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(This is meant as a comment to @JoshuaZ 's answer.) Here is a list of found cycles for a lot of $m \ne 3$ in the $m x+1$ - generalization. (I've not done any analysis about divergent sequences here, …

commen: this is a copy of my MSE-answer which I linked to by the other comment. Because of the existent discussion in the comments here I thought today, it would be convenient to have it explicitely h …

Lagarias' bibliographies give some references from where you might search forward. For instance see this This is from the year-2004 version; Lagarias updated this up to version 2011; the Lagarias bib …

The article of L. Szalay as mentioned by user Random, gives the solution. We can reformulate $$ \begin{array} {} \{b^2 - \{b-1\}_2 \}_2 & \overset{?}= 1 &&& (1.1)\\ b^2 - \{b-1\}_2 & \overset{?}= …

For the complete extraction of the factor $p$ and its powers from a natural number $n$ let's define the notation $$ \{n\}_p := { n \over p^{\nu_p(n)}} \tag 1$$ $ \qquad \qquad $ Here $\nu_p(n)$ means …

A list of predecessors as mentioned in my comment. I document pairs of $(m,n)$ for consecutive $m$ and their 1-step predecessors $n$ such that $f(n)=m$. The value $n=0$ indicates, that $m$ has n …

The expression you've coined reflects the orbit of one initial number $s$ towards $1$ by (the Syracuse-notation of) the Collatz-transformation. A perhaps better expression for this is $$ a_{N+1} = \sm …

this is a copy of my answer in MSE Let me rewrite the letters for your variables due to my long-time practice. I usually write $N$ for $c$ , $S$ for $c+b$ such that $S = \lceil N \cdot \log_2(3) \rcei …

This is not an answer, just an illustration of the comments of Asutora and Wojowu I looked at the "partial densities" of the sequence in the $24$ subsequences according to the residues $\pmod{24}$ of …

This is not an answer, just a comment to give another display of the results for smaller $n$. It is produced using Pari/GP and its inherent "ispseudoprime()" function. I document the (odd) initial num …

For the general focus of the question: "how many invariant sets" I don't remember any article dealing explicitely with this. Surely my readings are incomplete, but I also don't think that there is som …

The answers for b) and c) came out to be trivial and have likely nothing to to with Zsigmondy, so possibly I should retract my question. For the definition of a cycle for some $a_1=T_\rho(a_1;E …

Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll giv …

This is no answer, just some more illustrative material triggered by the numberlist of @Stefan Kohl. I consider the numbers $m$ from Stefan's list in base-4 representation. By that …

It might be a nice illustration of the general behaviour of increasing/decreasing by iterations from a purely statistical view. Consider some number odd number $a_0$ Then in the $mx+1$-problem, $a_1 = …