Bizarre Bag of Balls

Probability Level 4

Two bags of balls each contain balls numbered $1, 2, \ldots , n$ . For how many values of $n$ such that $1 \leq n \leq 500$ , does there exist a $k$ such that the probability of getting a sum of $n+1$ when a ball is selected from each bag is equal to the probability that the sum is $\leq k$ when a ball is selected from each bag?

Details and assumptions

Since a ball is selected from each bag, there are 2 balls selected, and we are looking at the sum of these values.