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Check if the given number is Ore number or not

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  • Last Updated : 01 Aug, 2022
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Given a positive integer n, check if it is an Ore number or not. Print ‘YES’ if n is an ore number otherwise print ‘NO’.
Ore Number: In mathematics, Ore numbers are positive integers whose divisors have an integer harmonic value. Ore numbers are often called harmonic divisor numbers. Ore numbers are named after Øystein Ore.
For example, 6 has four divisors namely 1, 2, 3, and 6. 
The harmonic mean of the divisors is- 
 

Harmonic mean of  6

The harmonic mean of divisors of 6 is 2, an integer. So, 6 is an Ore number or harmonic divisor number.
First, a few Ore numbers or harmonic divisor numbers are: 

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 
 

Examples:  

Input : N = 6
Output : Yes

Input : N = 4
Output: No
Explanation : Harmonic mean of divisors of 4 
              is not an Integer. 

Prerequisite: 

The idea is to generate all divisors of the given number and then check if the harmonic mean of the divisor is an Integer or not. 

  1. Generate All Divisors of the given number – ‘n’
  2. Calculate the Harmonic mean of the divisors of n
  3. Check if the Harmonic mean is an Integer or not
  4. If Yes, Then the number is an Ore Number otherwise Not

Below is the implementation of the above approach: 

C++




// CPP program to check if the given number is
// Ore number
 
#include <bits/stdc++.h>
using namespace std;
 
vector<int> arr;
 
// Function that returns harmonic mean
void generateDivisors(int n)
{
    // Note that this loop runs till square root
    for (int i = 1; i <= sqrt(n); i++) {
        if (n % i == 0) {
 
            // If divisors are equal, store 'i'
            if (n / i == i)
                arr.push_back(i);
 
            else // Otherwise store 'i' and 'n/i' both
            {
                arr.push_back(i);
                arr.push_back(n / i);
            }
        }
    }
}
 
// Utility function to calculate harmonic
// mean of the divisors
double harmonicMean(int n)
{
    generateDivisors(n);
 
    // Declare sum variables and initialize
    // with zero.
 
    double sum = 0.0;
 
    int len = arr.size();
 
    // calculate denominator
    for (int i = 0; i < len; i++)
        sum = sum + double(n / arr[i]);
 
    sum = double(sum / n);
 
    // Calculate harmonic mean and return
 
    return double(arr.size() / sum);
}
 
// Function to check if a number is ore number
bool isOreNumber(int n)
{
    // Calculate Harmonic mean of divisors of n
    double mean = harmonicMean(n);
 
    // Check if harmonic mean is an integer or not
    if (mean - int(mean) == 0)
        return true;
    else
        return false;
}
 
// Driver Code
int main()
{
    int n = 28;
 
    if (isOreNumber(n))
        cout << "YES";
    else
        cout << "NO";
 
    return 0;
}


Java




// Java program to check if the given
// number is Ore number
 
import java.util.*;
class GFG {
 
    static Vector<Integer> arr = new Vector<Integer>();
 
    // Function that returns harmonic mean.
    static void generateDivisors(int n)
    {
        // Note that this loop runs till square root
        for (int i = 1; i <= Math.sqrt(n); i++) {
            if (n % i == 0) {
 
                // If divisors are equal, store 'i'
                if (n / i == i)
                    arr.add(i);
 
                else // Otherwise store 'i' and 'n/i' both
                {
                    arr.add(i);
                    arr.add(n / i);
                }
            }
        }
    }
 
    // Utility function to calculate harmonic mean
    // of the divisors
    static double harmonicMean(int n)
    {
        generateDivisors(n);
 
        // Declare sum variables and initialize
        // with zero.
 
        double sum = 0.0;
 
        int len = arr.size();
 
        // calculate denominator
        for (int i = 0; i < len; i++)
            sum = sum + n / arr.get(i);
 
        sum = sum / n;
 
        // Calculate harmonic mean and return
 
        return arr.size() / sum;
    }
 
    // Function to check if a number
    // is Ore number
    static boolean isOreNumber(int n)
    {
        // Calculate Harmonic mean of divisors of n
        double mean = harmonicMean(n);
 
        // Check if Harmonic mean is an Integer or not
        if (mean - Math.floor(mean) == 0)
            return true;
        else
            return false;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int n = 28;
 
        if (isOreNumber(n))
            System.out.println("YES");
        else
            System.out.println("NO");
    }
}


Python3




# Python3 program to check if the
# given number is Ore number
arr = []
 
# Function that returns harmonic mean
def generateDivisors(n):
 
    # Note that this loop runs till square root
    for i in range(1, int(n**(0.5)) + 1):
        if n % i == 0:
 
            # If divisors are equal, store 'i'
            if n // i == i:
                arr.append(i)
             
            # Otherwise store 'i' and 'n/i' both
            else:
                arr.append(i)
                arr.append(n // i)
         
# Utility function to calculate harmonic
# mean of the divisors
def harmonicMean(n):
 
    generateDivisors(n)
 
    # Declare sum variables and initialize
    # with zero.
    Sum = 0
    length = len(arr)
 
    # calculate denominator
    for i in range(0, length):
        Sum = Sum + (n / arr[i])
 
    Sum = Sum / n
 
    # Calculate harmonic mean and return
    return length / Sum
 
# Function to check if a number
# is ore number
def isOreNumber(n):
 
    # Calculate Harmonic mean of
    # divisors of n
    mean = harmonicMean(n)
 
    # Check if harmonic mean is an
    # integer or not
    if mean - int(mean) == 0:
        return True
    else:
        return False
 
# Driver Code
if __name__ == "__main__":
 
    n = 28
 
    if isOreNumber(n) == True:
        print("YES")
    else:
        print("NO")
 
# This code is contributed
# by Rituraj Jain


C#




// C# program to check if the given
// number is Ore number
using System;
using System.Collections;
 
class GFG
{
 
static ArrayList arr = new ArrayList();
 
// Function that returns harmonic mean.
static void generateDivisors(int n)
{
    // Note that this loop runs
    // till square root
    for (int i = 1; i <= Math.Sqrt(n); i++)
    {
        if (n % i == 0)
        {
 
            // If divisors are equal,
            // store 'i'
            if (n / i == i)
                arr.Add(i);
 
            else // Otherwise store 'i'
                 // and 'n/i' both
            {
                arr.Add(i);
                arr.Add(n / i);
            }
        }
    }
}
 
// Utility function to calculate
// harmonic mean of the divisors
static double harmonicMean(int n)
{
    generateDivisors(n);
 
    // Declare sum variables and
    // initialize with zero.
    double sum = 0.0;
 
    int len = arr.Count;
 
    // calculate denominator
    for (int i = 0; i < len; i++)
        sum = sum + n / (int)arr[i];
 
    sum = sum / n;
 
    // Calculate harmonic mean
    // and return
    return arr.Count / sum;
}
 
// Function to check if a number
// is Ore number
static bool isOreNumber(int n)
{
    // Calculate Harmonic mean of
    // divisors of n
    double mean = harmonicMean(n);
 
    // Check if Harmonic mean is
    // an Integer or not
    if (mean - Math.Floor(mean) == 0)
        return true;
    else
        return false;
}
 
// Driver Code
public static void Main()
{
    int n = 28;
 
    if (isOreNumber(n))
        Console.WriteLine("YES");
    else
        Console.WriteLine("NO");
}
}
 
// This code is contributed by mits


Javascript




<script>
 
// Javascript program to check
// if the given number is
// Ore number
 
var arr = [];
 
// Function that returns harmonic mean
function generateDivisors(n)
{
    // Note that this loop runs till square root
    for (var i = 1; i <= Math.sqrt(n); i++) {
        if (n % i == 0) {
 
            // If divisors are equal, store 'i'
            if (n / i == i)
                arr.push(i);
 
            else // Otherwise store 'i' and 'n/i' both
            {
                arr.push(i);
                arr.push(n / i);
            }
        }
    }
}
 
// Utility function to calculate harmonic
// mean of the divisors
function harmonicMean(n)
{
    generateDivisors(n);
 
    // Declare sum variables and initialize
    // with zero.
 
    var sum = 0.0;
 
    var len = arr.length;
 
    // calculate denominator
    for (var i = 0; i < len; i++)
        sum = sum + (n / arr[i]);
 
    sum = (sum / n);
 
    // Calculate harmonic mean and return
 
    return (arr.length / sum);
}
 
// Function to check if a number is ore number
function isOreNumber(n)
{
    // Calculate Harmonic mean of divisors of n
    var mean = harmonicMean(n);
 
    // Check if harmonic mean is an integer or not
    if (mean - parseInt(mean) == 0)
        return true;
    else
        return false;
}
 
// Driver Code
var n = 28;
if (isOreNumber(n))
    document.write( "YES");
else
    document.write( "NO");
 
</script>


Output

YES

Time Complexity: O(sqrt(n)), Where n is the given number.
Auxiliary Space: O(sqrt(n)), for storing the divisor of n in the array


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