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# Optimized Euler Totient Function for Multiple Evaluations

• Difficulty Level : Medium
• Last Updated : 17 Sep, 2022

Euler Totient Function (ETF) Φ(n) for an input n is 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.
Examples:

```Φ(5) = 4
gcd(1, 5) is 1, gcd(2, 5) is 1,
gcd(3, 5) is 1 and gcd(4, 5) is 1

Φ(6) = 2
gcd(1, 6) is 1 and gcd(5, 6) is 1,```

Recommended Practice

We have discussed different methods to compute Euler Totient function that work well for single input. In problems where we have to call Euler’s Totient Function many times like 10^5 times, simple solution will result in TLE(Time limit Exceeded). The idea is to use Sieve of Eratosthenes.
Find all prime numbers upto maximum limit say 10^5 using Sieve of Eratosthenes
To compute Φ(n), we do following.

1. Initialize result as n.
2. Iterate through all primes smaller than or equal to square root of n (This is where it is different from simple methods. Instead of iterating through all numbers less than or equal to square root, we iterate through only primes). Let the current prime number be p. We check if p divides n, if yes, we remove all occurrences of p from n by repeatedly dividing it with n. We also reduce our result by n/p (these many numbers will not have GCD as 1 with n).
3. Finally we return result.

## C++

 `// C++ program to efficiently compute values``// of euler totient function for multiple inputs.``#include ``using` `namespace` `std;` `#define ll long long``const` `int` `MAX = 100001;` `// Stores prime numbers upto MAX - 1 values``vector p;` `// Finds prime numbers upto MAX-1 and``// stores them in vector p``void` `sieve()``{``    ``ll isPrime[MAX+1];` `    ``for` `(ll i = 2; i<= MAX; i++)``    ``{``        ``// if prime[i] is not marked before``        ``if` `(isPrime[i] == 0)``        ``{``            ``// fill vector for every newly``            ``// encountered prime``            ``p.push_back(i);` `            ``// run this loop till square root of MAX,``            ``// mark the index i * j as not prime``            ``for` `(ll j = 2; i * j<= MAX; j++)``                ``isPrime[i * j]= 1;``        ``}``    ``}``}` `// function to find totient of n``ll phi(ll n)``{``    ``ll res = n;` `    ``// this loop runs sqrt(n / ln(n)) times``    ``for` `(ll i=0; p[i]*p[i] <= n; i++)``    ``{``        ``if` `(n % p[i]== 0)``        ``{``            ``// subtract multiples of p[i] from r``            ``res -= (res / p[i]);` `            ``// Remove all occurrences of p[i] in n``            ``while` `(n % p[i]== 0)``               ``n /= p[i];``        ``}``    ``}` `    ``// when n has prime factor greater``    ``// than sqrt(n)``    ``if` `(n > 1)``       ``res -= (res / n);` `    ``return` `res;``}` `// Driver code``int` `main()``{``    ``// preprocess all prime numbers upto 10 ^ 5``    ``sieve();``    ``cout << phi(11) << ``"\n"``;``    ``cout << phi(21) << ``"\n"``;``    ``cout << phi(31) << ``"\n"``;``    ``cout << phi(41) << ``"\n"``;``    ``cout << phi(51) << ``"\n"``;``    ``cout << phi(61) << ``"\n"``;``    ``cout << phi(91) << ``"\n"``;``    ``cout << phi(101) << ``"\n"``;``    ``return` `0;``}`

## Java

 `// Java program to efficiently compute values``// of euler totient function for multiple inputs.``import` `java.util.*;` `class` `GFG{``static` `int` `MAX = ``100001``;` `// Stores prime numbers upto MAX - 1 values``static` `ArrayList p = ``new` `ArrayList();` `// Finds prime numbers upto MAX-1 and``// stores them in vector p``static` `void` `sieve()``{``    ``int``[] isPrime=``new` `int``[MAX+``1``];` `    ``for` `(``int` `i = ``2``; i<= MAX; i++)``    ``{``        ``// if prime[i] is not marked before``        ``if` `(isPrime[i] == ``0``)``        ``{``            ``// fill vector for every newly``            ``// encountered prime``            ``p.add(i);` `            ``// run this loop till square root of MAX,``            ``// mark the index i * j as not prime``            ``for` `(``int` `j = ``2``; i * j<= MAX; j++)``                ``isPrime[i * j]= ``1``;``        ``}``    ``}``}` `// function to find totient of n``static` `int` `phi(``int` `n)``{``    ``int` `res = n;` `    ``// this loop runs sqrt(n / ln(n)) times``    ``for` `(``int` `i=``0``; p.get(i)*p.get(i) <= n; i++)``    ``{``        ``if` `(n % p.get(i)== ``0``)``        ``{``            ``// subtract multiples of p[i] from r``            ``res -= (res / p.get(i));` `            ``// Remove all occurrences of p[i] in n``            ``while` `(n % p.get(i)== ``0``)``            ``n /= p.get(i);``        ``}``    ``}` `    ``// when n has prime factor greater``    ``// than sqrt(n)``    ``if` `(n > ``1``)``    ``res -= (res / n);` `    ``return` `res;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``// preprocess all prime numbers upto 10 ^ 5``    ``sieve();``    ``System.out.println(phi(``11``));``    ``System.out.println(phi(``21``));``    ``System.out.println(phi(``31``));``    ``System.out.println(phi(``41``));``    ``System.out.println(phi(``51``));``    ``System.out.println(phi(``61``));``    ``System.out.println(phi(``91``));``    ``System.out.println(phi(``101``));`` ` `}``}``// this code is contributed by mits`

## Python3

 `# Python3 program to efficiently compute values``# of euler totient function for multiple inputs.` `MAX` `=` `100001``;` `# Stores prime numbers upto MAX - 1 values``p ``=` `[];` `# Finds prime numbers upto MAX-1 and``# stores them in vector p``def` `sieve():` `    ``isPrime ``=` `[``0``] ``*` `(``MAX` `+` `1``);` `    ``for` `i ``in` `range``(``2``, ``MAX` `+` `1``):``        ` `        ``# if prime[i] is not marked before``        ``if` `(isPrime[i] ``=``=` `0``):``            ` `            ``# fill vector for every newly``            ``# encountered prime``            ``p.append(i);` `            ``# run this loop till square root of MAX,``            ``# mark the index i * j as not prime``            ``j ``=` `2``;``            ``while` `(i ``*` `j <``=` `MAX``):``                ``isPrime[i ``*` `j]``=` `1``;``                ``j ``+``=` `1``;` `# function to find totient of n``def` `phi(n):` `    ``res ``=` `n;` `    ``# this loop runs sqrt(n / ln(n)) times``    ``i ``=` `0``;``    ``while` `(p[i] ``*` `p[i] <``=` `n):``        ``if` `(n ``%` `p[i]``=``=` `0``):``            ` `            ``# subtract multiples of p[i] from r``            ``res ``-``=` `int``(res ``/` `p[i]);` `            ``# Remove all occurrences of p[i] in n``            ``while` `(n ``%` `p[i]``=``=` `0``):``                ``n ``=` `int``(n ``/` `p[i]);``        ``i ``+``=` `1``;` `    ``# when n has prime factor greater``    ``# than sqrt(n)``    ``if` `(n > ``1``):``        ``res ``-``=` `int``(res ``/` `n);` `    ``return` `res;` `# Driver code` `# preprocess all prime numbers upto 10 ^ 5``sieve();``print``(phi(``11``));``print``(phi(``21``));``print``(phi(``31``));``print``(phi(``41``));``print``(phi(``51``));``print``(phi(``61``));``print``(phi(``91``));``print``(phi(``101``));` `# This code is contributed by mits`

## C#

 `// C# program to efficiently compute values``// of euler totient function for multiple inputs.``using` `System;``using` `System.Collections;``class` `GFG{``static` `int` `MAX = 100001;` `// Stores prime numbers upto MAX - 1 values``static` `ArrayList p = ``new` `ArrayList();` `// Finds prime numbers upto MAX-1 and``// stores them in vector p``static` `void` `sieve()``{``    ``int``[] isPrime=``new` `int``[MAX+1];` `    ``for` `(``int` `i = 2; i<= MAX; i++)``    ``{``        ``// if prime[i] is not marked before``        ``if` `(isPrime[i] == 0)``        ``{``            ``// fill vector for every newly``            ``// encountered prime``            ``p.Add(i);` `            ``// run this loop till square root of MAX,``            ``// mark the index i * j as not prime``            ``for` `(``int` `j = 2; i * j<= MAX; j++)``                ``isPrime[i * j]= 1;``        ``}``    ``}``}` `// function to find totient of n``static` `int` `phi(``int` `n)``{``    ``int` `res = n;` `    ``// this loop runs sqrt(n / ln(n)) times``    ``for` `(``int` `i=0; (``int``)p[i]*(``int``)p[i] <= n; i++)``    ``{``        ``if` `(n % (``int``)p[i]== 0)``        ``{``            ``// subtract multiples of p[i] from r``            ``res -= (res / (``int``)p[i]);` `            ``// Remove all occurrences of p[i] in n``            ``while` `(n % (``int``)p[i]== 0)``            ``n /= (``int``)p[i];``        ``}``    ``}` `    ``// when n has prime factor greater``    ``// than sqrt(n)``    ``if` `(n > 1)``    ``res -= (res / n);` `    ``return` `res;``}` `// Driver code``static` `void` `Main()``{``    ``// preprocess all prime numbers upto 10 ^ 5``    ``sieve();``    ``Console.WriteLine(phi(11));``    ``Console.WriteLine(phi(21));``    ``Console.WriteLine(phi(31));``    ``Console.WriteLine(phi(41));``    ``Console.WriteLine(phi(51));``    ``Console.WriteLine(phi(61));``    ``Console.WriteLine(phi(91));``    ``Console.WriteLine(phi(101));` `}``}``// this code is contributed by mits`

## PHP

 ` 1)``    ``\$res` `-= (int)(``\$res` `/ ``\$n``);` `    ``return` `\$res``;``}` `// Driver code`` ` `    ``// preprocess all prime numbers upto 10 ^ 5``    ``sieve();``    ``print``(phi(11).``"\n"``);``    ``print``(phi(21).``"\n"``);``    ``print``(phi(31).``"\n"``);``    ``print``(phi(41).``"\n"``);``    ``print``(phi(51).``"\n"``);``    ``print``(phi(61).``"\n"``);``    ``print``(phi(91).``"\n"``);``    ``print``(phi(101).``"\n"``);` `// this code is contributed by mits``?>`

## Javascript

 ``

Output:

```10
12
30
40
32
60
72
100```

Time Complexity: O(MAX*log(MAX)+sqrt(n/log(n)))

Auxiliary Space: O(MAX)

This article is contributed by Abhishek Rajput. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.