This is a reference-request question.
The Set Up:
Fix $n\in\Bbb N$.
Suppose
$$C(x)=\begin{cases} x/2&:x\text{ is even},\\ 3x+1&:x\text{ is odd}. \end{cases}$$
Let $m\in\Bbb N\cup\{0\}$. Denote by $a^{(n)}_m$ the number $C^m(n)$.
Let $\left(p^{(n)}_j\right)_{j\in J}$ be all the prime divisors of all the $a^{(n)}_m$, in increasing order.
For example, if $n=11$, then
$$\begin{array}{c|c|c} i & a^{(n)}_i & \text{Prime divisors of }a_i^{(n)}\\ \hline 0 & 11 & 11\\ 1 & 34 & 2,17\\ 2 & 17 & 17\\ 3 & 52 & 2, 2, 13\\ 4 & 26 & 2, 13\\ 5 & 13 & 13\\ 6 & 40 & 2, 2, 2, 5\\ 7 & 20 & 2, 2, 5\\ 8 & 10 & 2, 5\\ 9 & 5 & 5\\ 10 & 16 & 2,2,2,2\\ 12 & 8 & 2,2,2\\ 13 & 4 & 2, 2\\ 14 & 2 & 2\\ 15 & 1 & \text{n/a} \end{array}.$$
Then $\left(p^{(11)}_j\right)_{j\in J}=(2, 5, 11, 13, 17)$.
Consider the group presentation
$$P(n):=\left\langle \left(p^{(n)}_j\right)_{j\in J} \,\middle |\, \left(a_m^{(n)}\right)_{m\in M}\right\rangle,$$
where each $a_m^{(n)}$ is written in its standard decomposition into primes, in ascending order.
For example,
$$\begin{align} P(11)&=\langle 2, 5, 11, 13, 17\mid 11, 2\cdot 17, 17, 2^2\cdot 13, 2\cdot 13, 13, 2^3\cdot 5, 2^2\cdot 5, 2\cdot 5, 5, 2^4, 2^3, 2^2, 2\rangle\\ &\cong E, \end{align}$$
where $E$ is the trivial group.
The Question:
Has $P(n)$ been studied before? If so, where?
Thoughts:
This sort of thing is difficult to search for, not least because of the vast amount of material on the Collatz Conjecture.
My suspicion is that the group defined by the presentation is trivial for "most" $n\in\Bbb N$.
It looks like $P(n)$ lends itself well to experimentation in GAP.
Another Example:
$$\begin{align} P(21)&=\langle 2, 3, 7\mid 3\cdot 7, 2^6, 2^5, 2^4, 2^3, 2^2, 2\rangle\\ &=\langle 3, 7\mid 3\cdot 7\rangle \\ &=\langle 3\mid \rangle\\ &\cong\Bbb Z. \end{align}$$
Please help :)