Periodic sequences of integers generated by $a_{n+1}=\frac{\operatorname{rad}(pa_{n})}{p}+\frac{\operatorname{rad}(qa_{n-1})}{q}$

Question

Let's define the radical of the positive integer $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$ and consider the sequence $$a_{n+1}=\frac{\operatorname{rad}(p\cdot a_{n})}{p}+\frac{\operatorname{rad}(q\cdot a_{n-1})}{q}$$

with $\,a_1=a_2=1\,$ and $\,p,\,q\,$ odd primes.

In some cases the sequence is cyclic, that is $$a_{n+\tau}=a_n$$ for all $\,n\gt n_0$, being $\,\tau\,$ the cycle length.

Just two examples:

• for $\,(p,q)=(31,31),\;(n_0,\tau)=(5,207)$
• for $\,(p,q)=(5,17),\;(n_0,\tau)=(6,159)$

It is quite easy to find dozens of periodic sequences using small prime numbers: in all cases, the length of the cycle turns out to be a multiple of 3. Is it possible to explain this singular behavior?

Answer 1

For any odd $p$, $q$ (not necessarily prime) the values modulo $2$ follow a cycle of order 3.