I found a sequence in OEIS (A180076) where a subsection is similar (but not entirely) to trajectory of $27$ in Collatz function. In the comments section I read: "Permutation of the natural numbers with inverse A180077;"
Here is a slice of the sequence
$\{ \mathbf{27}, 82, 41, 124, 31, 94, 47, 142, 71, 214, \mathbf{53},...\}$
But if I start with 27 of the formal collatz function:
$\{ \mathbf{27}, 82, 41, 124, \mathbf{62}, 31, 94, 47, 142, 71, 214, \mathbf{71},...\}$
How is the (topmost sequence) permutation generated? I found it hard to understand how it works by that page. I see there is one missing number ($62$) in that sequence and the last value is also different. Seems the last value is divided by $2$ when it is odd. It is of curiousity that looks so similar to a sequence that i supposed to be a permutation of the natural numbers with inverse. Is it because of the $3n+1$ formula that these are similar? Also what are the similarities by this sequence and the sequence generated by:
$f(n)=\{3n+1 \}\quad \text{if odd or} \quad \{n/2\} \quad \text{if even} $ ?