S-smooth number ($S$-smooth number)

For a finite set $S$ of primes, an integer is $S$-smooth when every prime dividing it belongs to $S$. Bounding the largest prime factor is equivalent to restricting the integer to be $S$-smooth for a suitable finite $S$.

If $S$ is a finite set of prime numbers, every S-smooth number has a unique expression $n=a{b}^2$, where $a$ is squarefree and its prime factors lie in $S$. There are at most $2^{|S|}$ choices for $a$ and at most $\lfloor \sqrt{x}\rfloor$ choices for $b$ when $n\le x$. Consequently $f_S(x)\le 2^{|S|}\sqrt{x}$.