How slowly can a sequence with no power-of-two sums grow?
Source: Bundeswettbewerb Mathematik 2024, Round 2 - Problem 2
September 13, 2024
algebraalgebra proposedSequenceSequencesinequalities
Problem Statement
Determine the set of all real numbers for which there exists an infinite sequence of positive integers satisfying the following three properties:
(1) No number occurs more than once in the sequence.
(2) The sum of two different elements of the sequence is never a power of two.
(3) For all positive integers , we have .