Count integers in a range which are divisible by their euler totient value

Given 2 integers L and R, the task is to find out the number of integers in the range [L, R] such that they are completely divisible by their Euler totient value.

Examples:

Input: L = 2, R = 3
Output: 1
\phi(2) = 2 => 2 % \phi(2) = 0
\phi(3) = 2 => 3 % \phi(3) = 1
Hence 2 satisfies the condition.

Input: L = 12, R = 21
Output: 3
Only 12, 16 and 18 satisfy the condition.



Approach: We know that the euler totient function of a number is given as follows:

 \phi(n) = n * (1 - \frac{1}{p_1}) * (1 - \frac{1}{p_2}) * ... * (1 - \frac{1}{p_k}) 

Rearranging the terms, we get:

 \frac{n}{\phi(n)} = \frac{p_1 * p_2 * ... * p_k}{(p_1 - 1) * (p_2 -1) * ... * (p_k -1)} 

If we take a close look at the RHS, we observe that only 2 and 3 are the primes that satisfy n % \phi(n) = 0. This is because for primes p1 = 2 and p2 = 3, p1 – 1 = 1 and p2 – 1 = 2. Hence, only numbers of the form 2p3q where p >= 1 and q >= 0 need to be counted while lying in the range [L, R].

Below is the implementation of the above approach:

C++

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// C++ implementation of the above approach.
#include <bits/stdc++.h>
  
#define ll long long
using namespace std;
  
// Function to return a^n
ll power(ll a, ll n)
{
    if (n == 0)
        return 1;
  
    ll p = power(a, n / 2);
    p = p * p;
  
    if (n & 1)
        p = p * a;
  
    return p;
}
  
// Function to return count of integers
// that satisfy n % phi(n) = 0
int countIntegers(ll l, ll r)
{
  
    ll ans = 0, i = 1;
    ll v = power(2, i);
  
    while (v <= r) {
  
        while (v <= r) {
  
            if (v >= l)
                ans++;
            v = v * 3;
        }
  
        i++;
        v = power(2, i);
    }
  
    if (l == 1)
        ans++;
  
    return ans;
}
  
// Driver Code
int main()
{
    ll l = 12, r = 21;
    cout << countIntegers(l, r);
  
    return 0;
}

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Java

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// Java implementation of the above approach.
class GFG 
{
  
// Function to return a^n
static long power(long a, long n)
{
    if (n == 0)
        return 1;
  
    long p = power(a, n / 2);
    p = p * p;
  
    if (n%2== 1)
        p = p * a;
  
    return p;
}
  
// Function to return count of integers
// that satisfy n % phi(n) = 0
static int countIntegers(long l, long r)
{
  
    long ans = 0, i = 1;
    long v = power(2, i);
  
    while (v <= r) 
    {
        while (v <= r) 
        {
  
            if (v >= l)
                ans++;
            v = v * 3;
        }
  
        i++;
        v = power(2, i);
    }
  
    if (l == 1)
        ans++;
  
    return (int) ans;
}
  
// Driver Code
public static void main(String[] args)
{
    long l = 12, r = 21;
    System.out.println(countIntegers(l, r));
}
}
  
// This code contributed by Rajput-Ji

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Python3

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# Python3 implementation of the approach 
  
# Function to return a^n 
def power(a, n): 
  
    if n == 0
        return 1
  
    p = power(a, n // 2
    p = p *
  
    if n & 1
        p = p *
  
    return
  
# Function to return count of integers 
# that satisfy n % phi(n) = 0 
def countIntegers(l, r): 
  
    ans, i = 0, 1
    v = power(2, i) 
  
    while v <= r: 
  
        while v <= r: 
  
            if v >= l: 
                ans += 1
              
            v = v * 3
  
        i += 1
        v = power(2, i) 
      
    if l == 1
        ans += 1
  
    return ans 
  
# Driver Code 
if __name__ == "__main__":
  
    l, r = 12, 21
    print(countIntegers(l, r)) 
      
# This code is contributed 
# by Rituraj Jain

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C#

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// C# implementation of the above approach.
using System;
  
class GFG 
{
  
// Function to return a^n
static long power(long a, long n)
{
    if (n == 0)
        return 1;
  
    long p = power(a, n / 2);
    p = p * p;
  
    if (n % 2 == 1)
        p = p * a;
  
    return p;
}
  
// Function to return count of integers
// that satisfy n % phi(n) = 0
static int countIntegers(long l, long r)
{
  
    long ans = 0, i = 1;
    long v = power(2, i);
  
    while (v <= r) 
    {
        while (v <= r) 
        {
  
            if (v >= l)
                ans++;
            v = v * 3;
        }
  
        i++;
        v = power(2, i);
    }
  
    if (l == 1)
        ans++;
  
    return (int) ans;
}
  
// Driver Code
public static void Main()
{
    long l = 12, r = 21;
    Console.WriteLine(countIntegers(l, r));
}
}
  
/* This code contributed by PrinciRaj1992 */

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PHP

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<?php
// PHP implementation of the above approach
  
// Function to return a^n 
function power($a, $n
    if ($n == 0) 
        return 1; 
  
    $p = power($a, $n / 2); 
    $p = $p * $p
  
    if ($n & 1) 
        $p = $p * $a
  
    return $p
  
// Function to return count of integers 
// that satisfy n % phi(n) = 0 
function countIntegers($l, $r
    $ans = 0 ;
    $i = 1; 
    $v = power(2, $i); 
  
    while ($v <= $r)
    
        while ($v <= $r
        
            if ($v >= $l
                $ans++; 
            $v = $v * 3; 
        
  
        $i++; 
        $v = power(2, $i); 
    
  
    if ($l == 1) 
        $ans++; 
  
    return $ans
  
// Driver Code 
$l = 12;
$r = 21; 
  
echo countIntegers($l, $r); 
  
// This code is contributed by Ryuga
?>

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Output:

3


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