Given an integer N, the task is to find the Nth pure number.
A pure number has to satisfy three conditions:
1) It has an even number of digits.
2) All digits are either 4 or 5.
3) And the number is a palindrome.
The Pure number series is: 44, 55, 4444, 4554, 5445, 5555, 444444, 445544, 454454, 455554 and so on.
Examples:
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.
Approach: We will assume that 2 numbers make one single block. For each block, there is a 2block number of pure numbers. For pure numbers with 1 block, there are 21 pure numbers; for numbers with 2 blocks, there are 22 numbers, and so on.
- Pure numbers starting with 4, start at position 2block – 1 for example, 4444 is at (22 -1 = 3) which means it is, at third position in the series.
- Pure numbers starting with 5 starts at position 2block + 2(block-1) -1 for example, 5555 is at (2^2 + 2^1 -1 =5) which means it is at the fifth position in the series.
A pure number in a block is essentially sandwiched between two 4’s or 5’s and is a combination of all previous block numbers. To understand it better, let’s consider the example below:
- The first pure number is 44 and the second pure number is 55.
- 4444 (“4″+ “44” + “4”) 44 from previous block
- 4554 (“4″+ “55” + “4”) 55 from previous block
- 5445 (“5″+ “44” + “5”) 44 from previous block
- 5555 (“5″+ “55” + “5”) 55 from previous block
This pattern repeats for all the numbers in the series.
Below is the implementation of the above approach:
C++
#include<bits/stdc++.h>using namespace std;// CPP program to find// the Nth pure num// Function to check if it// is a power of 2 or notbool 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) -1bool 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 numberstring 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 Codeint 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 numberimport 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); }} |
Python3
# Python program to find# the Nth pure num# Function to check if it# is a power of 2 or notimport mathdef isPowerOfTwo(N): number = math.log(N)/math.log(2) checker = math.floor(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) -1def isSeriesFour(N, digits): upperBound = math.floor(math.pow(2, digits) + math.pow(2, digits - 1)-1) lowerBound = math.floor(math.pow(2, digits)-1) return (N >= lowerBound) and (N < upperBound)# Method to find pure numberdef getPureNumber(N): numbers = ["" for i in range(N + 1)] numbers[0] = "" blocks = 0 displacement = 0 # Iterate from 1 to N for i in range(1,N + 1): # Check if number is power of two if (isPowerOfTwo(i + 1)): blocks = blocks + 1 if (isSeriesFour(i, blocks)): displacement = math.floor(math.pow(2, blocks - 1)) # Distance to previous # block numbers numbers[i] = f"4{numbers[i - displacement]}4" else: displacement = math.floor(math.pow(2, blocks)) # Distance to previous # block numbers numbers[i] = f"5{numbers[i - displacement]}5" return numbers[N]# Driver CodeN = 5pure = getPureNumber(N)print(pure)# This code is contributed by shinjanpatra |
C#
// C# program to find// the Nth pure numberusing 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
<script>// Javascript program to find// the Nth pure num// Function to check if it// is a power of 2 or notfunction isPowerOfTwo(N){ let number = Math.log(N)/Math.log(2); let checker = Math.floor(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) -1function isSeriesFour(N, digits){ let upperBound = Math.floor(Math.pow(2, digits) + Math.pow(2, digits - 1)-1); let lowerBound = Math.floor(Math.pow(2, digits)-1); return (N >= lowerBound) && (N < upperBound);}// Method to find pure numberfunction getPureNumber(N){ let numbers = new Array(N + 1); numbers[0] = ""; let blocks = 0; let displacement = 0; // Iterate from 1 to N for (let 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 = Math.floor(Math.pow(2, blocks - 1)); // Distance to previous // block numbers numbers[i] = "4" + numbers[i - displacement] + "4"; } else { displacement = Math.floor(Math.pow(2, blocks)); // Distance to previous // block numbers numbers[i] = "5" + numbers[i - displacement] + "5"; } } return numbers[N];}// Driver Codelet N = 5;let pure = getPureNumber(N);document.write(pure + "<br>");// This code is contributed by _saurabh_jaiswal</script> |
Output:
5445
Time Complexity: O(NlogN) as using pow function in a loop
Auxiliary Space: O(N)
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