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If I have points on a line at positions (0), (1), (3), and (7) the set of intervals between any two points is the set of integers (1,2,3,4,6,7). In-general, a set of N unique integers can be represented by approximately log(N) points on a line. However, I want to represent a contiguous set of integers (1 to N) this way, which seems to be a bit more challenging. So far I have only come up with sequences that solve this problem with around 2*√N points and I think that might be the hard lower bound.

If I loosen the requirements of the problem to say that I want to represent all integers from 1 to N or their multiples, more efficient sequences are possible. In fact, this can always be done with just two points: (0) and (N!). However, calculating N! is slower than O(nlogn) which is slower than the previous solution.

Is there an algorithm that can generate a sequence of points on a line that solves that second problem and is faster than O(√N)?

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