Summation of floor of harmonic progression

Given an integer N, the task is to find the summation of the harmonic series .

Examples:

Input: N = 5 
Output: 10 
floor(3/1) + floor(3/2) + floor(3/3) = 3 + 1 + 1 = 5
Input: N = 20 
Output: 66 

Naive approach: Run a loop from 1 to N and find the summation of the floor values of N / i. Time complexity of this approach will be O(n).
Efficient approach: Use the following formula to calculate the summation of the series: 

Now, the loop needs to be run from 1 to sqrt(N) and the time complexity gets reduced to O(sqrt(N))
Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function to return the summation of
// the given harmonic series
long long int getSum(int n)
{

    // To store the summation
    long long int sum = 0;

    // Floor of sqrt(n)
    int k = sqrt(n);

    // Summation of floor(n / i)
    for (int i = 1; i <= k; i++) {
        sum += floor(n / i);
    }

    // From the formula
    sum *= 2;
    sum -= pow(k, 2);

    return sum;
}

// Driver code
int main()
{
    int n = 5;

    cout << getSum(n);

    return 0;
}

Java

// Java implementation of the approach
class GFG
{

    // Function to return the summation of
    // the given harmonic series
    static long getSum(int n)
    {

        // To store the summation
        long sum = 0;

        // Floor of sqrt(n)
        int k = (int)Math.sqrt(n);

        // Summation of floor(n / i)
        for (int i = 1; i <= k; i++)
        {
            sum += Math.floor(n / i);
        }

        // From the formula
        sum *= 2;
        sum -= Math.pow(k, 2);

        return sum;
    }

    // Driver code
    public static void main (String[] args)
    {
        int n = 5;

        System.out.println(getSum(n));
    }
}

// This code is contributed by AnkitRai01

Python3

# Python3 implementation of the approach
from math import floor, sqrt, ceil

# Function to return the summation of
# the given harmonic series
def getSum(n):

    # To store the summation
    summ = 0

    # Floor of sqrt(n)
    k =(n)**(.5)

    # Summation of floor(n / i)
    for i in range(1, floor(k) + 1):
        summ += floor(n / i)

    # From the formula
    summ *= 2
    summ -= pow(floor(k), 2)

    return summ

# Driver code
n = 5

print(getSum(n))

# This code is contributed by Mohit Kumar

C#

// C# implementation of the approach
using System;

class GFG
{

    // Function to return the summation of
    // the given harmonic series
    static double getSum(int n)
    {

        // To store the summation
        double sum = 0;

        // Floor of sqrt(n)
        int k = (int)Math.Sqrt(n);

        // Summation of floor(n / i)
        for (int i = 1; i <= k; i++)
        {
            sum += Math.Floor((double)n / i);
        }

        // From the formula
        sum *= 2;
        sum -= Math.Pow(k, 2);

        return sum;
    }

    // Driver code
    public static void Main (String[] args)
    {
        int n = 5;

        Console.WriteLine(getSum(n));
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

// Javascript implementation of the approach

// Function to return the summation of
// the given harmonic series
function getSum(n)
{

    // To store the summation
    let sum = 0;

    // Floor of sqrt(n)
    let k = parseInt(Math.sqrt(n));

    // Summation of floor(n / i)
    for (let i = 1; i <= k; i++) {
        sum += Math.floor(n / i);
    }

    // From the formula
    sum *= 2;
    sum -= Math.pow(k, 2);

    return sum;
}

// Driver code
    let n = 5;

    document.write(getSum(n));

</script>

Output:

10

Time Complexity: O(sqrt(n)), since the for loop runs for sqrt(n) times.
Auxiliary Space: O(1), since no extra space has been taken.