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The top voted answer in Is it possible to describe the Collatz function in one formula? looks like a really complicated function at first sight.

Let me restate the function in it's original form here; $$f(n)=\frac74n+\frac12+(-1)^{n+1} \left(\frac54n+\frac12\right)$$

I do not understand what is so special about this function when there are simpler non-modular functions than that.

Of course one author of another answer says, quote

..the way the Collatz function is written definitively influences how it's studied

is an acceptable (and somewhat good) thing to state, but the former author does not describe fully what the formula is all about, why are the fractions the way they are. When so many have accepted it as the "best" answer, then there must be something special about it or was it just a fad of the moment? Maybe someone could explain to me in deeper detail why this function is of interest.

  • Has anyone studied it in more detail?
  • What can be said about each term and their long term behaviour?
  • Why is it so popular?

https://math.stackexchange.com/q/1449997

Edit: Here are a few simpler looking functions:

  1. $C(n)=n-\lceil \frac{n(-1)^n}{2}\rceil$
  2. $f(n)=\frac{\lceil\frac{3n}{2}\rceil}{2+(-1)^n}$
  3. $f(n)=\lceil{2+(-1)^{n-1}\over2}n\rceil$
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    Why "this function" is of interest? You mean the specific form in your $f(x)$? Hardly anyone is interested in that form. As for interest in the Collatz conjecture: a 10-second Internet search will bring you to thousands of hits and over a dozen books on Collatz. Just note that one of the greatest mathematicians of our time, Paul Erdös, stated "Mathematics is not ready for such problems." – David G. Stork Aug 08 '19 at 20:25
  • What are the simpler formulas you have in mind? – Théophile Aug 08 '19 at 20:26
  • @DavidG.Stork Seems many people were interested in that form when $60$ people upvoted it. I just referred to that thread and asked all these questions based on that. –  Aug 08 '19 at 20:30
  • @Théophile I can get back to you on that since my notes are a bit messy atm. But I know there are a few variants that are simpler. –  Aug 08 '19 at 20:31
  • @Théophile $C(n)=n-\lfloor \frac{n(-1)^n}{2}\rfloor$ –  Aug 08 '19 at 20:37
  • And $f(n)=\frac{\lceil\frac{3n}{2}\rceil}{2+(-1)^n}$. I have some more, but I won't post all of them here. I just give a few examples of simpler looking functions. –  Aug 08 '19 at 20:42
  • One more: $f(n)=\lceil{2+(-1)^{n-1}\over2}n\rceil$ –  Aug 08 '19 at 20:49
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    Sometimes an answer is just a message, and the message was nicely sent – Collag3n Aug 09 '19 at 06:11
  • "Why is it so popular?" Similar problem: Consider the old JPEG format from 1991. The compression performance is poor, it suffers from blocking compression artifacts, there is almost no error resilience, etc. On the other hand, we have the JPEG 2000 format with superior performance, not suffering from artifacts, having having good error resilience, etc. So why is the old JPEG so popular? Because it appeared first! – DaBler Aug 09 '19 at 13:36

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I think I can answer the question "Why is it so popular?". People like formulas that involve only elementary operations: addition, multiplication, exponentiation.

When you have a modulo 2 expression in your formula, there is a trick to remove it and replace it with an exponential term: namely, $(-1)^n$. In this case, since the piecewise expressions are rather simple, we can use a simple strategy. We write our expression as: $$ f(n)= an+b + (-1)^n (cn+d) $$ Then for $n$ even, we get $f(n)=(a+c)n+(b+d)$, and for $n$ odd, we get $f(n)=(a-c)n+(b-d)$. Then we just have to solve for $a,b,c,d$. In the case of the Collatz function, we want $a+c=1/2$, $b+d=0$, $a-c=3$, $b-d=1$. It is easy to see that this gives the coefficients in the formula you refer to.

Sambo
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  • Yes, I allready knew the trick. But what about the simpler functions I recently added? What can be said about them in relation to that formula? –  Aug 08 '19 at 20:57
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    Nothing really; they express the same quantity. People just tend to think of floor and ceiling functions as less "nice". – Sambo Aug 08 '19 at 21:01
  • Additionally: for the extrapolation to the reals the $(-1)^n$ can be generalized to some formula containing the cosine/sine instead and still keep the simple shape. – Gottfried Helms Aug 09 '19 at 08:31