Suppose we have $$$n$$$ vertices and we add edge $$$(i, j)$$$ with probability $$$p$$$. Is there a way to check if it is hamiltonian path/cycle or to find the longest non-self-intersecting path/cycle in polynomial time?

Let $$$p_i$$$ — minimal prime divisor of $$$i$$$.

$$$s(n) = \sum_{i=2}^n \lceil \log_2(p_i) \rceil$$$.

I checked that $$$s(n) \leq 4 \cdot n$$$ if $$$n \leq 10^{10}$$$.

What is actual estimation of this sum?