Totient function
You are encouraged to solve this task according to the task description, using any language you may know.
The totient function is also known as:
- Euler's totient function
- Euler's phi totient function
- phi totient function
- Ξ¦ 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 β€ k β€ n for which the greatest common divisor gcd(n,k) is equal to 1
- counts numbers β€ 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
- Also see
11l
<lang 11l>F f(n)
R sum((1..n).filter(k -> gcd(@n, k) == 1).map(k -> 1))
F is_prime(n)
R f(n) == n - 1
L(n) 1..25
print(β f(#.) == #.β.format(n, f(n))ββ(I is_prime(n) {β, is primeβ} E ββ))
V count = 0 L(n) 1..10'000
count += is_prime(n)
I n C (100, 1000, 10'000)
print(βPrimes up to #.: #.β.format(n, count))</lang>
- Output:
f(1) == 1 f(2) == 1, is prime f(3) == 2, is prime f(4) == 2 f(5) == 4, is prime f(6) == 2 f(7) == 6, is prime f(8) == 4 f(9) == 6 f(10) == 4 f(11) == 10, is prime f(12) == 4 f(13) == 12, is prime f(14) == 6 f(15) == 8 f(16) == 8 f(17) == 16, is prime f(18) == 6 f(19) == 18, is prime f(20) == 8 f(21) == 12 f(22) == 10 f(23) == 22, is prime f(24) == 8 f(25) == 20 Primes up to 100: 25 Primes up to 1000: 168 Primes up to 10000: 1229
Ada
<lang Ada>with Ada.Text_IO; with Ada.Integer_Text_IO;
procedure Totient is
function Totient (N : in Integer) return Integer is
Tot : Integer := N;
I : Integer;
N2 : Integer := N;
begin
I := 2;
while I * I <= N2 loop
if N2 mod I = 0 then
while N2 mod I = 0 loop
N2 := N2 / I;
end loop;
Tot := Tot - Tot / I;
end if;
if I = 2 then
I := 1;
end if;
I := I + 2;
end loop;
if N2 > 1 then
Tot := Tot - Tot / N2;
end if;
return Tot; end Totient;
Count : Integer := 0; Tot : Integer; Placeholder : String := " n Phi Is_Prime"; Image_N : String renames Placeholder ( 1 .. 3); Image_Phi : String renames Placeholder ( 6 .. 8); Image_Prime : String renames Placeholder (11 .. 17); use Ada.Text_IO; use Ada.Integer_Text_IO;
begin
Put_Line (Placeholder);
for N in 1 .. 25 loop
Tot := Totient (N);
if N - 1 = Tot then
Count := Count + 1;
end if;
Put (Image_N, N);
Put (Image_Phi, Tot);
Image_Prime := (if N - 1 = Tot then " True" else " False");
Put_Line (Placeholder);
end loop;
New_Line;
Put_Line ("Number of primes up to " & Integer'(25)'Image &" =" & Count'Image);
for N in 26 .. 100_000 loop
Tot := Totient (N);
if Tot = N - 1 then
Count := Count + 1;
end if;
if N = 100 or N = 1_000 or N mod 10_000 = 0 then
Put_Line ("Number of primes up to " & N'Image & " =" & Count'Image);
end if;
end loop;
end Totient;</lang>
- Output:
n Phi Is_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
AWK
<lang 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)
} </lang>
- Output:
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 <lang C> /*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; } </lang>
Output :
n Ο 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#
<lang csharp>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;
}
}</lang>
- Output:
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
C++
<lang cpp>#include <cassert>
- include <iomanip>
- include <iostream>
- include <vector>
class totient_calculator { public:
explicit totient_calculator(int max) : totient_(max + 1) {
for (int i = 1; i <= max; ++i)
totient_[i] = i;
for (int i = 2; i <= max; ++i) {
if (totient_[i] < i)
continue;
for (int j = i; j <= max; j += i)
totient_[j] -= totient_[j] / i;
}
}
int totient(int n) const {
assert (n >= 1 && n < totient_.size());
return totient_[n];
}
bool is_prime(int n) const {
return totient(n) == n - 1;
}
private:
std::vector<int> totient_;
};
int count_primes(const totient_calculator& tc, int min, int max) {
int count = 0;
for (int i = min; i <= max; ++i) {
if (tc.is_prime(i))
++count;
}
return count;
}
int main() {
const int max = 10000000;
totient_calculator tc(max);
std::cout << " n totient prime?\n";
for (int i = 1; i <= 25; ++i) {
std::cout << std::setw(2) << i
<< std::setw(9) << tc.totient(i)
<< std::setw(8) << (tc.is_prime(i) ? "yes" : "no") << '\n';
}
for (int n = 100; n <= max; n *= 10) {
std::cout << "Count of primes up to " << n << ": "
<< count_primes(tc, 1, n) << '\n';
}
return 0;
}</lang>
- Output:
n totient prime? 1 1 no 2 1 yes 3 2 yes 4 2 no 5 4 yes 6 2 no 7 6 yes 8 4 no 9 6 no 10 4 no 11 10 yes 12 4 no 13 12 yes 14 6 no 15 8 no 16 8 no 17 16 yes 18 6 no 19 18 yes 20 8 no 21 12 no 22 10 no 23 22 yes 24 8 no 25 20 no Count of primes up to 100: 25 Count of primes up to 1000: 168 Count of primes up to 10000: 1229 Count of primes up to 100000: 9592 Count of primes up to 1000000: 78498 Count of primes up to 10000000: 664579
D
<lang d>import std.stdio;
int totient(int n) {
int tot = n;
for (int 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;
}
void main() {
writeln(" n Ο prime");
writeln("---------------");
int count = 0;
for (int n = 1; n <= 25; n++) {
int tot = totient(n);
if (n - 1 == tot) {
count++;
}
writefln("%2d %2d %s", n,tot, n - 1 == tot);
}
writeln;
writefln("Number of primes up to %6d = %4d", 25, count);
for (int n = 26; n <= 100_000; n++) {
int tot = totient(n);
if (n - 1 == tot) {
count++;
}
if (n == 100 || n == 1_000 || n % 10_000 == 0) {
writefln("Number of primes up to %6d = %4d", n, count);
}
}
}</lang>
- Output:
n Ο 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
Delphi
See Pascal.
Dyalect
<lang 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)")
}
}</lang>
- Output:
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
<lang factor>USING: combinators formatting io kernel math math.primes.factors math.ranges sequences ; IN: rosetta-code.totient-function
- Ξ¦ ( n -- m )
{
{ [ dup 1 < ] [ drop 0 ] }
{ [ dup 1 = ] [ drop 1 ] }
[
dup unique-factors
[ 1 [ 1 - * ] reduce ] [ product ] bi / *
]
} cond ;
- show-info ( n -- )
[ Ξ¦ ] [ swap 2dup - 1 = ] bi ", prime" "" ? "Ξ¦(%2d) = %2d%s\n" printf ;
- totient-demo ( -- )
25 [1,b] [ show-info ] each nl 0 100,000 [1,b] [
[ dup Ξ¦ - 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</lang>
- Output:
Ξ¦( 1) = 1 Ξ¦( 2) = 1, prime Ξ¦( 3) = 2, prime Ξ¦( 4) = 2 Ξ¦( 5) = 4, prime Ξ¦( 6) = 2 Ξ¦( 7) = 6, prime Ξ¦( 8) = 4 Ξ¦( 9) = 6 Ξ¦(10) = 4 Ξ¦(11) = 10, prime Ξ¦(12) = 4 Ξ¦(13) = 12, prime Ξ¦(14) = 6 Ξ¦(15) = 8 Ξ¦(16) = 8 Ξ¦(17) = 16, prime Ξ¦(18) = 6 Ξ¦(19) = 18, prime Ξ¦(20) = 8 Ξ¦(21) = 12 Ξ¦(22) = 10 Ξ¦(23) = 22, prime Ξ¦(24) = 8 Ξ¦(25) = 20 25 primes <= 100 168 primes <= 1000 1229 primes <= 10000 9592 primes <= 100000
FreeBASIC
<lang freebasic>
- define esPar(n) (((n) And 1) = 0)
- define esImpar(n) (esPar(n) = 0)
Function Totient(n As Integer) As Integer
'delta son nΓΊmeros 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 rΓ‘pido.
'i = 2
result = n
If (2*2 <= n) Then
If Not(esImpar(n)) Then
' eliminar nΓΊmeros con factor 2,4,8,16,...
While Not(esImpar(n))
n = n \ 2
Wend
'eliminar los mΓΊltiplos 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 </lang>
- Output:
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. <lang 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)
}
}
}</lang>
- Output:
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:
<lang 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)
}
}
}</lang>
The output is the same as before.
Haskell
<lang Haskell>{-# LANGUAGE BangPatterns #-}
import Control.Monad (when) import Data.Bool (bool)
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 = bool tot (tot - (tot `div` m)) (1 < m)
| m `mod` i == 0 = loop m_ tot_ i_
| otherwise = loop m tot i_
where
i_
| i == 2 = 3
| otherwise = 2 + i
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_ = succ i
tot = totient i_
isPrime = tot == pred i_
count_
| isPrime = succ count
| otherwise = count
when (25 >= i_) $
putStrLn $ show i_ ++ "\t" ++ show tot ++ "\t" ++ show isPrime
when
(i_ `elem`
25 :
[ 10 ^ k
| k <- [2 .. 6] ]) $
putStrLn $ "Number of primes up to " ++ show i_ ++ " = " ++ show count_
loop (i + 1) count_
loop 0 0</lang>
- Output:
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
<lang 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
</lang>
Java
Efficiently pre-compute all needed values of phi up to 10,000,000 in .344 sec. <lang java> public class TotientFunction {
public static void main(String[] args) {
computePhi();
System.out.println("Compute and display phi for the first 25 integers.");
System.out.printf("n Phi IsPrime%n");
for ( int n = 1 ; n <= 25 ; n++ ) {
System.out.printf("%2d %2d %b%n", n, phi[n], (phi[n] == n-1));
}
for ( int i = 2 ; i < 8 ; i++ ) {
int max = (int) Math.pow(10, i);
System.out.printf("The count of the primes up to %,10d = %d%n", max, countPrimes(1, max));
}
}
private static int countPrimes(int min, int max) {
int count = 0;
for ( int i = min ; i <= max ; i++ ) {
if ( phi[i] == i-1 ) {
count++;
}
}
return count;
}
private static final int max = 10000000; private static final int[] phi = new int[max+1];
private static final void computePhi() {
for ( int i = 1 ; i <= max ; i++ ) {
phi[i] = i;
}
for ( int i = 2 ; i <= max ; i++ ) {
if (phi[i] < i) continue;
for ( int j = i ; j <= max ; j += i ) {
phi[j] -= phi[j] / i;
}
}
}
} </lang>
- Output:
Compute and display phi for the first 25 integers. 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 The count of the primes up to 100 = 25 The count of the primes up to 1,000 = 168 The count of the primes up to 10,000 = 1229 The count of the primes up to 100,000 = 9592 The count of the primes up to 1,000,000 = 78498 The count of the primes up to 10,000,000 = 664579
Julia
<lang julia>Ο(n) = sum(1 for k in 1:n if gcd(n, k) == 1)
is_prime(n) = Ο(n) == n - 1
function runphitests()
for n in 1:25
println(" Ο($n) == $(Ο(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()
</lang>
- Output:
Ο(1) == 1 Ο(2) == 1, is prime Ο(3) == 2, is prime Ο(4) == 2 Ο(5) == 4, is prime Ο(6) == 2 Ο(7) == 6, is prime Ο(8) == 4 Ο(9) == 6 Ο(10) == 4 Ο(11) == 10, is prime Ο(12) == 4 Ο(13) == 12, is prime Ο(14) == 6 Ο(15) == 8 Ο(16) == 8 Ο(17) == 16, is prime Ο(18) == 6 Ο(19) == 18, is prime Ο(20) == 8 Ο(21) == 12 Ο(22) == 10 Ο(23) == 22, is prime Ο(24) == 8 Ο(25) == 20 Primes up to 100: 25 Primes up to 1000: 168 Primes up to 10000: 1229 Primes up to 100000: 9592
Kotlin
<lang 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)
}
}
}</lang>
- Output:
Same as Go example.
Lua
Averages about 7 seconds under LuaJIT <lang 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</lang>
- Output:
n Ο 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
Mathematica
<lang Mathematica>Do[
tmp = EulerPhi[i];
If[i - 1 == tmp,
Print["\[CurlyPhi](", i, ")=", tmp, ", is prime"]
,
Print["\[CurlyPhi](", i, ")=", tmp]
]
,
{i, 25}
]
Count[EulerPhi[Range[100]] - Range[100], -1] Count[EulerPhi[Range[1000]] - Range[1000], -1] Count[EulerPhi[Range[10000]] - Range[10000], -1] Count[EulerPhi[Range[100000]] - Range[100000], -1] (*Alternative much faster way of findings the number primes up to a number*) (*PrimePi[100]*) (*PrimePi[1000]*) (*PrimePi[10000]*) (*PrimePi[100000]*)</lang>
- Output:
Ο(1)=1 Ο(2)=1, is prime Ο(3)=2, is prime Ο(4)=2 Ο(5)=4, is prime Ο(6)=2 Ο(7)=6, is prime Ο(8)=4 Ο(9)=6 Ο(10)=4 Ο(11)=10, is prime Ο(12)=4 Ο(13)=12, is prime Ο(14)=6 Ο(15)=8 Ο(16)=8 Ο(17)=16, is prime Ο(18)=6 Ο(19)=18, is prime Ο(20)=8 Ο(21)=12 Ο(22)=10 Ο(23)=22, is prime Ο(24)=8 Ο(25)=20 25 168 1229 9592
Nim
<lang 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 Ο 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}"</lang>
- Output:
n Ο 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 <lang 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.</lang>
- Output
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. <lang 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;</lang>
- Output
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 π
<lang perl>use utf8; binmode STDOUT, ":utf8";
sub gcd {
my ($u, $v) = @_;
while ($v) {
($u, $v) = ($v, $u % $v);
}
return abs($u);
}
push @π, 0; for $t (1..10000) {
push @π, scalar grep { 1 == gcd($_,$t) } 1..$t;
}
printf "π(%2d) = %3d%s\n", $_, $π[$_], $_ - $π[$_] - 1 ? : ' Prime' for 1 .. 25; print "\n";
for $limit (100, 1000, 10000) {
printf "Count of primes <= $limit: %d\n", scalar grep {$_ == $π[$_] + 1} 0..$limit;
} </lang>
- Output:
π( 1) = 1 π( 2) = 1 Prime π( 3) = 2 Prime π( 4) = 2 π( 5) = 4 Prime π( 6) = 2 π( 7) = 6 Prime π( 8) = 4 π( 9) = 6 π(10) = 4 π(11) = 10 Prime π(12) = 4 π(13) = 12 Prime π(14) = 6 π(15) = 8 π(16) = 8 π(17) = 16 Prime π(18) = 6 π(19) = 18 Prime π(20) = 8 π(21) = 12 π(22) = 10 π(23) = 22 Prime π(24) = 8 π(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.
<lang perl>use utf8; binmode STDOUT, ":utf8";
use ntheory qw(euler_phi);
my @π = euler_phi(0,10000); # Returns list of all values in range
printf "π(%2d) = %3d%s\n", $_, $π[$_], $_ - $π[$_] - 1 ? : ' Prime' for 1 .. 25; print "\n";
for $limit (100, 1000, 10000) {
printf "Count of primes <= $limit: %d\n", scalar grep {$_ == $π[$_] + 1} 0..$limit;
}</lang>
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
- Output:
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
<lang 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)) )</lang>
- Output:
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
<lang python>from math import gcd
def Ο(n):
return sum(1 for k in range(1, n + 1) if gcd(n, k) == 1)
if __name__ == '__main__':
def is_prime(n):
return Ο(n) == n - 1
for n in range(1, 26):
print(f" Ο({n}) == {Ο(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}")</lang>
- Output:
Ο(1) == 1 Ο(2) == 1, is prime Ο(3) == 2, is prime Ο(4) == 2 Ο(5) == 4, is prime Ο(6) == 2 Ο(7) == 6, is prime Ο(8) == 4 Ο(9) == 6 Ο(10) == 4 Ο(11) == 10, is prime Ο(12) == 4 Ο(13) == 12, is prime Ο(14) == 6 Ο(15) == 8 Ο(16) == 8 Ο(17) == 16, is prime Ο(18) == 6 Ο(19) == 18, is prime Ο(20) == 8 Ο(21) == 12 Ο(22) == 10 Ο(23) == 22, is prime Ο(24) == 8 Ο(25) == 20 Primes up to 100: 25 Primes up to 1000: 168 Primes up to 10000: 1229
Quackery
<lang Quackery > [ [ dup while
tuck mod again ] drop abs ] is gcd ( n n --> n )
[ 0 swap dup times
[ i over gcd
1 = rot + swap ]
drop ] is totient ( n --> n )
[ 0 temp put
times
[ i dup 1+ totient
= temp tally ]
temp take ] is primecount ( n --> n )
25 times
[ say "The totient of "
i^ 1+ dup echo
say " is "
dup totient dup echo
say ", so it is "
1+ != if say "not "
say "prime." cr ]
cr
' [ 100 1000 10000 100000 ]
witheach
[ say "There are "
dup primecount echo
say " primes up to " echo
say "." cr ]</lang>
((Out}}
The totient of 1 is 1, so it is not prime. The totient of 2 is 1, so it is prime. The totient of 3 is 2, so it is prime. The totient of 4 is 2, so it is not prime. The totient of 5 is 4, so it is prime. The totient of 6 is 2, so it is not prime. The totient of 7 is 6, so it is prime. The totient of 8 is 4, so it is not prime. The totient of 9 is 6, so it is not prime. The totient of 10 is 4, so it is not prime. The totient of 11 is 10, so it is prime. The totient of 12 is 4, so it is not prime. The totient of 13 is 12, so it is prime. The totient of 14 is 6, so it is not prime. The totient of 15 is 8, so it is not prime. The totient of 16 is 8, so it is not prime. The totient of 17 is 16, so it is prime. The totient of 18 is 6, so it is not prime. The totient of 19 is 18, so it is prime. The totient of 20 is 8, so it is not prime. The totient of 21 is 12, so it is not prime. The totient of 22 is 10, so it is not prime. The totient of 23 is 22, so it is prime. The totient of 24 is 8, so it is not prime. The totient of 25 is 20, so it is not prime. There are 25 primes up to 100. There are 168 primes up to 1000. There are 1229 primes up to 10000. There are 9592 primes up to 100000.
Racket
<lang racket>#lang racket
(require math/number-theory)
(define (prime*? n) (= (totient n) (sub1 n)))
(for ([n (in-range 1 26)])
(printf "Ο(~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)</lang>
- Output:
Ο(1) = 1 Ο(2) = 1 is prime (confirmed) Ο(3) = 2 is prime (confirmed) Ο(4) = 2 Ο(5) = 4 is prime (confirmed) Ο(6) = 2 Ο(7) = 6 is prime (confirmed) Ο(8) = 4 Ο(9) = 6 Ο(10) = 4 Ο(11) = 10 is prime (confirmed) Ο(12) = 4 Ο(13) = 12 is prime (confirmed) Ο(14) = 6 Ο(15) = 8 Ο(16) = 8 Ο(17) = 16 is prime (confirmed) Ο(18) = 6 Ο(19) = 18 is prime (confirmed) Ο(20) = 8 Ο(21) = 12 Ο(22) = 10 Ο(23) = 22 is prime (confirmed) Ο(24) = 8 Ο(25) = 20 Primes up to 100: 25 Primes up to 1000: 168 Primes up to 10000: 1229
Raku
(formerly Perl 6)
This is an incredibly inefficient way of finding prime numbers.
<lang perl6>use Prime::Factor;
my \π = 0, |(1..*).hyper.map: -> \t { t * [*] t.&prime-factors.squish.map: { 1 - 1/$_ } }
printf "π(%2d) = %3d %s\n", $_, π[$_], $_ - π[$_] - 1 ?? !! 'Prime' for 1 .. 25;
(1e2, 1e3, 1e4, 1e5).map: -> $limit {
say "\nCount of primes <= $limit: " ~ +(^$limit).grep: {$_ == π[$_] + 1}
}</lang>
- Output:
π( 1) = 1 π( 2) = 1 Prime π( 3) = 2 Prime π( 4) = 2 π( 5) = 4 Prime π( 6) = 2 π( 7) = 6 Prime π( 8) = 4 π( 9) = 6 π(10) = 4 π(11) = 10 Prime π(12) = 4 π(13) = 12 Prime π(14) = 6 π(15) = 8 π(16) = 8 π(17) = 16 Prime π(18) = 6 π(19) = 18 Prime π(20) = 8 π(21) = 12 π(22) = 10 π(23) = 22 Prime π(24) = 8 π(25) = 20 Count of primes <= 100: 25 Count of primes <= 1000: 168 Count of primes <= 10000: 1229 Count of primes <= 100000: 9592
REXX
idiomatic
<lang 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) " βββΊ " right(T, w) ' ' prime
end /*j*/
say say right(primes, w) ' primes detected for numbers up to and including ' N exit 0 /*stick a fork in it, we're all done. */ /*ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ*/ gcd: parse arg x,y; do until y==0; parse value x//y y with y x
end /*until*/; return x
/*ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ*/ phi: procedure; parse arg z; if z==1 then return 1
#= 1
do m=2 for z-2; if gcd(m, z)==1 then #= # + 1
end /*m*/; return #</lang>
- output when using the default input of: 25
totient number for 1 βββΊ 1 totient number for 2 βββΊ 1 (prime) totient number for 3 βββΊ 2 (prime) totient number for 4 βββΊ 2 totient number for 5 βββΊ 4 (prime) totient number for 6 βββΊ 2 totient number for 7 βββΊ 6 (prime) totient number for 8 βββΊ 4 totient number for 9 βββΊ 6 totient number for 10 βββΊ 4 totient number for 11 βββΊ 10 (prime) totient number for 12 βββΊ 4 totient number for 13 βββΊ 12 (prime) totient number for 14 βββΊ 6 totient number for 15 βββΊ 8 totient number for 16 βββΊ 8 totient number for 17 βββΊ 16 (prime) totient number for 18 βββΊ 6 totient number for 19 βββΊ 18 (prime) totient number for 20 βββΊ 8 totient number for 21 βββΊ 12 totient number for 22 βββΊ 10 totient number for 23 βββΊ 22 (prime) totient number for 24 βββΊ 8 totient number for 25 βββΊ 20 9 primes detected for numbers up to and including 25
- output when using the input of: -100
25 primes detected for numbers up to and including 100
- output when using the input of: -1000
168 primes detected for numbers up to and including 1000
- output when using the input of: -10000
1229 primes detected for numbers up to and including 10000
- output when using the input of: -1000000
9592 primes detected for numbers up to and including 100000
optimized
This REXX version moved the gcd function in-line.
It is about 7% faster than the 1st version.
<lang 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) " βββΊ " right(T, w) ' ' prime
end /*j*/
say say right(primes, w) ' primes detected for numbers up to and including ' N exit 0 /*stick a fork in it, we're all done. */ /*ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ*/ phi: procedure; parse arg z; if z==1 then return 1
#= 1
do m=2 for z-2; parse value m z with x y
do until y==0; parse value x//y y with y x
end /*until*/
if x==1 then #= # + 1
end /*m*/
return #</lang>
- output is identical to the 1st REXX version.
Ruby
<lang ruby> require "prime"
def π(n)
n.prime_division.inject(1) {|res, (pr, exp)| res *= (pr-1) * pr**(exp-1) }
end
1.upto 25 do |n|
tot = π(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-π(n) == 1} }"
end </lang>
- Output:
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
<lang 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()
}</lang>
Output is:
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: <lang 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</lang>
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: <lang 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)
}</lang>
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: <lang 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</lang>
To generate the output up to 100000, Longs are necessary.
- Output:
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
Shale
#!/usr/local/bin/shale
primes library
init dup var {
pl sieve type primes::()
100000 0 pl generate primes::()
} =
go dup var {
i var
c0 var
c1 var
c2 var
c3 var
p var
" N Phi Is prime" println
i 1 =
{ i 25 <= } {
i pl isprime primes::() { "True" } { "False" } if i pl phi primes::() i "%2d %3d %s\n" printf
i++
} while
c0 0 =
c1 0 =
c2 0 =
c3 0 =
i 0 =
{ i count pl:: < } {
p i pl get primes::() =
p 100 < { c0++ } ifthen
p 1000 < { c1++ } ifthen
p 10000 < { c2 ++ } ifthen
p 100000 < { c3 ++ } ifthen
i++
} while
"" println
" Limit Count" println
c0 " 100 %5d\n" printf
c1 " 1,000 %5d\n" printf
c2 " 10,000 %5d\n" printf
c3 "100,000 %5d\n" printf
} =
init()
go()
- Output:
N Phi Is 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
Limit Count
100 25
1,000 168
10,000 1229
100,000 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). <lang ruby>func π(n) {
n.factor_exp.prod {|p|
(p[0]-1) * p[0]**(p[1]-1)
}
}</lang>
<lang ruby>for n in (1..25) {
var totient = π(n)
printf("π(%2s) = %3s%s\n", n, totient, totient==(n-1) ? ' - prime' : )
}</lang>
- Output:
π( 1) = 1 π( 2) = 1 - prime π( 3) = 2 - prime π( 4) = 2 π( 5) = 4 - prime π( 6) = 2 π( 7) = 6 - prime π( 8) = 4 π( 9) = 6 π(10) = 4 π(11) = 10 - prime π(12) = 4 π(13) = 12 - prime π(14) = 6 π(15) = 8 π(16) = 8 π(17) = 16 - prime π(18) = 6 π(19) = 18 - prime π(20) = 8 π(21) = 12 π(22) = 10 π(23) = 22 - prime π(24) = 8 π(25) = 20
<lang ruby>[100, 1_000, 10_000, 100_000].each {|limit|
var pi = (1..limit -> count_by {|n| π(n) == (n-1) })
say "Number of primes <= #{limit}: #{pi}"
}</lang>
- Output:
Number of primes <= 100: 25 Number of primes <= 1000: 168 Number of primes <= 10000: 1229 Number of primes <= 100000: 9592
Tiny BASIC
1 indicates prime, 0 indicates not prime. <lang tinybasic> REM PRINT THE DATA FOR N=1 TO 25
LET N = 0
10 LET N = N + 1
IF N = 26 THEN GOTO 100
GOSUB 1000
IF T = N - 1 THEN LET P = 1
IF T <> N - 1 THEN LET P = 0
PRINT N," ", T," ",P
GOTO 10
100 REM COUNT PRIMES BELOW 10000
LET C = 0
LET N = 2
110 GOSUB 1000
IF T = N - 1 THEN LET C = C + 1
IF N = 100 THEN PRINT "There are ", C, " primes below 100."
IF N = 1000 THEN PRINT "There are ", C, " primes below 1000."
IF N = 10000 THEN PRINT "There are ", C, " primes below 10000."
LET N = N + 1
IF N < 10001 THEN GOTO 110
END
1000 REM TOTIENT FUNCTION OF INTEGER N
LET M = 1
LET T = 0
1010 IF M > N THEN RETURN
LET A = N
LET B = M REM NEED TO RENAME THESE SO THEY ARE PRESERVED
GOSUB 2000
IF G = 1 THEN LET T = T + 1
LET M = M + 1
GOTO 1010
2000 REM GCD OF INTEGERS A, B 2010 IF A>B THEN GOTO 2020
LET B = B - A
IF A=0 THEN LET G = B
IF A=0 THEN RETURN
2020 LET S = A
LET A = B
LET B = S
GOTO 2010</lang>
- Output:
1 1 02 1 1 3 2 1 4 2 0 5 4 1 6 2 0 7 6 1 8 4 0 9 6 0 10 4 0 11 10 1 12 4 0 13 12 1 14 6 0 15 8 0 16 8 0 17 16 1 18 6 0 19 18 1 20 8 0 21 12 0 22 10 0 23 22 1 24 8 0 25 20 0 There are 25 primes below 100. There are 168 primes below 1000.
There are 1229 primes below 10000.
VBA
<lang 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$(6 - Len(CStr(n)), " "); "="; Format(count, "@@@@@")
Case Else
End Select
Next n
End Sub</lang>
- Output:
n phi prime -------------- 1 1 False 2 1 True 3 2 True 4 2 False 5 4 True 6 2 False 7 6 True 8 4 False 9 6 False 10 4 False 11 10 True 12 4 False 13 12 True 14 6 False 15 8 False 16 8 False 17 16 True 18 6 False 19 18 True 20 8 False 21 12 False 22 10 False 23 22 True 24 8 False 25 20 False Number of primes up to 25 = 9 Number of primes up to 100 = 25 Number of primes up to 1000 = 168 Number of primes up to 10000 = 1229 Number of primes up to 100000 = 9592
Visual Basic .NET
<lang vbnet>Imports System.Linq.Enumerable
Module Module1
Sub Main()
For i = 1 To 25
Dim t = Totient(i)
Console.WriteLine("{0}{1}{2}{3}", i, vbTab, t, If(t = i - 1, vbTab & "prime", ""))
Next
Console.WriteLine()
Dim j = 100
While j <= 100000
Console.WriteLine($"{Range(1, j).Count(Function(x) Totient(x) + 1 = x):n0} primes below {j:n0}")
j *= 10
End While
End Sub
Function Totient(n As Integer) As Integer
If n < 3 Then
Return 1
End If
If n = 3 Then
Return 2
End If
Dim tot = n
If (n And 1) = 0 Then
tot >>= 1
Do
n >>= 1
Loop While (n And 1) = 0
End If
Dim i = 3
While i * i <= n
If n Mod i = 0 Then
tot -= tot \ i
Do
n \= i
Loop While (n Mod i) = 0
End If
i += 2
End While
If n > 1 Then
tot -= tot \ n
End If
Return tot End Function
End Module</lang>
- Output:
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
Wren
<lang ecmascript>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
}
System.print(" n phi prime") System.print("---------------") var count = 0 for (n in 1..25) {
var tot = totient.call(n)
var isPrime = (n - 1) == tot
if (isPrime) count = count + 1
System.print("%(Fmt.d(2, n)) %(Fmt.d(2, tot)) %(isPrime)")
} System.print("\nNumber of primes up to 25 = %(count)") for (n in 26..100000) {
var tot = totient.call(n)
if (tot == n - 1) count = count + 1
if (n == 100 || n == 1000 || n%10000 == 0) {
System.print("\nNumber of primes up to %(Fmt.d(-6, n)) = %(count)")
}
}</lang>
- Output:
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
zkl
<lang zkl>fcn totient(n){ [1..n].reduce('wrap(p,k){ p + (n.gcd(k)==1) }) } fcn isPrime(n){ totient(n)==(n - 1) }</lang> <lang zkl>foreach n in ([1..25]){
println("\u03c6(%2d) ==%3d %s"
.fmt(n,totient(n),isPrime(n) and "is prime" or ""));
}</lang>
- Output:
Ο( 1) == 1 Ο( 2) == 1 is prime Ο( 3) == 2 is prime Ο( 4) == 2 Ο( 5) == 4 is prime Ο( 6) == 2 Ο( 7) == 6 is prime Ο( 8) == 4 Ο( 9) == 6 Ο(10) == 4 Ο(11) == 10 is prime Ο(12) == 4 Ο(13) == 12 is prime Ο(14) == 6 Ο(15) == 8 Ο(16) == 8 Ο(17) == 16 is prime Ο(18) == 6 Ο(19) == 18 is prime Ο(20) == 8 Ο(21) == 12 Ο(22) == 10 Ο(23) == 22 is prime Ο(24) == 8 Ο(25) == 20
<lang 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));
}</lang>
- Output:
Primes <= 100 : 25 Primes <= 1,000 : 168 Primes <= 10,000 : 1,229