A sequence $\{a_i\}_{0}^{\infty}$ is defined as below:
$a_0 = 1$, and for $i \geq 1$, let $a_i = \text{ The smallest integer j } > a_{i-1} \text{ such that for every k } \leq \lfloor \sqrt{j} \rfloor , a^{-1}_{j-k^2}\text{exists}$
One olympiad question was proving that the sequence is infinite, it's trivial to prove by contradiction.
Now, I was experimenting with the density function, i.e the probability of taking a number not exceeding $i$ and it being a member of the sequence:
$$\displaystyle \rho_i := \frac{\#\{a_j | a_j \leq i\}}{i}$$
I wrote a python code (you can use it here), and computed upto $4*10^6$, and the density showed a graph similar to $$\rho_x \approx^?a \frac{log(x)}{x^.6}$$ (where $a$ is an constant)
Is the approximation to $\rho_x$ correct ? What is a better approximation to $\rho_x$ ?
