Maximum number of distinct positive integers that can be used to represent N

Given an integer N, the task is to find the maximum number of distinct positive integers that can be used to represent N.

Examples:

Input: N = 5
Output: 2
5 can be represented as 1 + 4, 2 + 3, 3 + 2, 4 + 1 and 5.
So maximum integers that can be used in the representation are 2.



Input: N = 10
Output: 4

Approach: We can always greedily choose distinct integers to be as small as possible to maximize the number of distinct integers that can be used. If we are using the first x natural numbers, let their sum be f(x).

So we need to find a maximum x such that f(x) < = n.

1 + 2 + 3 + … n < = n
x*(x+1)/2 < = n
x^2+x-2n < = 0
We can solve the above equation by using quadratic formula X = (-1 + sqrt(1+8*n))/2.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the required count
int count(int n)
{
    return int((-1 + sqrt(1 + 8 * n)) / 2);
}
  
// Driver code
int main()
{
    int n = 10;
  
    cout << count(n);
  
    return 0;
}

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Java

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// Java implementation of the approach
class GFG
{
    // Function to return the required count
    static int count(int n)
    {
        return (int)(-1 + Math.sqrt(1 + 8 * n)) / 2;
  
    }
      
    // Driver code
    public static void main (String[] args) 
    {
        int n = 10;
      
        System.out.println(count(n));
    }
}
  
// This code is contributed by ihritik

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Python3

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# Python3 implementation of the approach
from math import sqrt
  
# Function to return the required count
def count(n) :
  
    return (-1 + sqrt(1 + 8 * n)) // 2;
  
# Driver code
if __name__ == "__main__" :
  
    n = 10;
  
    print(count(n));
  
# This code is contributed by AnkitRai01

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C#

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// C# implementation of approach
using System;
  
class GFG 
      
    // Function to return the required count
    public static int count(int n)
    {
        return (-1 + (int)Math.Sqrt(1 + 8 * n)) / 2;
    }
  
    // Driver Code
    public static void Main() 
    
        int n = 10;
      
        Console.Write(count(n)); 
    
  
// This code is contributed by Mohit Kumar

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Output:

4

Time Complexity: O(1)

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