Sum of all even factors of numbers in the range [l, r]

Given a range [l, r], the task is to find the sum of all the even factors of the numbers from the given range.

Examples:

Input: l = 6, r = 8
Output: 22
factors(6) = 1, 2, 3, 6, evenfactors(6) = 2, 6 sumEvenFactors(6) = 2 + 6 = 8
factors(7) = 1, 7, No even factors
factors(8) = 1, 2, 4, 8, evenfactors(8) = 2, 4, 8 sumEvenFactors(8) = 2 + 4 + 8 = 14
Therefore sum of all even factors = 8 + 14 = 22

Input: l = 1, r = 10
Output: 42

Approach: We can modify Sieve Of Eratosthenes to store the sum of all even factors of a number at it’s corresponding index. Then we will make a prefix array to store sum upto that index. And now each query can be answered in O(1) using prefix[r] – prefix[l – 1].

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
#define ll long long int
  
const int MAX = 100000;
  
ll prefix[MAX];
  
// Function to calculate the prefix sum
// of all the even factors
void sieve_modified()
{
    for (int i = 2; i < MAX; i += 2) {
  
        // Add i to all the multiples of i
        for (int j = i; j < MAX; j += i)
            prefix[j] += i;
    }
  
    // Update the prefix sum
    for (int i = 1; i < MAX; i++)
        prefix[i] += prefix[i - 1];
}
  
// Function to return the sum of
// all the even factors of the
// numbers in the given range
ll sumEvenFactors(int L, int R)
{
    return (prefix[R] - prefix[L - 1]);
}
  
// Driver code
int main()
{
    sieve_modified();
    int l = 6, r = 10;
    cout << sumEvenFactors(l, r);
  
    return 0;
}
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// Java implementation of the approach
class GFG {
  
    static final int MAX = 100000;
    static long prefix[] = new long[MAX];
  
    // Function to calculate the prefix sum
    // of all the even factors
    static void sieve_modified()
    {
        for (int i = 2; i < MAX; i += 2) {
  
            // Add i to all the multiples of i
            for (int j = i; j < MAX; j += i)
                prefix[j] += i;
        }
  
        // Update the prefix sum
        for (int i = 1; i < MAX; i++)
            prefix[i] += prefix[i - 1];
    }
  
    // Function to return the sum of
    // all the even factors of the
    // numbers in the given range
    static long sumEvenFactors(int L, int R)
    {
        return (prefix[R] - prefix[L - 1]);
    }
  
    // Driver code
    public static void main(String args[])
    {
        sieve_modified();
        int l = 6, r = 10;
        System.out.print(sumEvenFactors(l, r));
    }
}
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# Python3 implementation of the approach. 
  
# Function to calculate the prefix sum 
# of all the even factors 
def sieve_modified(): 
  
    for i in range(2, MAX, 2): 
  
        # Add i to all the multiples of i 
        for j in range(i, MAX, i): 
            prefix[j] +=
  
    # Update the prefix sum 
    for i in range(1, MAX): 
        prefix[i] += prefix[i - 1
  
# Function to return the sum of 
# all the even factors of the 
# numbers in the given range 
def sumEvenFactors(L, R): 
  
    return (prefix[R] - prefix[L - 1]) 
  
# Driver code 
if __name__ == "__main__":
      
    MAX = 100000
    prefix = [0] * MAX
    sieve_modified() 
    l, r = 6, 10
    print(sumEvenFactors(l, r)) 
  
# This code is contributed by Rituraj Jain
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using System;
// C# implementation of the approach
using System;
  
class GFG
{
  
    public const int MAX = 100000;
    public static long[] prefix = new long[MAX];
  
    // Function to calculate the prefix sum
    // of all the even factors
    public static void sieve_modified()
    {
        for (int i = 2; i < MAX; i += 2)
        {
  
            // Add i to all the multiples of i
            for (int j = i; j < MAX; j += i)
            {
                prefix[j] += i;
            }
        }
  
        // Update the prefix sum
        for (int i = 1; i < MAX; i++)
        {
            prefix[i] += prefix[i - 1];
        }
    }
  
    // Function to return the sum of
    // all the even factors of the
    // numbers in the given range
    public static long sumEvenFactors(int L, int R)
    {
        return (prefix[R] - prefix[L - 1]);
    }
  
    // Driver code
    public static void Main(string[] args)
    {
        sieve_modified();
        int l = 6, r = 10;
        Console.Write(sumEvenFactors(l, r));
    }
}
  
// This code is contributed by Shrikant13
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<?php
// PHP implementation of the approach
$MAX = 10000;
  
$prefix = array_fill(0, $MAX, 0);
  
// Function to calculate the prefix sum
// of all the even factors
function sieve_modified()
{
    global $MAX, $prefix;
    for ($i = 2; $i < $MAX; $i += 2) 
    {
  
        // Add i to all the multiples of i
        for ($j = $i; $j < $MAX; $j += $i)
            $prefix[$j] += $i;
    }
  
    // Update the prefix sum
    for ($i = 1; $i < $MAX; $i++)
        $prefix[$i] += $prefix[$i - 1];
}
  
// Function to return the sum of
// all the even factors of the
// numbers in the given range
function sumEvenFactors($L, $R)
{
    global $MAX, $prefix;
    return ($prefix[$R] - $prefix[$L - 1]);
}
  
// Driver code
sieve_modified();
$l = 6;
$r = 10;
echo sumEvenFactors($l, $r);
  
// This code is contributed by mits
?>
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Output:
34



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