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Nth term of Ruler Function Series

  • Difficulty Level : Hard
  • Last Updated : 25 Jun, 2021

Given a positive integer N, the task is to find the Nth term of the Ruler Function Series.

The Ruler Function Series is a series, having 1 as the first term, formed by performing the following two operations:

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  • Append the smallest positive integer not present in the series.
  • Then, append all the numbers before the last number.

Examples:



Input: N = 5
Output: 1
Explanation: The Ruler Function Series is given by {1, 2, 1, 3, 1, 2, 1, 4, ….. }. Therefore, the 5th term of the series is 1.

Input: N = 8
Output: 4

Naive Approach: The simplest approach to solve the given problem is to generate the series till the Nth term and then print the Nth term. 

Time Complexity: O(N)
Auxiliary Space: O(N)

Efficient Approach: The above approach can also be optimized based on the following observation that the Nth element in the Ruler Function Series is equal to the number of set bits in (N^(N – 1)) as illustrated below:

  • Number of set bits in the bitwise XOR of 0 and 1 = 1
  • Number of set bits in the bitwise XOR of 1 and 2 = 2
  • Number of set bits in the bitwise XOR of 2 and 3 = 1
  • Number of set bits in the bitwise XOR of 3 and 4 = 3
  • Number of set bits in the bitwise XOR of 4 and 5 = 1
  • Number of set bits in the bitwise XOR of 5 and 6 = 2
  • Number of set bits in the bitwise XOR of 6 and 7 = 1
  • Number of set bits in the bitwise XOR of 7 and 8 = 4
  • and so on…

Therefore, from the above observations, the Nth term of the Ruler Function Series is given by the Bitwise XOR of N and (N – 1)

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count the number
// of set bits in the number N
int setBits(long n)
{
    // Store the number of setbits
    int count = 0;
 
    while (n > 0) {
 
        // Update the value of n
        n = n & (n - 1);
 
        // Update the count
        count++;
    }
 
    // Return the total count
    return count;
}
 
// Function to find the Nth term of
// the Ruler Function Series
void findNthTerm(int N)
{
    // Store the result
    int x = setBits(N ^ (N - 1));
 
    // Print the result
    cout << x;
}
 
// Driver Code
int main()
{
    int N = 8;
    findNthTerm(N);
 
    return 0;
}

Java




// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;
 
class GFG{
 
// Function to count the number
// of set bits in the number N
static int setBits(long n)
{
     
    // Store the number of setbits
    int count = 0;
 
    while (n > 0)
    {
         
        // Update the value of n
        n = n & (n - 1);
 
        // Update the count
        count++;
    }
 
    // Return the total count
    return count;
}
 
// Function to find the Nth term of
// the Ruler Function Series
static void findNthTerm(int N)
{
     
    // Store the result
    int x = setBits(N ^ (N - 1));
 
    // Print the result
    System.out.println(x);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 8;
     
    findNthTerm(N);
}
}
 
// This code is contributed by Kingash

Python3




# Python3 program for the above approach
 
# Function to count the number
# of set bits in the number N
def setBits(n):
     
    # Store the number of setbits
    count = 0
 
    while (n > 0):
         
        # Update the value of n
        n = n & (n - 1)
 
        # Update the count
        count += 1
         
    # Return the total count
    return count
 
# Function to find the Nth term of
# the Ruler Function Series
def findNthTerm(N):
     
    # Store the result
    x = setBits(N ^ (N - 1))
 
    # Print the result
    print(x)
 
# Driver Code
N = 8
 
findNthTerm(N)
 
# This code is contributed by Ankita saini

C#




// C# program for the above approach
using System;
class GFG {
 
    // Function to count the number
    // of set bits in the number N
    static int setBits(long n)
    {
 
        // Store the number of setbits
        int count = 0;
 
        while (n > 0) {
 
            // Update the value of n
            n = n & (n - 1);
 
            // Update the count
            count++;
        }
 
        // Return the total count
        return count;
    }
 
    // Function to find the Nth term of
    // the Ruler Function Series
    static void findNthTerm(int N)
    {
 
        // Store the result
        int x = setBits(N ^ (N - 1));
 
        // Print the result
        Console.WriteLine(x);
    }
 
    // Driver Code
    public static void Main(string[] args)
    {
        int N = 8;
 
        findNthTerm(N);
    }
}
 
// This code is contributed by ukasp.

Javascript




<script>
 
// Javascript program for the above approach
 
// Function to count the number
// of set bits in the number N
function setBits(n)
{
     
    // Store the number of setbits
    var count = 0;
 
    while (n > 0)
    {
         
        // Update the value of n
        n = n & (n - 1);
 
        // Update the count
        count++;
    }
 
    // Return the total count
    return count;
}
 
// Function to find the Nth term of
// the Ruler Function Series
function findNthTerm(N)
{
     
    // Store the result
    var x = setBits(N ^ (N - 1));
 
    // Print the result
    document.write(x);
}
 
// Driver code
var N = 8;
     
findNthTerm(N);
 
// This code is contributed by Khushboogoyal499
 
</script>
Output: 
4

 

Time Complexity: O(log N)
Auxiliary Space: O(1)




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