# A relation of Kolomogrov complexity

Given a positive strictly monotonically increasing infinite sequence $$n_1, n_2, \dots$$ with Kolmogorov complexity

$$K(n_i)\geq\lceil\log_2 n_i\rceil/2\,.$$

If $$q_i$$ is the greatest prime number that divides $$n_i$$, I've been asked to show to show that the set $$Q=\{q_i\}_{i>0}$$ is infinite. How do I do this?

My idea was to assume if it is finite, then we can write all $$n_i=p_1^{r_1}...p_k^{r_k}$$. So all the the information is encoded by $$(r_1,...,r_k)$$. We can recover the number by a program merely with input $$r_1,...,r_k$$. And I want to deduce such program can have length shorter that the given function.