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\sum_{i=1}^n \frac{1}{x_i} includes all natural numbers

Source: Norwegian Mathematical Olympiad 1995 - Abel Competition p3

February 11, 2020
Sumalgebranatural

Problem Statement

Show that there exists a sequence x1,x2,...x_1,x_2,... of natural numbers in which every natural number occurs exactly once, such that the sums ∑i=1n1xi\sum_{i=1}^n \frac{1}{x_i}, n=1,2,3,...n = 1,2,3,..., include all natural numbers.
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