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

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

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++

 `// C++ implementation of the approach ` `#include ` `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; ` `} `

Java

 `// 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 `

Python3

 `# 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 `

C#

 `// 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 `

Output:

```4
```

Time Complexity: O(1)

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