标尺函数系列的第N个术语

给定正整数N ，任务是找到标尺函数系列的^第N^个项。

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

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

例子：

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

Input: N = 8
Output: 4

天真的方法：解决给定问题的最简单方法是生成序列，直到第N^个项，然后打印第N^个项。
时间复杂度： O(N)
辅助空间： O(N)

高效方法：还可以根据以下观察结果对上述方法进行优化，即观察到标尺函数系列中的^第N^个元素等于(N ^(N – 1))中的设置位数，如下所示：

• 0和1 = 1的按位XOR设置位的数目
• 1和2的按位XOR设置位的位数= 2
• 2和3的按位XOR设置位的位数
• 3和4的按位XOR设置位的位数= 3
• 4和5的按位XOR设置位的位数= 1
• 5和6的按位XOR设置位的位数= 2
• 6和7的按位XOR设置位的位数= 1
• 7和8的按位XOR设置位的位数= 4
• 等等…

因此，根据上述观察，标尺函数系列的^第N^个项由N与(N – 1)的按位XOR给出。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
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) {
 
        // Upate 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)
    {
         
        // Upate 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

输出：

4

时间复杂度： O(log N)
辅助空间： O(1)