Note: I will use the abbreviation RCF for the Reduced Collatz Function.
The arrangement of certain specific digits produce a particular pattern on the next iteration of the Reduced Collatz Function. Before dwelving into the pattern, lets have a recap of the function.
The RCF applies only to odd number, and is defined as: $$R(n)=\frac{3n+1}{2^r}$$ where $r$ denotes the largest exponent of the largest power of two that divides $3n+1$. Denote $i$ as the $i$'th iteration of the function. Define also $r$ and $n\in\mathbb{Z_0^+}$.
Lets do some examples (the initial numbers are explicitly chosen):
\begin{array}\hline R(23)\to35\to53\to5\to1 \\ R(27)\to...\to395\to593\to... \\ R(359)\to539\to809\to607\to... \\ R(3559)\to5339\to8009\to6007\to... \\ R(35559)\to53339\to80009\to60007\to... \\ \end{array}
Note these are just observations of mine. You see the patterns? Digits $3$, $5$ and $9$ are noticeable. $5$ and $3$ swapped places. $9$ is at the same digit-position. However, in $R^i(n)$, the digits $9$ and $5$ can't be the first digit or else $R^{i+1}(n)$ will not produce the desired effect.
Q: Im sure these patterns have been researched on, any literature about these specifics?
When $R^i(n) = 539$ then $n=359$ for $i-1$. This particular iteration of the function looks like it has an inverse. There might be examples that I am not aware of that work for other numbers but I have not found any. I've seen these numbers in other sequences, namely prime numbers.
The following formula gives primes where $n$ is defined by a sequence that can be found here:
$$\frac{320\cdot10^n+31}{9}$$
Set $n=7$
This results in the number: $355555559$. Whatever $n$ we set, the count of $5$'s between $3$ and $9$ will be same as $n$.
If we put this into the RCF as input? Well we get $R^i(355555559)\to R^{i+1}(533333339)$.
There does not seem to be a relation between the number of digits between $5$ and $9$ and the primes in the collatz function, but the most significant and least significant digit might have?
Q: Are there any core math on this subject of the Primes and Collatz? I am open to listen.