Consider a function $\pi_{C}(x)$ that counts the amount of numbers less than or equal to $x$ in a given Collatz sequence.
If the Collatz conjecture is false because a Collatz sequence has been found with values that increase without bound, what statements can be made about the asymptotic behavior of $\pi_{C}(x)$ as $x \rightarrow \infty $ for that sequence?
Some comments:
If the Collatz conjecture is true, then the asymptotic behavior of this function for any Collatz sequence must be constant: $\pi_{C}(x) = O(1)$ as $x \rightarrow \infty$.
If the Collatz conjecture is false because a closed loop in a Collatz sequence has been found, then $\pi_{C}(x)$ will necessarily jump to infinity at some finite number for that sequence.