The rules are simple:

Take any number $n$. If $n$ is even divide it by two, if $n$ is odd triple it and subtract one. Repeat indefinitely. (Note that this is a variation, in the original Collatz conjecture you add one.)

Like the original Collatz conjecture seems to always get to one, this variation always seems to get to either $1, 7$ or $17$. I checked that it does for initial values of $n$ up to 443 million.

Can you give a number that doesn't get to $1$, $7$ or $17$, or if not, at least show that such number exists?

  • $\begingroup$ $\{5,14,7,20,10\}$ is the cycle, and it clearly goes through seven. $\endgroup$ Sep 4 '16 at 18:26
  • $\begingroup$ @AlgorithmsX The question is to find a trajectory that never passes through 1, 7, or 17. $\endgroup$
    – user326210
    Sep 4 '16 at 18:37
  • 3
    $\begingroup$ @user326210 there was a comment which suggested n = 5 and the comment from AlgorithmsX is an answer to that now deleted comment $\endgroup$
    – pavelkomin
    Sep 4 '16 at 18:40
  • 4
    $\begingroup$ I think mathematics isn't ready for this thing either (borrowing Erdős)... but I do suspect that any Collatz-like iteration has a finite number of cycles. I'm moving on from here. $\endgroup$ Sep 4 '16 at 18:54
  • 1
    $\begingroup$ Zero, negative numbers, for example. $\endgroup$
    – DaBler
    Dec 2 '19 at 9:59

Your function (let's call it $g: \mathbb Z \rightarrow \mathbb Z$) is just $g(x)=-f(-x)$ with $f$ being the normal Collatz function.

Your cycles are known, they correspond to $\{-1\rightarrow-2\}$, $\{-5\rightarrow-14\rightarrow-7\rightarrow-20\rightarrow-10\}$ and $\{-17\rightarrow-50\rightarrow-25\rightarrow-74\rightarrow-37\rightarrow-110\rightarrow-55\rightarrow-164\rightarrow-82\rightarrow-41\rightarrow-122\rightarrow-61\rightarrow-182\rightarrow-91\rightarrow-272\rightarrow-136\rightarrow-68\rightarrow-34\}$ for $f$.

It is unknown whether the Collatz hypothesis is true on the naturals, even less is known about extensions. See also https://en.wikipedia.org/wiki/Collatz_conjecture#Iterating_on_all_integers


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