Generalizing the Collatz $T_{3,+}(n) = \left\{ \begin{array} {cl}
{3n+1 \over 2} & \text{ when } n=2k+1 \\
\frac n{2^B} & \text{with maximal } B \gt 0 \text{ where } 2^B | n \end{array} \right.$
to
$\qquad T_{m,+}(n) $ with odd $m>3$ and $"+1"$ in the numerator $\qquad \qquad $ as well as to
$\qquad T_{m,-}(n) $ analoguously but with $-1$ in the numerator
then it is somehow trivial to show, that for all $m=2^k-1$ the transformation $T_{m,+}(1)=1$ exists and thus $T_{2^k-1,+}(1)=1$ allows the "trivial cycle" for all $k>1$ .
In the same way trivial is it that
$\qquad T_{2^k+1,-}(1)=(1)$ exists for all $k$, and thus we might say:
$\qquad \qquad $ For all $m \in \{3,7,15,31,63,...\}$ in $T_{m,+}$ we have the trivial cycle $[1,1,1,1...]$
$\qquad \qquad $ For all $m \in \{3,5,9,17,33,65,...\}$ in $T_{m,-}$ we have the trivial cycle $[1,1,1,1...]$
Of course it is equivalent to say $T_{m,-}(n)$ gives the same map for the positive $n$ as the map $T_{m,+}(n)$ gives for the negative $n$ and I prefer the latter interpretation - but leave it at the moment as it is.
The "systematic" $m=3$ and $m=5$ give a couple of additional "non-trivial" cycles. Let's denote the number of involved different odd numbers as length $N$, such that $T_{m,+}^{[N]}(n) = n$ and the number of involved divisions by $2$ as "sum-of-exponents" $S$.
The for $m=3$ the transformation $T_{3,-}(n)$ gives cycles
$\qquad \qquad [5,7,5,...]$ with length $N=2$ and
$\qquad \qquad [17, 25, 37, 55, 41, 61, 91, 17, ...]$ with length $N=7$
For $m=5$ the transformation $T_{5,+}(n)$ gives cycles
$\qquad \qquad [1, 3, 1, ...]$ with length $N=2$
$\qquad \qquad [13, 33, 83, 13, ...]$ with length $N=3$ and
$\qquad \qquad [17, 43, 27, 17, ...]$ with length $N=3$
After brute-force checking all odd $m$ with $3 \le m \le 19999$ and initial $n$ with $1 \le n \le 999 $ it seems, there are no other constellations for "non-systematic" $m \ne 2^k \pm1$ which allow cycles of length $ N \lt 30$ except one example:
I found only one more multiplicator $m=181$ which allows two nontrivial cycles for $T_{181,+}(n)$:
- $[27,611,27,611,...]$ with length $N=2$
- $[35,99,35,99,...]$ with length $N=2$
The only relevant property that I found for this $m=181$ so far is that $m^2$ is very near to $2^{15}$ in that $2^{15} - m^2 = 7$ or with the symbols $S$ and $N$ that $2^S - m^N = 7$
Now the distance of such expressions in terms of $S,N$ can be discussed with focus on exceptional high partial quotients in the continued fraction of $\log(m) / \log(2)$ which provide that specific combinations of $S$ and $N$.
So here is a list of $m$ with the according continued fractions of $\log(m)/\log(2)$ which have "high" cf-coefficients at early stage, and thus have small distances $2^S - m^N$ (which is critical for the possibility of cycles):
m cont-frac
----- ---------------------------------------------------------------------
181 [7, 2, 1621, 1, 2, 5, 12, 1, 1, 3, 2, 4, 7, 2, 1, 2, 3, 9, 1, 1,...]
741 [9, 1, 1, 7, 1234, 12, 6, 1, 1, 1, 7, 5, 1, 14, 1, 18, 1, 4, 1, 15,...]
1505 [10, 1, 1, 4, 1585, 1, 4, 1, 5, 1, 6, 4, 23, 2, 5, 19, 1, 1, 3, 1,...]
2047 [10, 1, 1418, 4, 1, 1, 3, 12, 1, 11, 4, 1, 1, 2, 15, 1, 1, 1, 4, 2,...]
2049 [11, 1419, 1, 10, 2, 1, 3, 1, 1, 18, 2, 10, 3, 3, 2, 1, 6, 1, 1, 2,...]
2195 [11, 9, 1, 1913, 5, 2, 1, 8, 2, 20, 1, 39, 2, 2, 4, 33, 1, 1, 8, 1,...]
2989 [11, 1, 1, 5, 1122, 7, 1, 1, 1, 1, 1, 1, 3, 5, 11, 1, 3, 1, 6, 2,...]
3251 [11, 1, 2, 94640, 1, 19, 1, 8, 5, 1, 4, 14, 38, 1, 5, 11, 2, 18, 2, 5,...]
3891 [11, 1, 12, 2, 2074, 1, 11, 3, 2, 1, 4, 6, 1, 2, 12, 1, 3, 3, 2, 1,...]
and the most likely cases which allow cycles are possibly that of the form, where the 3'rd coefficient is above 1000 and the second is above 1. A slightly expanded list of such m is this
m cont-frac
----- ---------------------------------------------------------------------
181 [7, 2, 1621, 1, 2, 5, 12, 1, 1, 3]
4705 [12, 5, 1905, 3, 1, 20, 1, 1, 26, 22]
10321 [13, 3, 2908, 1, 4, 4, 15, 1, 41, 1]
11585 [13, 2, 8452, 4, 2, 1, 2, 4, 2, 2]
20641 [14, 3, 1027, 1, 1, 1, 5, 3, 2, 8]
23167 [14, 2, 1154, 1, 5, 1, 1, 12, 1, 5]
23169 [14, 2, 2721, 1, 1, 7, 1, 90, 1, 1]
38967 [15, 4, 1798, 23, 1, 1, 3, 2, 1, 4]
By some small tries I've not found anything else remarkable so far, and I expect, that the condition for the possibility of occurence of cycles can be much sharpened/improved.
Q1: Is this multiplicator $m$ known in some literature and are some more multiplicators known for wider search arrays? (I already tried google with some obvious search patterns but did not yet find a comparable discussion; most frequently the summand in the numerator was generalized instead of the multiplicator)
Q2: Are there any more properties of this multiplicator known to be relevant which makes it specific compared to other $m$ whose squares/cubes/... are near to powers of $2$ but do not admit nontrivial cycles (well, of course without already proving the Collatz for the positive integers... ;-))
I've reread and found a remark in the 1978-article of R.E.Crandall,
Here he mentions in chap 8.1 the bases $m=5$ and $m=181$ as known to have a nontrivial cycle. But he doesn't mention the second cycle of $m=181$, so it seems that occurences were not found by analytical reasons but by short heuristics and there's no documentation about the bounds or other properties of his heuristics.
Interestingly (but with a bit different focus) he also mentions the (wieferich-prime) $m=1093$ and says, that it is known that it has divergent trajectories (again I don't have an idea why he does not mention the second known wieferich prime $m=3511$ whose structure regarding this problem should give similar solutions) - however he does not say, there is or there is not a cycle . (Perhaps this follows from something in the neighborhood of the statement, but I didn't manage to recognize any hint).