Alice and Bob own an integer N. Neither of them is happy with it. Last night they went to a cocktail party and found out that another couple had the exact same integer, so they are getting a new one.
Bob wants to impress the other couple, so he thinks the new integer has to be strictly larger than N.
Alice is fond of one specific integer k. She therefore thinks that whatever integer they pick, it must be possible to write it as a sum of k distinct powers of 2.
Bob is also a cheapskate and wants to spend as little money as possible. The price of an integer grows with its size, so he wants the smallest integer that satisfies both conditions.