Collatz stopping time curves

Question

We define the Collatz stopping-time of an integer $n$ to be the number of iterations of the Collatz function on $n$ untill we reach $1$. A corollary of the Colatz conjecture is that this time is finite for all $n$.

This is a plot of the stopping-time for numbers up to $10^4$.

The plot seems to consist of a family of decreasing curves. Is there a known formula for these curves?

Answer 1

considering that the stopping time=$\lceil \log_2(6^i\cdot n)\rceil$ (with $i$ the number of odd steps), you should have these k-family of curves: $$f(k,n)=\log_{\frac{8}{9}}(6^{5-k}n^5)$$

e.g. with $k=30$

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