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:
- 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.
- 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"?