Define a set of numbers with small radicals ([A341645][1] in OEIS) by
$$A_2=\{n\in\mathbb{N} \;|\; \text{rad}(n)^2\le n\}$$

The asymptotic density of $A_2\cap \{1,\dots N\}$ is $\sqrt{N}\times e^{2(1+o(1))\sqrt{\log N / \log \log N}}$, as per Lucia's answer [here][3].

1) Main question: does the sumset $A=A_2+A_2$ contain all sufficiently large integers? In other words, is $\mathbb{N}\setminus A$ finite?

The number of misses is initially large but becomes sparse very rapidly. I didn't find any after $86931723$,  up to $10^9$. $A$ is not in OEIS (it's strictly a superset of [A085253][2] there).


Other questions:

2) for any prime $p$, do the elements not divisible by $p$ have relative asymptotic density $0$ in $A_2$?

3) Computing (up to $10^9$) the subset $B\subset A$ of sums of **coprime** pairs in $A_2$, points to the misses thinning out very slowly (still above $13\%$ near $10^9$). Are there euristic arguments for or against $\mathbb{N}\setminus B$ being finite?

Is anything else known, or worth asking, about $A_2$, $A$ and $B$?

[1]: http://oeis.org/A341645
[2]: http://oeis.org/A085253
[3]: https://mathoverflow.net/q/296473