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Everybody knows or at least heard about Collatz or $3x+1$ conjecture.

Let us now define something like:

Definition 1: Number $m \in \mathbb{N}$ is called $k-Collatz$ number if in its sequence leading to $1$ (including 1) following the Collatz rule there are less or equal $k$ odd numbers.

Definition 2: Number $m \in \mathbb{N}$ is called $Collatz$ number if it is k-Collatz for some $k\leq m$.

Example:

  1. Let $m=42$, then its Collatz sequence is $( 42, 21, 64, 32, 16, 8, 4, 2, 1 )$. So we have only $2$ odd numbers in it. Hence $42$ is a $2-Collatz$ number. And indeed it is a "$Collatz$" number.
  2. Let $m=27$. I will not write the whole sequence, but it has $112$ elements, and $42$ are odd. Hence $27$ is $42-Collatz$ and is a "non-Collatz-number".

After doing some computations

I find that “non-Collatz” numbers are $27$ and $31$. Are there any other "non-Collatz-numbers"?

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    $\begingroup$ Not below $10^7$, so I expect these two to be the only two. However, if the Collatz conjecture is false, there are infinitely many $\endgroup$ – vrugtehagel May 15 '17 at 14:26
  • $\begingroup$ @vrugtehagel If there is any other, I would of course like to know which it is. If there is no other, it will be great if we can show anything like proof of it. $\endgroup$ – Mesmerized student May 15 '17 at 14:26
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    $\begingroup$ If mathematicians through the ages haven't been able to prove that a number's Collatz sequence even ends, I'm not sure if we should expect a Math SE hobbyist to prove that there are no numbers with a finite amount of odd numbers in its Collatz sequence (which means it ends, so actually this statement is stronger than the Collatz conjecture) $\endgroup$ – vrugtehagel May 15 '17 at 14:33
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    $\begingroup$ Even if Collatz conjecture is false, it doesn't imply that there infinitely many non Collatz numbers. Suppose it has some non-trivial cycle, then only elements of this cycle would be non Collatz. Of course in most general way it is hard problem to determine this numbers, but may be under some conditions we can say at least something about those Collatz and non-Collatz number. @vrugtehagel $\endgroup$ – Mesmerized student May 15 '17 at 15:10
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    $\begingroup$ You might look at "stopping time" and "total stopping time" and record holders at Eric Roosendaal's site ericr.nl/wondrous/index.html He has a lot of properties of numbers under the Collatz-iteration and perhaps you even find a discussion of what you call "non-collatz". $\endgroup$ – Gottfried Helms May 16 '17 at 6:47
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Call the function $\text{OddHeight}(n)$ which counts the odd values in the Collatz-iteration of (odd) $n$ down to $1$. Then the simple picture of the first $20000$ odd $n$ on a logarithmic scale suggests very clearly, that you won't find any more cases where $\text{OddHeight}(n) \gt n$

picture

The magenta curve indicates $n = \text{OddHeight}(n)$. I've added two more lines to focus two general trends

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