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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 $10^4$.

Collatz stopping time plot

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

Rd Basha
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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 $k=30$ e.g. k=30

You can try here SageMath

Collag3n
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  • Thanks! Can you link to a source? – Rd Basha Apr 14 '23 at 17:33
  • @Rd Basha, I don't know if this has already been studied. I just used the (unproven) stopping time formula $st\sim i\cdot log_26 + \log_2n$ and the above plot in logarithmic form (using $\log_2n$ instead of $n$ axis) and transposed to a non-logarithmic plot – Collag3n Apr 14 '23 at 18:09
  • Do you have a source for the unproven stopping time formula? Where does it come from? Just numerical fitting? – Rd Basha Apr 14 '23 at 18:38
  • @Rd Basha, Same thing. I am not sure it exists somewhere (Perhaps Lagarias paper), but it comes from $3^i\Big(1+\frac{1}{3n_0}\Big)\Big(1+\frac{1}{3n_1}\Big)\cdots\Big(1+\frac{1}{3n_{i-1}}\Big)=\frac{n_i}{n_0}\prod_{j=1}^i2^{k_j}$ that you can find here: https://math.stackexchange.com/questions/3212129/a-possible-way-to-prove-non-cyclicity-of-eventual-counterexamples-of-the-collatz/3215501#3215501 where you just take the $log_2$ of both sides and you end up with $k_1+k_2+k_3+\ldots+k_i=i\cdot \log_23+\log_2{n_0}+\log_2\prod_{j=0}^{i-1}\Big(1+\frac{1}{3n_j}\Big)$ ... – Collag3n Apr 14 '23 at 18:52
  • ... then with $k$ the sum of $k_j$: $k=\lceil i\cdot \log_23+\log_2{n_0} \rceil$ knowing that $st=k+i$. It relies on the fact that the "+1" terms never exceed $3^i n_0$ – Collag3n Apr 14 '23 at 18:53
  • but the last sentence of the above comment is unproven – Collag3n Apr 14 '23 at 20:25