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 $$\log\left(S(x)\right)=\log\left(\sum_{n\le x} \frac{1}{rad(n)}\right)\sim \left(\frac{8\log n}{\log x}{\log \log n}\right)^{1/2}$$ x}\right)^{1/2}$$and that$$\sum_{n\le x} \frac{n}{rad(n)}=o\left(xS(x)\right).$$See also the article "Idempotents and Nilpotents Modulo n" for a discussion of this and similar problems (and more complete references). 1 This problem was studied by De Bruijn, see N. G. de Bruijn and J. H. van Lint, "On the number of integers \le x whose prime factors divide n", Acta Arith. 8 (1963) 349–356 The main result is that$$\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).