A "Take a Square Root When You Can" conjecture related to the prime factorization

Question

I would tend to think that the following has already been investigated.

But as implied from the title, I have no idea how to even start looking for it.

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Let $P_n$ denote the sum of the squares of the prime factors of $n$ with multiplicity; i.e.

let $\ S\ $ be a finite set of primes. Let $\ e: S\rightarrow\{1\ 2\ ...\}\ $ represents positive integer exponents in $\ n\ :=\ \prod_{p\in S}\ p^{e(p)}.\ $ Then

$$P_n\ :=\ \sum_{p\in S}\ e(p)\cdot p^2$$

Let $A_n$ denote the following sequence (defined for every $n>1$):

$ A_n= \begin{cases} \sqrt{P_n} & \text{$\sqrt{P_n} \in\mathbb{N}$}\\ {P_n} & \text{$\sqrt{P_n}\not\in\mathbb{N}$}\\ \end{cases} $

A few examples:

• $A_{5}=\sqrt{5^2}=5$
• $A_{30}=2^2+3^2+5^2=38$
• $A_{48}=\sqrt{2^2+2^2+2^2+2^2+3^2}=5$
• $A_{60}=2^2+2^2+3^2+5^2=42$

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Now, form a sequence by performing this operation repeatedly, beginning with any positive integer $k>1$, and taking the result at each step as the input to the next. My conjecture is that this process will eventually reach a number $n$ such that $A_n=n$, regardless of which positive integer $k$ is chosen initially.

For example, an initial value of $k=9$ yields the following process:

• $A_{9}=3^2+3^2=18$
• $A_{18}=2^2+3^2+3^2=22$
• $A_{22}=2^2+11^2=125$
• $A_{125}=5^2+5^2+5^2=75$
• $A_{75}=3^2+5^2+5^2=59$
• $A_{59}=\sqrt{59^2}=59$

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It's easy to observe that $A_n=n$ for every prime number $n$ ("landing" on the 1st case).

In addition to that, $A_n=n$ also for the non-prime $n=27$ ("landing" on the 2nd case).

I am pretty sure that $A_n\neq{n}$ for any other value of $n$, but I do not know how to prove this.

It would be nice if someone suggested a proof, although it is not the primary question at hand.

In any case, this conjecture seems much more probable than the $3n+1$ conjecture, since the latter has to "land" on a power of $2$, while the former can "land" on any prime number (or $27$), which is significantly more likely to occur within any given range.

However, the process seems to be growing up to rather large figures for certain initial values of $k$, such as $30$ and $60$ (and many others), so it may be possible to refute this conjecture by showing an initial value of $k$ for which the process never terminates.

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So at this point I can consider three different approaches:

1. Prove that the process converges for every initial value
2. Prove that the process diverges for some initial value
3. Show that the process loops for some initial value

The first two options seem quite difficult, and I wouldn't even know where to start.

The third option is the easiest, but I find it very unlikely that such initial value exists.

So any ideas or points of reference to related researches will be highly appreciated.