SIMON-MARAIS 2024 A2

A positive integer $n$ is tripairable if it is possible to partition the set $\{1, 2, \ldots, n\}$ into disjoint pairs such that the sum of the two elements in each pair is a power of 3. For example, 6 is tripairable because $\{1, 2, 3, 4, 5, 6\} = \{1, 2\} \cup \{3, 6\} \cup \{4, 5\}$ and

$$1 + 2 = 3^1, \quad 3 + 6 = 3^2 \quad \text{and} \quad 4 + 5 = 3^2$$

are all powers of 3.

How many positive integers less than or equal to 2024 are tripairable?