Define a set of numbers with small radicals (A341645 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.
- 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 (its complement is strictly a subset of A085253 there).
Other questions:
for any prime $p$, do the elements not divisible by $p$ have relative asymptotic density $0$ in $A_2$?
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 $B$?
