This problem was studied by De Bruijn, see

N. G. de Bruijn and J. H. van Lint, "On the number of integers $\le x$ whose prime factors divide $n$", Acta Arith. 8 (1963) 349–356

The main result is that $$\log\left(S(x)\right)=\log\left(\sum_{n\le x} \frac{1}{rad(n)}\right)\sim \left(\frac{8\log n}{\log \log n}\right)^{1/2}$$ and that $$\sum_{n\le x} \frac{n}{rad(n)}=o\left(xS(x)\right).$$ See also the article "Idempotents and Nilpotents Modulo n" for a discussion of this and similar problems (and more complete references).

This problem was studied by De Bruijn, see

N. G. de Bruijn and J. H. van Lint, "On the number of integers $\le x$ whose prime factors divide $n$", Acta Arith. 8 (1963) 349–356

The main result is that $$\log\left(S(x)\right)=\log\left(\sum_{n\le x} \frac{1}{rad(n)}\right)\sim \left(\frac{8\log x}{\log \log x}\right)^{1/2}$$ and that $$\sum_{n\le x} \frac{n}{rad(n)}=o\left(xS(x)\right).$$ See also the article "Idempotents and Nilpotents Modulo n" for a discussion of this and similar problems (and more complete references).