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I was playing with the Collatz conjecture and decided to check it for the first 100,000,000 natural numbers. I divided this range in equal intervals of 100,000 and calculated the average number of steps to reach 1 in each one of these intervals. By graphing these averages, I got the following result enter image description here

Which seems to show a very clear logarithmic behavior. By doing a least squares regression (shown in blue), I derived that the average number of steps ($S$) for a number $n$ to reach 1 is $$S\approx10.45\ln(n)-3$$ Is there a way to derive this logarithmic relation?

ordptt
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    Well, of course not; if we knew anything like this it would prove the Collatz conjecture! It would be interesting to see even a heuristic argument, though. – Qiaochu Yuan Oct 16 '22 at 04:39
  • I believe that a heuristic or probabilistic argument would be the case here, as it is about an average behavior. – ordptt Oct 16 '22 at 04:43
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    Well, sometimes it's possible to prove, rigorously, that some function on the positive integers has some asymptotic behavior when averaged. The point is that if the average is finite then all of the numbers being averaged are finite, so if one could prove this average rigorously that immediately implies that all Collatz orbits are finite. – Qiaochu Yuan Oct 16 '22 at 04:45
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    The fluctuations are however heavy. – Peter Oct 16 '22 at 16:41
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    Each odd integer in a Collatz sequence is on average ~3/4 of the previous odd integer using a probabilistic argument. Going from one odd integer to the next takes 3 steps on average (1 multiply add, 2 divides) so the average number of steps to get to 1 starting with an odd integer should be about -3ln(n)/ln(3/4)) ~ 10.428ln(n). – Eric Shumard Oct 16 '22 at 17:21
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    I wanted reproduce your statistics, but did not know, what you were counting. Odd steps - (type: $3x+1$or type $(3x+1)/2$ ) ? Even steps ? their sum? please insert a statement in your question. Moreover, a logarithmic scale for x-axis gave a better (?) pattern. Even more, if I used the logarithm of the meanvalue-of-category instead of the lower bound of the category this seemed to come out better... – Gottfried Helms Oct 16 '22 at 18:48

2 Answers2

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The function $S(n)$ can have its value estimated by considering the expected value $s(n)$, which corresponds to the average number of steps near $n$. For some "random" $n$, there is a $1/2$ probability that it is even or odd, therefore, $$s(n)=\frac{1}{2}\left(1+s\left(\frac{n}{2}\right)\right)+\frac{1}{2}\left(2+s\left(\frac{3n+1}{2}\right)\right)$$ If $s(n)=b\ln(n)$, then $$b\ln(n)=\frac{1}{2}\left(1+b\ln\left(\frac{n}{2}\right)\right)+\frac{1}{2}\left(2+b\ln\left(\frac{3n+1}{2}\right)\right)$$ Solving for $b$, $$b=\frac{3}{\ln\left(\frac{4}{3+1/n}\right)}$$ which, for $n\rightarrow \infty$, $$b=\frac{3}{\ln\left(4/3\right)}\approx10.43$$ Which matches the value found by the regression.

Note that $s(n)=b\ln(n)+c$ is also a solution to the first equation for any constant $c$, and the value of $b$ does not change as all terms with $c$ are cancelled out. Determining its value is addressed in another post I made.

ordptt
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This is not an answer but an extended comment

As @Peter already observed, the fluctuations are heavy. So I tried to reproduce your statistic, hoping that the $S=$ number-of-even-steps in the transformation rule "$T(x):$ $3x+1$ when $x$ is odd, $x/2$ when $x$ is even" is what you've evaluated.

I only could go up to $10 \; 000 \; 000$ values at all, so I show the scattering in bins of size=1000 (blue), size=10000 (green), size=100000 (red). (The trendline is in red, based only on the 1000-size bins).
Different from your picture I scale the x-axis logarithmically and get a near linear trend. However, the fluctuations increase with the index of the bins - independently, how large the bins are: an interesting observation on its own.

image1

After the fluctuations increase with the index-number of the bins, I think, a variable/an increasing binsize might be appropriate. Here I show the average number of even steps, when the size of bins are doubled with the index. Moreover, I took as x-coordinate the geometric mean value of the bin-bounds. See this:

image2

Here we get a nice nearly linear trend. The slope $7.216$ divided by $\ln(2)$ gives about $10.4104$ which agrees roughly with the slope that has been found in the OP.
Again: the increasing fluctuations over the bins of equal size seem a remarkable effect to me.

Gottfried Helms
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  • I posted another question which links and shows the solution to this one, if you are interested. – ordptt Oct 21 '22 at 03:34
  • @ordptt - yes, I've already seen this; but I've so far no real idea for the consturction of the constant $c$. When I change the set of points-to-regress-on I only observed that it comes out much variable and my guess is it would tend to zero for larger and larger $n$ - but I've no idea of its true structure. I'm not going deeper in this, but am curious as to what you'll find... so, thanks for your notification! – Gottfried Helms Oct 21 '22 at 08:49