# Finding the range of $r$ for a sequence avoiding power of two sums

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,
• $$a_n < rn$$ for each positive integer $$n$$.

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 $$a_n = 3n$$. 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.