Let us take a natural number x > 1. Then define a sequence $x_n$ as follows:
- $x_0=x$;
- if $x_n = p_1\cdots p_s$, where $p_1\leqslant\dots\leqslant p_s$ are prime numbers, then $x_{n+1}$ is the number given by the concatenation of the decimal digits of $p_1,\dots,p_s$ in the same order.
Examples:
- $x=6$, then $x_1=23$, then $x_2=23$, and so on;
- $x=10$, then $x_1=25$, then $x_2=55$, then $x_3=511$, then $x_4=773$, then $x_5=773$, and so on;
- $x=8$, then $x_1=222$, then $x_2=2337$, then ..., finally we obtain $x_{10}=11613496501723$, and this is a prime number (if I haven't made a mistake).
One can change from decimal to any other base, for example binary or hexadecimal. And we'll have an analogous result: this sequence tends to be stable.
It is obvious that such a sequence is always increasing.
Moreover, let us denote $q_b(p)=b^{\lfloor\log_b p\rfloor+1}$, where the power means the number of digits in the number $p$ on base $b$ (w/o leading zeros). Then we obtain that $$ x_{n+1}=p_s + p_{s-1}q_{10}(p_s) + p_{s-2}q_{10}(p_{s-1})q_{10}(p_{s})+\dots+p_1q_{10}(p_2)\cdots q_{10}(p_s), $$ if $x_n=p_1\cdots p_s$. And it is easy to see that $p\leqslant q_b(p)\leqslant pb$, so one can estimate $x_{n+1}$ as follows: $$ p_s + p_sp_{s-1} + p_sp_{s-1}p_{s-2}+\dots+x_n\leqslant x_{n+1}\leqslant p_s + p_sp_{s-1}10 + p_sp_{s-1}p_{s-2}10^2+\dots+x_n10^{s-1}. $$
My questions are:
- Is it True that for all natural $x>1$ the sequence $x_n$ becomes stable?
- If we change from decimal to another base, will the sequence be stable if and only if it is stable for a decimal base?
- If (1) holds, can it be proved in PA, ACA${}_0$ or RCA${}_0$?
- Are there any connections between this problem with well-known problems like the Collatz conjecture, or with Goodstein sequences?
