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Count of numbers in range [L, R] which can be represented as sum of two perfect powers

  • Last Updated : 30 Sep, 2021

Given a range [L, R], the task is to find the count of numbers in the range [L, R] that can be expressed as a sum of two perfect powers.

Examples:

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Input: L = 0, R = 1
Output: 2
Explanation:
The valid numbers are:

  1. 1 as it can be expressed as, 1 = 12 + 02.
  2. 0 as it can be expressed as, 0 = 02 + 02.

Therefore, the count of such numbers is 2.



Input: L = 5, R = 8
Output: 2
Explanation:
The valid numbers are:

  1. 5 as it can be expressed as, 5 = 12 + 22.
  2. 8 as it can be expressed as, 0 = 02 + 23.

Therefore, the count of such numbers is 2.

Approach: The given problem can be solved by using some mathematical observations. Follow the steps below to solve the problem:

Below is the implementation of the above approach: 

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the number of numbers
// that can be expressed in the form of
// the sum of two perfect powers
int TotalPerfectPowerSum(long long L,
                         long long R)
{
    // Stores all possible powers
    vector<long long> pows;
 
    // Push 1 and 0 in it
    pows.push_back(0);
    pows.push_back(1);
 
    // Iterate over all the exponents
    for (int p = 2; p < 25; p++) {
 
        // Iterate over all possible numbers
        long long int num = 2;
 
        // This loop will run for a
        // maximum of sqrt(R) times
        while ((long long int)(pow(num, p) + 0.5) <= R) {
 
            // Push this power in
            // the array pows[]
            pows.push_back(
                (long long int)(pow(num, p) + 0.5));
 
            // Increase the number
            num++;
        }
    }
 
    // Stores if i can be expressed as
    // the sum of perfect power or not
    int ok[R + 1];
    memset(ok, 0, sizeof(ok));
 
    // Iterate over all possible pairs
    // of the array pows[]
    for (int i = 0;
         i < pows.size(); i++) {
 
        for (int j = 0;
             j < pows.size(); j++) {
 
            if (pows[i] + pows[j] <= R
                and pows[i] + pows[j] >= L) {
 
                // The number is valid
                ok[pows[i] + pows[j]] = 1;
            }
        }
    }
 
    // Find the prefix sum of the
    // array ok[]
    for (int i = 0; i <= R; i++) {
        ok[i] += ok[i - 1];
    }
 
    // Return the count of required number
    return ok[R] - ok[L - 1];
}
 
// Driver Code
signed main()
{
    int L = 5, R = 8;
    cout << TotalPerfectPowerSum(L, R);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
 
class GFG {
 
    // Function to find the number of numbers
    // that can be expressed in the form of
    // the sum of two perfect powers
    static int TotalPerfectPowerSum(int L, int R)
    {
        // Stores all possible powers
        ArrayList<Integer> pows = new ArrayList<Integer>();
 
        // Push 1 and 0 in it
        pows.add(0);
        pows.add(1);
 
        // Iterate over all the exponents
        for (int p = 2; p < 25; p++) {
 
            // Iterate over all possible numbers
            int num = 2;
 
            // This loop will run for a
            // maximum of sqrt(R) times
            while ((int)(Math.pow(num, p) + 0.5) <= R) {
 
                // Push this power in
                // the array pows[]
                pows.add((int)(Math.pow(num, p) + 0.5));
 
                // Increase the number
                num++;
            }
        }
 
        // Stores if i can be expressed as
        // the sum of perfect power or not
        int[] ok = new int[R + 2];
        // memset(ok, 0, sizeof(ok));
 
        // Iterate over all possible pairs
        // of the array pows[]
        for (int i = 0; i < pows.size(); i++) {
 
            for (int j = 0; j < pows.size(); j++) {
 
                if (pows.get(i) + pows.get(j) <= R
                    && pows.get(i) + pows.get(j) >= L) {
 
                    // The number is valid
                    ok[pows.get(i) + pows.get(j)] = 1;
                }
            }
        }
 
        // Find the prefix sum of the
        // array ok[]
        for (int i = 1; i <= R; i++) {
            ok[i] += ok[i - 1];
        }
 
        // Return the count of required number
        return ok[R] - ok[L - 1];
    }
   
    // Driver Code
    public static void main(String args[])
    {
 
        int L = 5, R = 8;
        System.out.print(TotalPerfectPowerSum(L, R));
    }
}
 
// This code is contributed by avijitmondal1998.

Python3




# python program for the above approach
 
# Function to find the number of numbers
# that can be expressed in the form of
# the sum of two perfect powers
def TotalPerfectPowerSum(L, R):
   
    # Stores all possible powers
    pows = []
 
    # Push 1 and 0 in it
    pows.append(0)
    pows.append(1)
 
    # Iterate over all the exponents
    for p in range(2, 25):
 
                # Iterate over all possible numbers
        num = 2
 
        # This loop will run for a
        # maximum of sqrt(R) times
        while ((int)(pow(num, p) + 0.5) <= R):
 
                        # Push this power in
                        # the array pows[]
            pows.append((int)(pow(num, p) + 0.5))
 
            # Increase the number
            num = num + 1
 
        # Stores if i can be expressed as
        # the sum of perfect power or not
    ok = [0 for _ in range(R + 1)]
     
    # int ok[R + 1];
    # memset(ok, 0, sizeof(ok));
    # Iterate over all possible pairs
    # of the array pows[]
    for i in range(0, int(len(pows))):
        for j in range(0, len(pows)):
            if (pows[i] + pows[j] <= R and pows[i] + pows[j] >= L):
 
                                # The number is valid
                ok[pows[i] + pows[j]] = 1
 
        # Find the prefix sum of the
        # array ok[]
    for i in range(0, R+1):
        ok[i] += ok[i - 1]
 
        # Return the count of required number
    return ok[R] - ok[L - 1]
 
# Driver Code
if __name__ == "__main__":
    L = 5
    R = 8
    print(TotalPerfectPowerSum(L, R))
     
    # This code is contributed by rakeshsahni

C#




// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {
 
    // Function to find the number of numbers
    // that can be expressed in the form of
    // the sum of two perfect powers
    static int TotalPerfectPowerSum(long L, long R)
    {
        // Stores all possible powers
        List<long> pows = new List<long>();
 
        // Push 1 and 0 in it
        pows.Add(0);
        pows.Add(1);
 
        // Iterate over all the exponents
        for (int p = 2; p < 25; p++) {
 
            // Iterate over all possible numbers
            long num = 2;
 
            // This loop will run for a
            // maximum of sqrt(R) times
            while ((long)(Math.Pow(num, p) + 0.5) <= R) {
 
                // Push this power in
                // the array pows[]
                pows.Add((long)(Math.Pow(num, p) + 0.5));
 
                // Increase the number
                num++;
            }
        }
 
        // Stores if i can be expressed as
        // the sum of perfect power or not
        int[] ok = new int[R + 2];
        // memset(ok, 0, sizeof(ok));
 
        // Iterate over all possible pairs
        // of the array pows[]
        for (int i = 0; i < pows.Count; i++) {
 
            for (int j = 0; j < pows.Count; j++) {
 
                if (pows[i] + pows[j] <= R
                    && pows[i] + pows[j] >= L) {
 
                    // The number is valid
                    ok[pows[i] + pows[j]] = 1;
                }
            }
        }
 
        // Find the prefix sum of the
        // array ok[]
        for (int i = 1; i <= R; i++) {
            ok[i] += ok[i - 1];
        }
 
        // Return the count of required number
        return ok[R] - ok[L - 1];
    }
 
    // Driver Code
    public static void Main()
    {
        int L = 5, R = 8;
        Console.WriteLine(TotalPerfectPowerSum(L, R));
    }
}
 
// This code is contributed by ukasp.

Javascript




<script>
       // JavaScript Program to implement
       // the above approach
 
       // Function to find the number of numbers
       // that can be expressed in the form of
       // the sum of two perfect powers
       function TotalPerfectPowerSum(L,
           R)
       {
        
           // Stores all possible powers
           let pows = [];
 
           // Push 1 and 0 in it
           pows.push(0);
           pows.push(1);
 
           // Iterate over all the exponents
           for (let p = 2; p < 25; p++) {
 
               // Iterate over all possible numbers
               let num = 2;
 
               // This loop will run for a
               // maximum of sqrt(R) times
               while (Math.floor(Math.pow(num, p) + 0.5) <= R) {
 
                   // Push this power in
                   // the array pows[]
                   pows.push(
                       Math.floor(Math.pow(num, p) + 0.5));
 
                   // Increase the number
                   num++;
               }
           }
 
           // Stores if i can be expressed as
           // the sum of perfect power or not
 
           let ok = new Array(R + 1).fill(0);
 
 
           // Iterate over all possible pairs
           // of the array pows[]
           for (let i = 0;
               i < pows.length; i++) {
 
               for (let j = 0;
                   j < pows.length; j++) {
 
                   if (pows[i] + pows[j] <= R
                       && pows[i] + pows[j] >= L) {
 
                       // The number is valid
                       ok[pows[i] + pows[j]] = 1;
                   }
               }
           }
 
           // Find the prefix sum of the
           // array ok[]
           for (let i = 1; i <= R; i++) {
               ok[i] += ok[i - 1];
 
           }
 
           // Return the count of required number
           return ok[R] - ok[L - 1];
       }
 
       // Driver Code
       let L = 5, R = 8;
       document.write(TotalPerfectPowerSum(L, R));
 
    // This code is contributed by Potta Lokesh
   </script>
Output: 
2

 

Time Complexity: O(R*log(R))
Auxiliary Space: O(R)




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