Count of Perfect Numbers in given range for Q queries

Given an array arr[] consisting of N pairs, where each pair represents a query of the form {L, R}, the task is to find the count of perfect numbers in the given range for each query.

Examples:

Input: arr[][] = {{1, 10}, {10, 20}, {20, 30}}
Output: 1 1 1
Explanation:

1. Query(1, 10): Perfect numbers in the range is only 6.
2. Query(10, 20): There are no perfect numbers in the range.
3. Query(20, 30): The perfect number in the range is only 28.

Input: arr[][] = {{1, 1000}, {1000, 2000}, {2000, 3000}
Output: 3 0 0
Explanation:

1. Query(1, 1000): Perfect numbers in the range are 6, 28, and 496.
2. Query(1000, 2000): There are no perfect numbers in the range.
3. Query(2000, 3000): There are no perfect numbers in the range.

Naive Approach: The simplest approach is, iterate over the range in each query and check if a number is a perfect number or not, and then print the count of perfect numbers in the range for the respective query.

Time Complexity: O(N*M*?M)), where M is the largest size of a range. 
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized by prestoring the count of perfect Numbers from 1 to every other number using the prefix sum array technique, which results in constant time calculation of each query. Follow the steps below to solve the problem:

• Find the maximum value of the right boundary of a range by traversing the array arr[] and store it in a variable, say MAX.
• Initialize an array, say prefix[] of size MAX+1 with the value of each element as 0, where the prefix[i] stores the count of perfect numbers up to i.
• Iterate over the range [2, MAX] using the variable i and do the following:
  • Update prefix[i] to prefix[i-1] and then check if current integer i is a perfect number then increase prefix[i] by 1.
• Traverse the array arr[] and in each iteration print the count of perfect numbers in the current range, [L, R] as prefix[R] – prefix[L-1].

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
const int MAX = 100005;

// Function to check whether a number
// is perfect Number
bool isPerfect(long long int N)
{
    // Stores sum of divisors
    long long int sum = 1;

    // Iterate over the range[2, sqrt(N)]
    for (long long int i = 2; i * i <= N; i++) {
        if (N % i == 0) {
            if (i * i != N)
                sum = sum + i + N / i;
            else
                sum = sum + i;
        }
    }
    // If sum of divisors is equal to
    // N, then N is a perfect number
    if (sum == N && N != 1)
        return true;

    return false;
}

// Function to find count of perfect
// numbers in a given range
void Query(int arr[][2], int N)
{
    // Stores the count of perfect Numbers
    // upto a every number less than MAX
    int prefix[MAX + 1] = { 0 };

    // Iterate over the range [1, MAX]
    for (int i = 2; i <= MAX; i++) {
        prefix[i] = prefix[i - 1] + isPerfect(i);
    }

    // Traverse the array arr[]
    for (int i = 0; i < N; i++) {
        // Print the count of perfect numbers
        // in the range [arr[i][0], arr[i][1]]
        cout << prefix[arr[i][1]] - prefix[arr[i][0] - 1]
             << " ";
    }
}

// Driver Code
int main()
{
    int arr[][2]
        = { { 1, 1000 }, { 1000, 2000 }, { 2000, 3000 } };
    int N = sizeof(arr) / sizeof(arr[0]);

    Query(arr, N);
}

Java

// C++ program for the above approach
import java.util.*;
public class MyClass
{

static int MAX = 100005;

// Function to check whether a number
// is perfect Number
static int isPerfect(long N)
{

    // Stores sum of divisors
    long  sum = 1;

    // Iterate over the range[2, sqrt(N)]
    for (long i = 2; i * i <= N; i++) {
        if (N % i == 0) {
            if (i * i != N)
                sum = sum + i + N / i;
            else
                sum = sum + i;
        }
    }

    // If sum of divisors is equal to
    // N, then N is a perfect number
    if (sum == N && N != 1)
        return 1;

    return 0;
}

// Function to find count of perfect
// numbers in a given range
static void Query(int arr[][], int N)
{

    // Stores the count of perfect Numbers
    // upto a every number less than MAX
    int []prefix = new int [MAX + 1];
    Arrays.fill(prefix,0);

    // Iterate over the range [1, MAX]
    for (int i = 2; i <= MAX; i++) {
        prefix[i] = prefix[i - 1] + isPerfect(i);
    }

    // Traverse the array arr[]
    for (int i = 0; i < N; i++)
    {

        // Print the count of perfect numbers
        // in the range [arr[i][0], arr[i][1]]
       System.out.print( prefix[arr[i][1]] - prefix[arr[i][0] - 1]+ " ");
    }
}

// Driver Code
public static void main(String args[])
{
    int [][]arr = { { 1, 1000 }, { 1000, 2000 }, { 2000, 3000 } };
    int N = arr.length;

    Query(arr, N);
}
}

// This code is contributed by SoumikMondal

Python3

# python 3 program for the above approach
MAX = 100005

from math import sqrt

# Function to check whether a number
# is perfect Number
def isPerfect(N):

    # Stores sum of divisors
    sum = 1

    # Iterate over the range[2, sqrt(N)]
    for i in range(2,int(sqrt(N))+1,1):
        if (N % i == 0):
            if (i * i != N):
                sum = sum + i + N // i
            else:
                sum = sum + i

    # If sum of divisors is equal to
    # N, then N is a perfect number
    if (sum == N and N != 1):
        return True

    return False

# Function to find count of perfect
# numbers in a given range
def Query(arr, N):

    # Stores the count of perfect Numbers
    # upto a every number less than MAX
    prefix = [0 for i in range(MAX + 1)]

    # Iterate over the range [1, MAX]
    for i in range(2,MAX+1,1):
        prefix[i] = prefix[i - 1] + isPerfect(i)

    # Traverse the array arr[]
    for i in range(N):

        # Print the count of perfect numbers
        # in the range [arr[i][0], arr[i][1]]
        print(prefix[arr[i][1]] - prefix[arr[i][0] - 1],end= " ")

# Driver Code
if __name__ == '__main__':
    arr = [[1, 1000],[1000, 2000],[2000, 3000]]
    N = len(arr)
    Query(arr, N)

    # This code is contributed by SURENDRA_GANGWAR.

C#

// C# program for the above approach
using System;
public class MyClass {

    static int MAX = 100005;

    // Function to check whether a number
    // is perfect Number
    static int isPerfect(long N)
    {

        // Stores sum of divisors
        long sum = 1;

        // Iterate over the range[2, sqrt(N)]
        for (long i = 2; i * i <= N; i++) {
            if (N % i == 0) {
                if (i * i != N)
                    sum = sum + i + N / i;
                else
                    sum = sum + i;
            }
        }

        // If sum of divisors is equal to
        // N, then N is a perfect number
        if (sum == N && N != 1)
            return 1;

        return 0;
    }

    // Function to find count of perfect
    // numbers in a given range
    static void Query(int[, ] arr, int N)
    {

        // Stores the count of perfect Numbers
        // upto a every number less than MAX
        int[] prefix = new int[MAX + 1];
        // Arrays.fill(prefix,0);

        // Iterate over the range [1, MAX]
        for (int i = 2; i <= MAX; i++) {
            prefix[i] = prefix[i - 1] + isPerfect(i);
        }

        // Traverse the array arr[]
        for (int i = 0; i < N; i++) {

            // Print the count of perfect numbers
            // in the range [arr[i][0], arr[i][1]]
            Console.Write(prefix[arr[i, 1]]
                          - prefix[arr[i, 0] - 1] + " ");
        }
    }

    // Driver Code
    public static void Main()
    {
        int[, ] arr = { { 1, 1000 },
                        { 1000, 2000 },
                        { 2000, 3000 } };
        int N = arr.GetLength(0);

        Query(arr, N);
    }
}

// This code is contributed by ukasp.

Javascript

<script>
        // JavaScript program for the above approach


        const MAX = 100005;

        // Function to check whether a number
        // is perfect Number
        function isPerfect(N) {
            // Stores sum of divisors
            let sum = 1;

            // Iterate over the range[2, sqrt(N)]
            for (let i = 2; i * i <= N; i++) {
                if (N % i == 0) {
                    if (i * i != N)
                        sum = sum + i + Math.floor(N / i);
                    else
                        sum = sum + i;
                }
            }
            // If sum of divisors is equal to
            // N, then N is a perfect number
            if (sum == N && N != 1)
                return true;

            return false;
        }

        // Function to find count of perfect
        // numbers in a given range
        function Query(arr, N) {
            // Stores the count of perfect Numbers
            // upto a every number less than MAX
            let prefix = new Array(MAX + 1).fill(0);

            // Iterate over the range [1, MAX]
            for (let i = 2; i <= MAX; i++) {
                prefix[i] = prefix[i - 1] + isPerfect(i);
            }

            // Traverse the array arr[]
            for (let i = 0; i < N; i++) {
                // Print the count of perfect numbers
                // in the range [arr[i][0], arr[i][1]]
                document.write(prefix[arr[i][1]] - prefix[arr[i][0] - 1] + " ");
            }
        }

        // Driver Code

        let arr
            = [[1, 1000], [1000, 2000], [2000, 3000]];
        let N = arr.length;

        Query(arr, N);

    // This code is contributed by Potta Lokesh

    </script>

Output

3 0 0 

Time Complexity: O(M*?M+ N), where M is the largest right boundary.
Auxiliary Space: O(M)