📜  找到第 N 个纯数

📅  最后修改于: 2021-09-06 05:22:13             🧑  作者: Mango

给定一个整数N ,任务是找到第 N 个纯数。

例子:

Input: 5
Output: 5445
Explanation: 
5445 is the 5th pure number in the series.

Input: 19
Output: 45444454
Explanation: 
45444454 is the 19th pure number in the series. 

方法:我们假设 2 个数字构成一个块。对于每个区块,有 2 个区块编号的纯数字。对于1个块的纯数,有2个1纯数;对于有 2 个块的数字,有 2 2 个数字,依此类推。

  • 以 4 开头的纯数字,从位置2开始例如,4444 位于 (2 2 -1 = 3) 处,这意味着它在系列中的第三个位置。
  • 以 5 开头的纯数字从位置2 block + 2 (block-1) -1 开始,例如,5555 在 (2^2 + 2^1 -1 =5) 这意味着它在系列中的第五个位置。

块中的纯数字本质上夹在两个 4 或 5 之间,并且是所有先前块编号的组合。为了更好地理解它,让我们考虑以下示例:

  • 第一个纯数是 44,第二个纯数是 55。
  • 4444(“4”+“44”+“4”)来自前一个块的44
  • 4554 (“4”+“55”+“4”) 55 来自前一个区块
  • 5445 (“5”+“44”+“5”) 44 来自前一个区块
  • 5555 (“5”+“55”+“5”) 55 来自前一个区块

该模式对系列中的所有数字重复。
下面是上述方法的实现:

C++
#include
using namespace std;
 
// CPP program to find
// the Nth pure num
 
// Function to check if it
// is a power of 2 or not
bool isPowerOfTwo(int N)
{
    double number = log(N)/log(2);
    int checker = int(number);
    return number - checker == 0;
}
 
// if a number belongs to 4 series
// it should lie between 2^blocks -1 to
// 2^blocks + 2^(blocks-1) -1
bool isSeriesFour(int N, int digits)
{
    int upperBound = int(pow(2, digits)+pow(2, digits - 1)-1);
    int lowerBound = int(pow(2, digits)-1);
    return (N >= lowerBound) && (N < upperBound);
}
 
// Method to find pure number
string getPureNumber(int N)
{
    string numbers[N + 1];
 
    numbers[0] = "";
 
    int blocks = 0;
    int displacement = 0;
 
    // Iterate from 1 to N
    for (int i = 1; i < N + 1; i++) {
 
        // Check if number is power of two
        if (isPowerOfTwo(i + 1)) {
            blocks = blocks + 1;
        }
 
        if (isSeriesFour(i, blocks)) {
            displacement
                = int(pow(2, blocks - 1));
 
            // Distance to previous
            // block numbers
            numbers[i] = "4" + numbers[i - displacement] + "4";
        }
 
        else {
 
            displacement = int(pow(2, blocks));
 
            // Distance to previous
            // block numbers
            numbers[i] = "5" + numbers[i - displacement] + "5";
        }
    }
 
    return numbers[N];
}
 
// Driver Code
int main()
{
    int N = 5;
 
    string pure = getPureNumber(N);
 
    cout << pure << endl;
}
 
// This code is contributed by Surendra_Gangwar


Java
// Java program to find
// the Nth pure number
 
import java.io.*;
 
class PureNumbers {
 
    // Function to check if it
    // is a power of 2 or not
    public boolean isPowerOfTwo(int N)
    {
        double number
            = Math.log(N) / Math.log(2);
        int checker = (int)number;
        return number - checker == 0;
    }
 
    // if a number belongs to 4 series
    // it should lie between 2^blocks -1 to
    // 2^blocks + 2^(blocks-1) -1
    public boolean isSeriesFour(
        int N, int digits)
    {
        int upperBound
            = (int)(Math.pow(2, digits)
                    + Math.pow(2, digits - 1)
                    - 1);
        int lowerBound
            = (int)(Math.pow(2, digits)
                    - 1);
        return (N >= lowerBound)
            && (N < upperBound);
    }
 
    // Method to find pure number
    public String getPureNumber(int N)
    {
        String[] numbers
            = new String[N + 1];
 
        numbers[0] = "";
 
        int blocks = 0;
        int displacement = 0;
 
        // Iterate from 1 to N
        for (int i = 1; i < N + 1; i++) {
 
            // Check if number is power of two
            if (isPowerOfTwo(i + 1)) {
                blocks = blocks + 1;
            }
 
            if (isSeriesFour(i, blocks)) {
                displacement
                    = (int)Math.pow(
                        2, blocks - 1);
 
                // Distance to previous
                // block numbers
                numbers[i]
                    = "4"
                      + numbers[i - displacement]
                      + "4";
            }
            else {
 
                displacement
                    = (int)Math.pow(
                        2, blocks);
 
                // Distance to previous
                // block numbers
                numbers[i]
                    = "5"
                      + numbers[i - displacement]
                      + "5";
            }
        }
 
        return numbers[N];
    }
 
    // Driver Code
    public static void main(String[] args)
        throws Exception
    {
        int N = 5;
 
        // Create an object of the class
        PureNumbers ob = new PureNumbers();
 
        // Function call to find the
        // Nth pure number
        String pure = ob.getPureNumber(N);
 
        System.out.println(pure);
    }
}


C#
// C# program to find
// the Nth pure number
using System;
  
class PureNumbers {
  
    // Function to check if it
    // is a power of 2 or not
    public bool isPowerOfTwo(int N)
    {
        double number
            = Math.Log(N) / Math.Log(2);
        int checker = (int)number;
        return number - checker == 0;
    }
  
    // if a number belongs to 4 series
    // it should lie between 2^blocks -1 to
    // 2^blocks + 2^(blocks-1) -1
    public bool isSeriesFour(
        int N, int digits)
    {
        int upperBound
            = (int)(Math.Pow(2, digits)
                    + Math.Pow(2, digits - 1)
                    - 1);
        int lowerBound
            = (int)(Math.Pow(2, digits)
                    - 1);
        return (N >= lowerBound)
            && (N < upperBound);
    }
  
    // Method to find pure number
    public string getPureNumber(int N)
    {
        string[] numbers
            = new string[N + 1];
  
        numbers[0] = "";
  
        int blocks = 0;
        int displacement = 0;
  
        // Iterate from 1 to N
        for (int i = 1; i < N + 1; i++) {
  
            // Check if number is power of two
            if (isPowerOfTwo(i + 1)) {
                blocks = blocks + 1;
            }
  
            if (isSeriesFour(i, blocks)) {
                displacement
                    = (int)Math.Pow(
                        2, blocks - 1);
  
                // Distance to previous
                // block numbers
                numbers[i]
                    = "4"
                      + numbers[i - displacement]
                      + "4";
            }
            else {
  
                displacement
                    = (int)Math.Pow(
                        2, blocks);
  
                // Distance to previous
                // block numbers
                numbers[i]
                    = "5"
                      + numbers[i - displacement]
                      + "5";
            }
        }
  
        return numbers[N];
    }
  
    // Driver Code
    public static void Main()
    {
        int N = 5;
  
        // Create an object of the class
        PureNumbers ob = new PureNumbers();
  
        // Function call to find the
        // Nth pure number
        string pure = ob.getPureNumber(N);
  
        Console.Write(pure);
    }
}
 
// This code is contributed by chitranayal


Javascript


输出:
5445