Take a set $\mathcal P$ of primes and denote by $\langle \mathcal P\rangle $ the set of all natural numbers composed of primes from $\mathcal P$. If \[ \sum _{p\in \mathcal P}\frac {1}{p}\] converges then it can be shown without too much trouble that \[ \sum _{n\leq x\atop {n\in \langle \mathcal P\rangle }}1\ll x/\log x.\] Is is possible to get an asymptotic result for this count? I suspect that if we know something about the sizes of the elements of $\mathcal P$ then we can say something with sieve methods, but in this generality is it still possible to say something?

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