Given a set $S=\{s_1,s_2,\ldots\}$ of pairwise coprime positive integers greater than 1, define $T$ as the set of products of zero or more elements of $S$ so $T$ contains $1, s_1, s_2, s_1^2, s_1s_2,$ etc. If the growth of $S$ is $f(x)$, i.e., $$ \lim_{n\to\infty}\frac{\#(S\cap\{1,2,\ldots,n\})}{f(n)} = 1 $$ where $f$ is sufficiently nice (monotone, smooth, etc.), what can be said about the growth $g(x)$ of $T$?
Example 1: If $S$ is the set of prime numbers, then $f(x)\sim x\log x$ and $g(x)\sim x$ since $T$ is just the set of positive integers.
Example 2: If $S$ is the set of primes congruent to 1 mod 4 together with 2 and the squares of the primes congruent to 3 mod 4, then $T$ is the set of numbers that are the sum of two squares and $f(x)\sim 2x\log x$ and $g(x)\sim kx\sqrt{\log x}$ with a constant $k$ (the inverse of the Landau-Ramanujan constant, to be precise).
Example 3: If $S$ is finite with $n$ elements, then $f(x)=n$ for large enough $x$ and $\log g(x) \sim \log^nx$.
Example 4: Bending the rules to allow non-coprime elements, if $S$ is the set of squares and cubes of primes, then $f(x) \sim x^2\log^2x$ and $g(x) \sim kx^2$ with $k=\zeta(3)/\zeta(3/2)$ and $T$ is the set of 2-full or powerful numbers.