phi function (n log (log(n)))

The phi function, denoted as φ(n) or Euler's totient function, calculates the count of positive integers less than or equal to n that are coprime (i.e., have no common factors) with n. In C++, you can implement the phi function using the following algorithm:

1. Start by initializing a variable called result to n, which will store the final value of φ(n).
2. Iterate over all prime numbers p less than or equal to the square root of n (i.e., p ≤ √n).
3. For each prime number p, check if it divides n evenly (i.e., n % p == 0). If it does, it means that p is a prime factor of n.
4. If p divides n, subtract n divided by p from the result (i.e., result -= result / p). This step accounts for the numbers that are not coprime with n because they have p as a factor.
5. Continue this process for all prime numbers p less than or equal to √n.
6. After the iteration, if n is still greater than 1, it means that n is a prime number and is coprime with all previous numbers. In this case, subtract n divided by n from the result (i.e., result -= result / n).
7. The final value of result is φ(n), which represents the count of positive integers less than or equal to n that are coprime with n.

Here is an example implementation of the phi function in C++:

#include <iostream>
#include <cmath>

int phi(int n) {
    int result = n;
    for (int p = 2; p <= sqrt(n); ++p) {
        if (n % p == 0) {
            while (n % p == 0) {
                n /= p;
            }
            result -= result / p;
        }
    }
    if (n > 1) {
        result -= result / n;
    }
    return result;
}

int main() {
    int n;
    std::cout << "Enter a number: ";
    std::cin >> n;
    int phi_n = phi(n);
    std::cout << "phi(" << n << ") = " << phi_n << std::endl;
    return 0;
}

This implementation calculates the value of φ(n) for a given number n and prints the result.