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.

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:

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

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:

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

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.

So at this point I can consider three different approaches:

- Prove that the process converges for every initial value
- Prove that the process diverges for some initial value
- 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.

onedownvote on this question. Not "down-votes";onedownvote. Get over it. $\endgroup$ – Gerry Myerson Jan 11 '15 at 5:17