# Nth non-Square number

Given n, find the nth number which is not a perfect square among natural numbers (1, 2, 3, 4, 5, 6, … )

Examples:

Input : 3
Output : 5
First three non-square numbers are 2, 3
and 5

Input : 5
Output : 7

Input : 16
Output : 20


## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Looking at the problem statement we can come up to a straight-forward brute-force approach. We can start from n = 1, and start to check if each of them is a perfect square or not. So we can come up to the nth non-square number.

However, the above approach is very slow as it searches for each in every number smaller than the target each time.

We can observe that the series under consideration is 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, … .
We can come upto the constant time formula for the nth number in this sequence, by inspection.
.

The correctness of the formula can be proved by the Principle of Mathematical Induction.

The implementation of the above formula is given below.

 // CPP program to find n-th non-square number.  #include     using namespace std;     // function to find the nth Non-Square Number  int findNthNonSquare(int n)  {      // conversion from int to long double is      // necessary in order to preserve decimal       // places after square root.      long double x = (long double)n;         // calculating the result      long double ans = x + floor(0.5 + sqrt(x));         return (int)ans;  }     // Driver code  int main()  {      // initializing the term number      int n = 16;         // Print the result      cout << "The " << n << "th Non-Square number is ";      cout << findNthNonSquare(n);         return 0;  }

 // Java program to find  // n-th non-square number.  import java.io.*;  import java.util.*;  import java.lang.*;     class GFG  {         // function to find the  // nth Non-Square Number  static int findNthNonSquare(int n)  {      // conversion from int to       // long double is necessary      // in order to preserve decimal       // places after square root.      double x = (double)n;         // calculating the result      double ans = x + Math.floor(0.5 +                        Math.sqrt(x));         return (int)ans;  }     // Driver code  public static void main(String[] args)  {      // initializing      // the term number      int n = 16;         // Print the result      System.out.print("The " + n +                        "th Non-Square number is ");      System.out.print(findNthNonSquare(n));  }  }

 # Python3 program to find n-th   # non-square number.  import math     # function to find the nth  # Non-Square Number  def findNthNonSquare(n):         # conversion from int to long       # double is necessary in order       # to preserve decimal places      # after square root.      x = n;         # calculating the result      ans = x + math.floor(0.5 + math.sqrt(x));         return int(ans);     # Driver code     # initializing the term number  n = 16;     # Print the result  print("The", n, "th Non-Square number is",                        findNthNonSquare(n));     # This code is contributed by mits

 // C# program to find  // n-th non-square number.  using System;     class GFG  {         // function to find the  // nth Non-Square Number  static int findNthNonSquare(int n)  {      // conversion from int       // to long double is       // necessary in order      // to preserve decimal       // places after square       // root.      double x = (double)n;         // calculating the result      double ans = x + Math.Floor(0.5 +                        Math.Sqrt(x));         return (int)ans;  }     // Driver code  public static void Main()  {      // initializing      // the term number      int n = 16;         // Print the result      Console.Write("The " + n +                     "th Non-Square " +                         "number is ");      Console.Write(findNthNonSquare(n));  }  }     // This code is contributed   // by anuj_67.

 

Output:

The 16th Non-Square number is 20


Time Complexity .
Space Complexity

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Improved By : vt_m, Mithun Kumar