# nt.number theory – Is almost every number the sum of two numbers with small radicals?

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

The asymptotic density of $$A_2cap {1,dots N}$$ is $$sqrt{N}times e^{2(1+o(1))sqrt{log N / log log N}}$$, as per Lucia’s answer here.

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 (its complement is strictly a subset of A085253 there).

Other questions:

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

2. Computing (up to $$10^9$$) the subset $$Bsubset A$$ 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 $$B$$?