We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$: $$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$ Where $p_1^{q_1}*\cdots *p_m^{q_m}$, with $p_1<\cdots<p_m$ is the ordered prime factorisation of $S_{n}$.
For example: $$S_{28,1}=28$$ Now since $28=2^2*7^1$: $$S_{28,2}=2271$$ Since $2271=3^1*757^1$: $$S_{28,3}=317571$$ Etcetera.
We see that, at least in this particular case, the sequence grows rapidly. This turns out to be a general phenomenon. However, sometimes the sequence can decrease: $$S_{512,1}=512=2^9\;\;\;\;\; S_{512,2}=29$$ Motivated by this, I am wondering wether or not there exists a $k\in\mathbb{N}$ such that the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$ has cycles (i.e. is periodic).
By doing computer research, I have found that there exist no cycles which have highest element $\leq 10^7$. My conjecture is that no cycles exist for any $k$, however I am not sure where to start a proof.
For further investigation we might also consider the problem in other number systems (with a different base). Is there any base for which there are cycles?
EDIT: The answer to the last question is yes. Thanks to Joffan we have found bases in which there even exist fixed points.
Some of my further attempt at fixed points:
Note that in order to find fixed points in base $x$, we will need to find an $n\in\mathbb{N}$ such that there exist prime numbers $p_1<\cdots<p_n$ and positive integers $q_1,\cdots,q_n$ satisfying:
$$p_1^{q_1}*\cdots*p_n^{q_n}=p_1x^{2n-1}+q_1x^{2n-2}+\cdots+p_nx+q_n$$ We are particularly interested in the case $x=10$.
Maybe it is easier to first solve the equation for $x$. For example, for $n=1$ the equation is: $$p^q=px+q$$ Which gives us $x=p^{q-1}-\frac{q}{p}$. So we would like to find a prime number $p$ and a positive integer $q$ such that $p^{q-1}-\frac{q}{p}=10$ (probably no luck here).
For $n=2$ we get: $$ p_1^{q_1}p_2^{q_2}=p_1x^3+q_1x^2+p_2x+q_2$$ Which gives us some nastier solutions for $x$, but will maybe give us more chance to end up with the magical $10$ since we have more $p_i$s and $q_i$s to choose from.
Maybe someone can even find a proof that there exist no fixed points for base $10$, which of course will also be a big step. Please do not hesitate to post any suggestions that come to mind.