckPrimeTotient(n:NativeUint):Boolean;inline;
begin
result := (Totient(n) = (n-1));
end;
procedure OutCountPrimes(n:NativeUInt);
var
i,cnt : NativeUint;
begin
cnt := 0;
For i := 1 to n do
inc(cnt,Ord(CheckPrimeTotient(i)));
writeln(n:10,cnt:8);
end;
procedure display(n:NativeUint);
var
idx,phi : NativeUint;
Begin
if n = 0 then
EXIT;
writeln('number n':5,'Totient(n)':11,'isprime':8);
For idx := 1 to n do
Begin
phi := Totient(idx);
writeln(idx:4,phi:10,(phi=(idx-1)):12);
end
end;
var
i : NativeUint;
Begin
display(25);
writeln('Limit primecount');
i := 100;
repeat
OutCountPrimes(i);
i := i*10;
until i >100000;
end.
```
;Output:
```txt
number n Totient(n) isprime
1 1 FALSE
2 1 TRUE
3 2 TRUE
4 2 FALSE
5 4 TRUE
6 2 FALSE
7 6 TRUE
8 4 FALSE
9 6 FALSE
10 4 FALSE
11 10 TRUE
12 4 FALSE
13 12 TRUE
14 6 FALSE
15 8 FALSE
16 8 FALSE
17 16 TRUE
18 6 FALSE
19 18 TRUE
20 8 FALSE
21 12 FALSE
22 10 FALSE
23 22 TRUE
24 8 FALSE
25 20 FALSE
Limit primecount
100 25
1000 168
10000 1229
100000 9592
real 3m39,745s
```
### alternative
changing Totient-funtion in program atop to Computing Euler's totient function on wikipedia, like GO and C.
Impressive speedup.Checking with only primes would be even faster.
```pascal
function totient(n:NativeUInt):NativeUInt;
const
//delta of numbers not divisible by 2,3,5 (0_1+6->7+4->11 ..+6->29+2->3_1
delta : array[0..7] of NativeUint = (6,4,2,4,2,4,6,2);
var
i, quot,idx: NativeUint;
Begin
// div mod by constant is fast.
//i = 2
result := n;
if (2*2 <= n) then
Begin
IF not(ODD(n)) then
Begin
// remove numbers with factor 2,4,8,16, ...
while not(ODD(n)) do
n := n DIV 2;
//remove count of multiples of 2
dec(result,result DIV 2);
end;
end;
//i = 3
If (3*3 <= n) AND (n mod 3 = 0) then
Begin
repeat
quot := n DIV 3;
IF n <> quot*3 then
BREAK
else
n := quot;
until false;
dec(result,result DIV 3);
end;
//i = 5
If (5*5 <= n) AND (n mod 5 = 0) then
Begin
repeat
quot := n DIV 5;
IF n <> quot*5 then
BREAK
else
n := quot;
until false;
dec(result,result DIV 5);
end;
i := 7;
idx := 1;
//i = 7,11,13,17,19,23,29, ...49 ..
while i*i <= n do
Begin
quot := n DIV i;
if n = quot*i then
Begin
repeat
IF n <> quot*i then
BREAK
else
n := quot;
quot := n DIV i;
until false;
dec(result,result DIV i);
end;
i := i + delta[idx];
idx := (idx+1) AND 7;
end;
if n> 1 then
dec(result,result div n);
end;
```
;Output:
```txt
number n Totient(n) isprime
1 1 FALSE
2 1 TRUE
3 2 TRUE
4 2 FALSE
5 4 TRUE
6 2 FALSE
7 6 TRUE
8 4 FALSE
9 6 FALSE
10 4 FALSE
11 10 TRUE
12 4 FALSE
13 12 TRUE
14 6 FALSE
15 8 FALSE
16 8 FALSE
17 16 TRUE
18 6 FALSE
19 18 TRUE
20 8 FALSE
21 12 FALSE
22 10 FALSE
23 22 TRUE
24 8 FALSE
25 20 FALSE
Limit primecount
100 25
1000 168
10000 1229
100000 9592
1000000 78498
10000000 664579
real 0m7,369s
```
## Perl
====Direct calculation of 𝜑====
{{trans|Perl 6}}
```perl
use utf8;
binmode STDOUT, ":utf8";
sub gcd {
my ($u, $v) = @_;
while ($v) {
($u, $v) = ($v, $u % $v);
}
return abs($u);
}
push @𝜑, 0;
for $t (1..10000) {
push @𝜑, scalar grep { 1 == gcd($_,$t) } 1..$t;
}
printf "𝜑(%2d) = %3d%s\n", $_, $𝜑[$_], $_ - $𝜑[$_] - 1 ? '' : ' Prime' for 1 .. 25;
print "\n";
for $limit (100, 1000, 10000) {
printf "Count of primes <= $limit: %d\n", scalar grep {$_ == $𝜑[$_] + 1} 0..$limit;
}
```
{{out}}
```txt
𝜑( 1) = 1
𝜑( 2) = 1 Prime
𝜑( 3) = 2 Prime
𝜑( 4) = 2
𝜑( 5) = 4 Prime
𝜑( 6) = 2
𝜑( 7) = 6 Prime
𝜑( 8) = 4
𝜑( 9) = 6
𝜑(10) = 4
𝜑(11) = 10 Prime
𝜑(12) = 4
𝜑(13) = 12 Prime
𝜑(14) = 6
𝜑(15) = 8
𝜑(16) = 8
𝜑(17) = 16 Prime
𝜑(18) = 6
𝜑(19) = 18 Prime
𝜑(20) = 8
𝜑(21) = 12
𝜑(22) = 10
𝜑(23) = 22 Prime
𝜑(24) = 8
𝜑(25) = 20
Count of primes <= 100: 25
Count of primes <= 1000: 168
Count of primes <= 10000: 1229
```
====Using 'ntheory' library====
Much faster. Output is the same as above.
{{libheader|ntheory}}
```perl
use utf8;
binmode STDOUT, ":utf8";
use ntheory qw(euler_phi);
my @𝜑 = euler_phi(0,10000); # Returns list of all values in range
printf "𝜑(%2d) = %3d%s\n", $_, $𝜑[$_], $_ - $𝜑[$_] - 1 ? '' : ' Prime' for 1 .. 25;
print "\n";
for $limit (100, 1000, 10000) {
printf "Count of primes <= $limit: %d\n", scalar grep {$_ == $𝜑[$_] + 1} 0..$limit;
}
```
## Perl 6
{{works with|Rakudo|2018.11}}
This is an ''incredibly'' inefficient way of finding prime numbers.
```perl6
my \𝜑 = 0, |(1..*).hyper(:8degree).map: -> $t { +(^$t).grep: * gcd $t == 1 };
printf "𝜑(%2d) = %3d %s\n", $_, 𝜑[$_], $_ - 𝜑[$_] - 1 ?? '' !! 'Prime' for 1 .. 25;
(100, 1000, 10000).map: -> $limit {
say "\nCount of primes <= $limit: " ~ +(^$limit).grep: {$_ == 𝜑[$_] + 1}
}
```
{{out}}
```txt
𝜑( 1) = 1
𝜑( 2) = 1 Prime
𝜑( 3) = 2 Prime
𝜑( 4) = 2
𝜑( 5) = 4 Prime
𝜑( 6) = 2
𝜑( 7) = 6 Prime
𝜑( 8) = 4
𝜑( 9) = 6
𝜑(10) = 4
𝜑(11) = 10 Prime
𝜑(12) = 4
𝜑(13) = 12 Prime
𝜑(14) = 6
𝜑(15) = 8
𝜑(16) = 8
𝜑(17) = 16 Prime
𝜑(18) = 6
𝜑(19) = 18 Prime
𝜑(20) = 8
𝜑(21) = 12
𝜑(22) = 10
𝜑(23) = 22 Prime
𝜑(24) = 8
𝜑(25) = 20
Count of primes <= 100: 25
Count of primes <= 1000: 168
Count of primes <= 10000: 1229
```
## Phix
{{trans|Go}}
```Phix
function totient(integer n)
integer tot = n, i = 2
while i*i<=n do
if mod(n,i)=0 then
while true do
n /= i
if mod(n,i)!=0 then exit end if
end while
tot -= tot/i
end if
i += iff(i=2?1:2)
end while
if n>1 then
tot -= tot/n
end if
return tot
end function
printf(1," n phi prime\n")
printf(1," --------------\n")
integer count = 0
for n=1 to 25 do
integer tot = totient(n),
prime = (n-1=tot)
count += prime
string isp = iff(prime?"true":"false")
printf(1,"%2d %2d %s\n",{n,tot,isp})
end for
printf(1,"\nNumber of primes up to 25 = %d\n",count)
for n=26 to 100000 do
count += (totient(n)=n-1)
if find(n,{100,1000,10000,100000}) then
printf(1,"Number of primes up to %-6d = %d\n",{n,count})
end if
end for
```
{{out}}
```txt
n phi prime
--------------
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 100000 = 9592
```
## PicoLisp
```PicoLisp
(gc 32)
(de gcd (A B)
(until (=0 B)
(let M (% A B)
(setq A B B M) ) )
(abs A) )
(de totient (N)
(let C 0
(for I N
(and (=1 (gcd N I)) (inc 'C)) )
(cons C (= C (dec N))) ) )
(de p? (N)
(let C 0
(for A N
(and
(cdr (totient A))
(inc 'C) ) )
C ) )
(let Fmt (3 7 10)
(tab Fmt "N" "Phi" "Prime?")
(tab Fmt "-" "---" "------")
(for N 25
(tab Fmt
N
(car (setq @ (totient N)))
(cdr @) ) ) )
(println
(mapcar p? (25 100 1000 10000 100000)) )
```
{{out}}
```txt
N Phi Prime?
- --- ------
1 1
2 1 T
3 2 T
4 2
5 4 T
6 2
7 6 T
8 4
9 6
10 4
11 10 T
12 4
13 12 T
14 6
15 8
16 8
17 16 T
18 6
19 18 T
20 8
21 12
22 10
23 22 T
24 8
25 20
(9 25 168 1229 9592)
```
## Python
```python
from math import gcd
def φ(n):
return sum(1 for k in range(1, n + 1) if gcd(n, k) == 1)
if __name__ == '__main__':
def is_prime(n):
return φ(n) == n - 1
for n in range(1, 26):
print(f" φ({n}) == {φ(n)}{', is prime' if is_prime(n) else ''}")
count = 0
for n in range(1, 10_000 + 1):
count += is_prime(n)
if n in {100, 1000, 10_000}:
print(f"Primes up to {n}: {count}")
```
{{out}}
```txt
φ(1) == 1
φ(2) == 1, is prime
φ(3) == 2, is prime
φ(4) == 2
φ(5) == 4, is prime
φ(6) == 2
φ(7) == 6, is prime
φ(8) == 4
φ(9) == 6
φ(10) == 4
φ(11) == 10, is prime
φ(12) == 4
φ(13) == 12, is prime
φ(14) == 6
φ(15) == 8
φ(16) == 8
φ(17) == 16, is prime
φ(18) == 6
φ(19) == 18, is prime
φ(20) == 8
φ(21) == 12
φ(22) == 10
φ(23) == 22, is prime
φ(24) == 8
φ(25) == 20
Primes up to 100: 25
Primes up to 1000: 168
Primes up to 10000: 1229
```
## Racket
```racket
#lang racket
(require math/number-theory)
(define (prime*? n) (= (totient n) (sub1 n)))
(for ([n (in-range 1 26)])
(printf "φ(~a) = ~a~a~a\n"
n
(totient n)
(if (prime*? n) " is prime" "")
(if (prime? n) " (confirmed)" "")))
(for/fold ([count 0] #:result (void)) ([n (in-range 1 10001)])
(define new-count (if (prime*? n) (add1 count) count))
(when (member n '(100 1000 10000))
(printf "Primes up to ~a: ~a\n" n new-count))
new-count)
```
{{out}}
```txt
φ(1) = 1
φ(2) = 1 is prime (confirmed)
φ(3) = 2 is prime (confirmed)
φ(4) = 2
φ(5) = 4 is prime (confirmed)
φ(6) = 2
φ(7) = 6 is prime (confirmed)
φ(8) = 4
φ(9) = 6
φ(10) = 4
φ(11) = 10 is prime (confirmed)
φ(12) = 4
φ(13) = 12 is prime (confirmed)
φ(14) = 6
φ(15) = 8
φ(16) = 8
φ(17) = 16 is prime (confirmed)
φ(18) = 6
φ(19) = 18 is prime (confirmed)
φ(20) = 8
φ(21) = 12
φ(22) = 10
φ(23) = 22 is prime (confirmed)
φ(24) = 8
φ(25) = 20
Primes up to 100: 25
Primes up to 1000: 168
Primes up to 10000: 1229
```
## REXX
```rexx
/*REXX program calculates the totient numbers for a range of numbers, and count primes. */
parse arg N . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 25 /*Not specified? Then use the default.*/
tell= N>0 /*N positive>? Then display them all. */
N= abs(N) /*use the absolute value of N for loop.*/
w= length(N) /*W: is used in aligning the output. */
primes= 0 /*the number of primes found (so far).*/
/*if N was negative, only count primes.*/
do j=1 for N; T= phi(j) /*obtain totient number for a number. */
prime= word('(prime)', 1 + (T \== j-1 ) ) /*determine if J is a prime number. */
if prime\=='' then primes= primes+1 /*if a prime, then bump the prime count*/
if tell then say 'totient number for ' right(j, w) " ──► " right(T, w) ' ' prime
end /*j*/
say
say right(primes, w) ' primes detected for numbers up to and including ' N
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: parse arg x,y; do until y==0; parse value x//y y with y x
end /*until*/; return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
phi: procedure; parse arg z; #= z==1
do m=1 for z-1; if gcd(m, z)==1 then #= # + 1
end /*m*/; return #
```
{{out|output|text= when using the default input of: 25 }}
```txt
totient number for 1 ──► 1
totient number for 2 ──► 1 (prime)
totient number for 3 ──► 2 (prime)
totient number for 4 ──► 2
totient number for 5 ──► 4 (prime)
totient number for 6 ──► 2
totient number for 7 ──► 6 (prime)
totient number for 8 ──► 4
totient number for 9 ──► 6
totient number for 10 ──► 4
totient number for 11 ──► 10 (prime)
totient number for 12 ──► 4
totient number for 13 ──► 12 (prime)
totient number for 14 ──► 6
totient number for 15 ──► 8
totient number for 16 ──► 8
totient number for 17 ──► 16 (prime)
totient number for 18 ──► 6
totient number for 19 ──► 18 (prime)
totient number for 20 ──► 8
totient number for 21 ──► 12
totient number for 22 ──► 10
totient number for 23 ──► 22 (prime)
totient number for 24 ──► 8
totient number for 25 ──► 20
9 primes detected for numbers up to and including 25
```
{{out|output|text= when using the input of: -100 }}
```txt
25 primes detected for numbers up to and including 100
```
{{out|output|text= when using the input of: -1000 }}
```txt
168 primes detected for numbers up to and including 1000
```
{{out|output|text= when using the input of: -10000 }}
```txt
1229 primes detected for numbers up to and including 10000
```
{{out|output|text= when using the input of: -1000000 }}
```txt
9592 primes detected for numbers up to and including 100000
```
## Ruby
```ruby
require "prime"
def 𝜑(n)
n.prime_division.inject(1) {|res, (pr, exp)| res *= (pr-1) * pr**(exp-1) }
end
1.upto 25 do |n|
tot = 𝜑(n)
puts "#{n}\t #{tot}\t #{"prime" if n-tot==1}"
end
[100, 1_000, 10_000, 100_000].each do |u|
puts "Number of primes up to #{u}: #{(1..u).count{|n| n-𝜑(n) == 1} }"
end
```
{{out}}
```txt
1 1
2 1 prime
3 2 prime
4 2
5 4 prime
6 2
7 6 prime
8 4
9 6
10 4
11 10 prime
12 4
13 12 prime
14 6
15 8
16 8
17 16 prime
18 6
19 18 prime
20 8
21 12
22 10
23 22 prime
24 8
25 20
Number of primes up to 100: 25
Number of primes up to 1000: 168
Number of primes up to 10000: 1229
Number of primes up to 100000: 9592
```
## Rust
```rust
use num::integer::gcd;
fn main() {
// Compute the totient of the first 25 natural integers
println!("N\t phi(n)\t Prime");
for n in 1..26 {
let phi_n = phi(n);
println!("{}\t {}\t {:?}", n, phi_n, phi_n == n - 1);
}
// Compute the number of prime numbers for various steps
[1, 100, 1000, 10000, 100000]
.windows(2)
.scan(0, |acc, tuple| {
*acc += (tuple[0]..=tuple[1]).filter(is_prime).count();
Some((tuple[1], *acc))
})
.for_each(|x| println!("Until {}: {} prime numbers", x.0, x.1));
}
fn is_prime(n: &usize) -> bool {
phi(*n) == *n - 1
}
fn phi(n: usize) -> usize {
(1..=n).filter(|&x| gcd(n, x) == 1).count()
}
```
Output is:
```txt
N phi(n) Prime
1 1 false
2 1 true
3 2 true
4 2 false
5 4 true
6 2 false
7 6 true
8 4 false
9 6 false
10 4 false
11 10 true
12 4 false
13 12 true
14 6 false
15 8 false
16 8 false
17 16 true
18 6 false
19 18 true
20 8 false
21 12 false
22 10 false
23 22 true
24 8 false
25 20 false
Until 100: 25 prime numbers
Until 1000: 168 prime numbers
Until 10000: 1229 prime numbers
Until 100000: 9592 prime numbers
```
## Scala
The most concise way to write the totient function in Scala is using a naive lazy list:
```scala
@tailrec
def gcd(a: Int, b: Int): Int = if(b == 0) a else gcd(b, a%b)
def totientLaz(num: Int): Int = LazyList.range(2, num).count(gcd(num, _) == 1) + 1
```
The above approach, while concise, is not very performant. It must check the GCD with every number between 2 and num, giving it an O(n*log(n)) time complexity. A better solution is to use the product definition of the totient, repeatedly extracting prime factors from num:
```scala
def totientPrd(num: Int): Int = {
@tailrec
def dTrec(f: Int, n: Int): Int = if(n%f == 0) dTrec(f, n/f) else n
@tailrec
def tTrec(ac: Int, i: Int, n: Int): Int = if(n != 1){
if(n%i == 0) tTrec(ac*(i - 1)/i, i + 1, dTrec(i, n))
else tTrec(ac, i + 1, n)
}else{
ac
}
tTrec(num, 2, num)
}
```
This version is significantly faster, but the introduction of multiple recursive methods makes it far less concise. We can, however, embed the recursion into a lazy list to obtain a function as fast as the second example yet almost as concise as the first, at the cost of some readability:
```scala
@tailrec
def scrub(f: Long, num: Long): Long = if(num%f == 0) scrub(f, num/f) else num
def totientLazPrd(num: Long): Long = LazyList.iterate((num, 2: Long, num)){case (ac, i, n) => if(n%i == 0) (ac*(i - 1)/i, i + 1, scrub(i, n)) else (ac, i + 1, n)}.find(_._3 == 1).get._1
```
To generate the output up to 100000, Longs are necessary.
{{out}}
```txt
1 (Not Prime): 1
2 (Prime) : 1
3 (Prime) : 2
4 (Not Prime): 2
5 (Prime) : 4
6 (Not Prime): 2
7 (Prime) : 6
8 (Not Prime): 4
9 (Not Prime): 6
10 (Not Prime): 4
11 (Prime) : 10
12 (Not Prime): 4
13 (Prime) : 12
14 (Not Prime): 6
15 (Not Prime): 8
16 (Not Prime): 8
17 (Prime) : 16
18 (Not Prime): 6
19 (Prime) : 18
20 (Not Prime): 8
21 (Not Prime): 12
22 (Not Prime): 10
23 (Prime) : 22
24 (Not Prime): 8
25 (Not Prime): 20
Prime Count <= N...
100: 25
1000: 168
10000: 1229
100000: 9592
```
## Sidef
The Euler totient function is built-in as '''Number.euler_phi()''', but we can easily re-implement it using its multiplicative property: '''phi(p^k) = (p-1)*p^(k-1)'''.
```ruby
func 𝜑(n) {
n.factor_exp.prod {|p|
(p[0]-1) * p[0]**(p[1]-1)
}
}
```
```ruby
for n in (1..25) {
var totient = 𝜑(n)
printf("𝜑(%2s) = %3s%s\n", n, totient, totient==(n-1) ? ' - prime' : '')
}
```
{{out}}
𝜑( 1) = 1
𝜑( 2) = 1 - prime
𝜑( 3) = 2 - prime
𝜑( 4) = 2
𝜑( 5) = 4 - prime
𝜑( 6) = 2
𝜑( 7) = 6 - prime
𝜑( 8) = 4
𝜑( 9) = 6
𝜑(10) = 4
𝜑(11) = 10 - prime
𝜑(12) = 4
𝜑(13) = 12 - prime
𝜑(14) = 6
𝜑(15) = 8
𝜑(16) = 8
𝜑(17) = 16 - prime
𝜑(18) = 6
𝜑(19) = 18 - prime
𝜑(20) = 8
𝜑(21) = 12
𝜑(22) = 10
𝜑(23) = 22 - prime
𝜑(24) = 8
𝜑(25) = 20
```
```ruby
[100, 1_000, 10_000, 100_000].each {|limit|
var pi = (1..limit -> count_by {|n| 𝜑(n) == (n-1) })
say "Number of primes <= #{limit}: #{pi}"
}
```
{{out}}
```txt
Number of primes <= 100: 25
Number of primes <= 1000: 168
Number of primes <= 10000: 1229
Number of primes <= 100000: 9592
```
## VBA
{{trans|Phix}}
```vb
Private Function totient(ByVal n As Long) As Long
Dim tot As Long: tot = n
Dim i As Long: i = 2
Do While i * i <= n
If n Mod i = 0 Then
Do While True
n = n \ i
If n Mod i <> 0 Then Exit Do
Loop
tot = tot - tot \ i
End If
i = i + IIf(i = 2, 1, 2)
Loop
If n > 1 Then
tot = tot - tot \ n
End If
totient = tot
End Function
Public Sub main()
Debug.Print " n phi prime"
Debug.Print " --------------"
Dim count As Long
Dim tot As Integer, n As Long
For n = 1 To 25
tot = totient(n)
prime = (n - 1 = tot)
count = count - prime
Debug.Print Format(n, "@@"); Format(tot, "@@@@@"); Format(prime, "@@@@@@@@")
Next n
Debug.Print
Debug.Print "Number of primes up to 25 = "; Format(count, "@@@@")
For n = 26 To 100000
count = count - (totient(n) = n - 1)
Select Case n
Case 100, 1000, 10000, 100000
Debug.Print "Number of primes up to"; n; String$(6 - Len(CStr(n)), " "); "="; Format(count, "@@@@@")
Case Else
End Select
Next n
End Sub
```
{{out}}
```txt
n phi prime
--------------
1 1 False
2 1 True
3 2 True
4 2 False
5 4 True
6 2 False
7 6 True
8 4 False
9 6 False
10 4 False
11 10 True
12 4 False
13 12 True
14 6 False
15 8 False
16 8 False
17 16 True
18 6 False
19 18 True
20 8 False
21 12 False
22 10 False
23 22 True
24 8 False
25 20 False
Number of primes up to 25 = 9
Number of primes up to 100 = 25
Number of primes up to 1000 = 168
Number of primes up to 10000 = 1229
Number of primes up to 100000 = 9592
```
## zkl
```zkl
fcn totient(n){ [1..n].reduce('wrap(p,k){ p + (n.gcd(k)==1) }) }
fcn isPrime(n){ totient(n)==(n - 1) }
```
```zkl
foreach n in ([1..25]){
println("\u03c6(%2d) ==%3d %s"
.fmt(n,totient(n),isPrime(n) and "is prime" or ""));
}
```
{{out}}
φ( 1) == 1
φ( 2) == 1 is prime
φ( 3) == 2 is prime
φ( 4) == 2
φ( 5) == 4 is prime
φ( 6) == 2
φ( 7) == 6 is prime
φ( 8) == 4
φ( 9) == 6
φ(10) == 4
φ(11) == 10 is prime
φ(12) == 4
φ(13) == 12 is prime
φ(14) == 6
φ(15) == 8
φ(16) == 8
φ(17) == 16 is prime
φ(18) == 6
φ(19) == 18 is prime
φ(20) == 8
φ(21) == 12
φ(22) == 10
φ(23) == 22 is prime
φ(24) == 8
φ(25) == 20
```
```zkl
count:=0;
foreach n in ([1..10_000]){ // yes, this is sloooow
count+=isPrime(n);
if(n==100 or n==1000 or n==10_000)
println("Primes <= %,6d : %,5d".fmt(n,count));
}
```
{{out}}
```txt
Primes <= 100 : 25
Primes <= 1,000 : 168
Primes <= 10,000 : 1,229
```