Bizarre Bag of Balls

Two bags of balls each contain balls numbered 1,2,,n1, 2, \ldots , n. For how many values of nn such that 1n5001 \leq n \leq 500, does there exist a kk such that the probability of getting a sum of n+1n+1 when a ball is selected from each bag is equal to the probability that the sum is k\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.

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