2020 CIME II Problems/Problem 5
A positive integer is said to be -consecutive if it can be written as the sum of consecutive positive integers. Find the number of positive integers less than that are either -consecutive or -consecutive (or both), but not -consecutive.
The smallest -consecutive positive integer is , and every multiple of greater than is also -consecutive, with the last one less than being . There are -consecutive positive integers less than . The smallest -consecutive positive integer is , and the largest one less than is . There are of these. However, we counted twice and we only wanted to count them once, so we subtract from our total, giving us a total of that are either - or -consecutive. The ones that are also -consecutive are for a total of integers to be removed. The answer is .
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