I'm working on a problem and I would appreciate some help.
The problem is to determine the set of all real numbers $ r $ for which there exists an infinite sequence of positive integers $ a_1, a_2, ... $ that satisfies the following properties: No number appears more than once in the sequence, The sum of any two distinct terms in the sequence is never a power of two, for all $n$ positive integers, such that $ a_n < r\times n$.
Here is my attempt at solving the problem: I suspect the answer is $ r\geq2$. I have considered constructing the sequence $ a_n $ using induction. The idea is to ensure that the sums of any two different terms do not form a power of two, while also satisfying the growth constraint. For example, I started by considering sequences where each term grows linearly with $ n $, such as $ a_n = 2n$.Starting with a base sequence, I would attempt to show that it's always possible to find a next term $ a_{n+1} $ such that the sequence continues to meet the properties for $ r\geq2$.
However I am unsure how to formally prove this for all $r\geq2$ and would appreciate guidance or a facilitated proof.