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Homework Statement
Say we have an infinite sequence of natural numbers A such that no K subsequences can be found adjacent such that the average of the elements in any subsequence is equal for all K subsequences. Sorry about my poor description, an example would be that {2, 3, 4, 1} wouldn't work for K=2 because {2, 3} and {4, 1} are adjacent and both their averages are 5/2. {2, 3, 10, 4, 1} would work however because {2, 3} and {4, 1} are no longer adjacent. Anyway, my question is: which sequence that follows this has the lowest sum for K? If that's too general, then which for K=2? Honestly, any information on the behavior of this sequence would be great.
Homework Equations
None that I know of.
The Attempt at a Solution
It looks like for k=2 the lowest A is {1, 2, 1, 3, 1, 2, 1, 4 ...} but I have no idea how to prove that it is.