Bad Tuples

A tuple of positive integers $(a, b, c)$ is said to be a *bad tuple* if $a, b, c$ are distinct, $a$ divides $b$, and $b$ divides $c$. For example, $(1, 2, 4)$ and $(7, 21, 42)$ are bad tuples while $(1, 2, 3)$ and $(2, 4, 4)$ are not. Chef has the $N$ integers from $1$ to $N$ with him. He wants to pick a [subset](https://en.wikipedia.org/wiki/Subset) of these numbers such that it doesn't contain