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.
Input: N = 5
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
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:
Time Complexity: O(1)
Auxiliary Space: O(1)