Euler’s Totient function Φ(n) for an input n is the count of numbers in {1, 2, 3, …, n} that are relatively prime to n, i.e., the numbers whose GCD (Greatest Common Divisor) with n is 1.
For example, Φ(4) = 2, Φ(3) = 2 and Φ(5) = 4. There are 2 numbers smaller or equal to 4 that are relatively prime to 4, 2 numbers smaller or equal to 3 that are relatively prime to 3. And 4 numbers smaller than or equal to 5 that are relatively prime to 5.
We have discussed different methods for the computation of Φ(n) in the previous post.
How to compute Φ for all numbers smaller than or equal to n?
Example:
Input: n = 5
Output: Totient of 1 is 1
Totient of 2 is 1
Totient of 3 is 2
Totient of 4 is 2
Totient of 5 is 4We strongly recommend you to minimize your browser and try this yourself first.
A Simple Solution is to call Φ(i) for i = 1 to n.
An Efficient Solution is to use an idea similar to the Sieve of Eratosthenes to precompute all values. The method is based on below product formula.
The formula basically says that the value of Φ(n) is equal to n multiplied by-product of (1 – 1/p) for all prime factors p of n. For example value of Φ(6) = 6 * (1-1/2) * (1 – 1/3) = 2.
Below is the complete algorithm:
1) Create an array phi[1..n] to store Φ values of all numbers
from 1 to n.
2) Initialize all values such that phi[i] stores i. This
initialization serves two purposes.
a) To check if phi[i] is already evaluated or not. Note that
the maximum possible phi value of a number i is i-1.
b) To initialize phi[i] as i is multiple in the above product
formula.
3) Run a loop for p = 2 to n
a) If phi[p] is p, means p is not evaluated yet and p is a
prime number (similar to Sieve), otherwise phi[p] must
have been updated in step 3.b
b) Traverse through all multiples of p and update all
multiples of p by multiplying with (1-1/p).
4) Run a loop from i = 1 to n and print all Ph[i] values.Below is the implementation of above algorithm.
C++
#include<iostream>
using namespace std;
void computeTotient(int n)
{
long long phi[n+1];
for (int i=1; i<=n; i++)
phi[i] = i;
for (int p=2; p<=n; p++)
{
if (phi[p] == p)
{
phi[p] = p-1;
for (int i = 2*p; i<=n; i += p)
{
phi[i] = (phi[i]/p) * (p-1);
}
}
}
for (int i=1; i<=n; i++)
cout << "Totient of " << i << " is "
<< phi[i] << endl;
}
int main()
{
int n = 12;
computeTotient(n);
return 0;
}
|
Java
import java.util.*;
class GFG {
static void computeTotient(int n) {
long phi[] = new long[n + 1];
for (int i = 1; i <= n; i++)
phi[i] = i;
for (int p = 2; p <= n; p++) {
if (phi[p] == p) {
phi[p] = p - 1;
for (int i = 2 * p; i <= n; i += p) {
phi[i] = (phi[i] / p) * (p - 1);
}
}
}
for (int i = 1; i <= n; i++)
System.out.println("Totient of " + i +
" is " + phi[i]);
}
public static void main(String[] args) {
int n = 12;
computeTotient(n);
}
}
|
Python3
def computeTotient(n):
phi=[]
for i in range(n + 2):
phi.append(0)
for i in range(1, n+1):
phi[i] = i
for p in range(2,n+1):
if (phi[p] == p):
phi[p] = p-1
for i in range(2*p,n+1,p):
phi[i] = (phi[i]//p) * (p-1)
for i in range(1,n+1):
print("Totient of ", i ," is ",
phi[i])
n = 12
computeTotient(n)
|
C#
using System;
class GFG
{
static void computeTotient(int n)
{
long []phi = new long[n + 1];
for (int i = 1; i <= n; i++)
phi[i] = i;
for (int p = 2; p <= n; p++)
{
if (phi[p] == p)
{
phi[p] = p - 1;
for (int i = 2 * p; i <= n; i += p)
{
phi[i] = (phi[i] / p) * (p - 1);
}
}
}
for (int i = 1; i <= n; i++)
Console.WriteLine("Totient of " + i +" is " + phi[i]);
}
public static void Main()
{
int n = 12;
computeTotient(n);
}
}
|
PHP
<?php
function computeTotient($n)
{
for($i = 1; $i <= $n; $i++)
$phi[$i] = $i;
for($p = 2; $p <= $n; $p++)
{
if ($phi[$p] == $p)
{
$phi[$p] = $p - 1;
for($i = 2 * $p; $i <= $n; $i += $p)
{
$phi[$i] = ($phi[$i] / $p) * ($p - 1);
}
}
}
for($i = 1; $i <= $n; $i++)
echo "Totient of " , $i , " is ",
$phi[$i] ,"\n";
}
$n = 12;
computeTotient($n);
?>
|
Javascript
<script>
function computeTotient(n)
{
let phi = new Array(n + 1);
for (let i = 1; i <= n; i++)
phi[i] = i;
for (let p = 2; p <= n; p++)
{
if (phi[p] == p)
{
phi[p] = p - 1;
for (let i = 2 * p; i <= n; i += p)
{
phi[i] = parseInt(phi[i] / p, 10) * (p - 1);
}
}
}
for (let i = 1; i <= n; i++)
document.write("Totient of " + i +" is " + phi[i] + "</br>");
}
let n = 12;
computeTotient(n);
</script>
|
OutputTotient of 1 is 1
Totient of 2 is 1
Totient of 3 is 2
Totient of 4 is 2
Totient of 5 is 4
Totient of 6 is 2
Totient of 7 is 6
Totient of 8 is 4
Totient of 9 is 6
Totient of 10 is 4
Totient of 11 is 10
Totient of 12 is 4
Time Complexity: O(n2)
Auxiliary Space: O(n)
The same solution can be used when we have a large number of queries for computing the totient function.
Another way to compute Euler’s totient function can also be done by using the below formula:
Euler’s totient function
Let us see an example to understand the above function, basically, it does the same job but in a different way:
For example, ϕ(12) = { (2^(2-1)) x (2-1) } x { (3^(1-1)) x (3-1) } =4
Note that ϕ(n) = n−1 if n is prime.
Below is the implementation of the above formula:
C++
#include <bits/stdc++.h>
using namespace std;
#define ll long long
ll Euler_totient_function(ll n)
{
ll result = 1;
for (ll i = 2; i * i <= n; i++) {
ll c = 0;
if (n % i == 0) {
while (n % i == 0) {
c++;
n /= i;
}
}
if (c > 0) {
ll power = (ll)pow(i, c - 1);
ll sm = (ll)pow(i, c - 1) * (i - 1);
result *= sm;
}
}
if (n > 1) {
result *= (n - 1);
}
return result;
}
int main()
{
for (ll i = 1; i < 13; i++) {
cout << "Euler_totient_function(" << i << "): ";
cout << Euler_totient_function(i) << endl;
}
}
#praveeny182
|
Java
import java.io.*;
class GFG{
static long Euler_totient_function(long n)
{
long result = 1;
for(long i = 2; i * i <= n; i++)
{
long c = 0;
if (n % i == 0)
{
while (n % i == 0)
{
c++;
n /= i;
}
}
if (c > 0)
{
long power = (long)Math.pow(i, c - 1);
long sm = (long)Math.pow(i, c - 1) * (i - 1);
result *= sm;
}
}
if (n > 1)
{
result *= (n - 1);
}
return result;
}
public static void main(String[] args)
{
for(long i = 1; i < 13; i++)
{
System.out.print("Euler_totient_function(" +
i + "): ");
System.out.println(Euler_totient_function(i));
}
}
}
|
Python3
import math
def Euler_totient_function(n):
result = 1
for i in range(2,n+1):
c = 0
if n % i == 0:
while (n % i == 0):
c+=1
n //= i
if (c > 0):
power = math.pow(i,c-1)
m = math.pow(i,c-1)*(i-1)
result*=m
if (n > 1):
result *= (n - 1)
return int(result)
for i in range(1,13):
print("Euler_totient_function(" , i , "): ",end="")
print(Euler_totient_function(i))
|
C#
using System;
class GFG {
static long Euler_totient_function(long n)
{
long result = 1;
for (long i = 2; i * i <= n; i++) {
long c = 0;
if (n % i == 0) {
while (n % i == 0) {
c++;
n /= i;
}
}
if (c > 0) {
long sm
= (long)Math.Pow(i, c - 1) * (i - 1);
result *= sm;
}
}
if (n > 1) {
result *= (n - 1);
}
return result;
}
public static void Main()
{
for (long i = 1; i < 13; i++) {
Console.Write("Euler_totient_function(" + i
+ "): ");
Console.WriteLine(Euler_totient_function(i));
}
}
}
|
Javascript
<script>
function Euler_totient_function(n)
{
let result = 1;
for (let i = 2; i * i <= n; i++) {
let c = 0;
if (n % i == 0) {
while (n % i == 0) {
c++;
n = parseInt(n / i);
}
}
if (c > 0) {
let power = Math.pow(i, c - 1);
let sm = Math.pow(i, c - 1) * (i - 1);
result *= sm;
}
}
if (n > 1) {
result *= (n - 1);
}
return result;
}
for (let i = 1; i < 13; i++) {
document.write("Euler_totient_function(" + i + "): ");
document.write(Euler_totient_function(i) + "<br>");
}
</script>
|
OutputEuler_totient_function(1): 1
Euler_totient_function(2): 1
Euler_totient_function(3): 2
Euler_totient_function(4): 2
Euler_totient_function(5): 4
Euler_totient_function(6): 2
Euler_totient_function(7): 6
Euler_totient_function(8): 4
Euler_totient_function(9): 6
Euler_totient_function(10): 4
Euler_totient_function(11): 10
Euler_totient_function(12): 4
Time Complexity: O(sqrt(n)*log(n))
Auxiliary Space: O(1)
This article is contributed by Ekta Goel. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above