Smooth values of certain sequences

Question

I have a sequence $ a_{n}$, and the set

$ A=\{ m \le x | m \in \{a_{n}\}_{n=0}^{\infty} \} $ satisfying

$ \#A= x^\alpha+O(x^{\alpha-\varepsilon}). $

Also I have

$ \#\{ m \le x : p|m , m \in \{a_{n}\}_{n=0}^{\infty} \}= \frac{x^\alpha}{p}+O(x^{\alpha-\varepsilon}). $ for all p prime (of course those $p \ll x^{\varepsilon}$).

Is there any sieve(or maybe direct) theoritical approach from which I can conclude that R-smooth numbers in the above set has positive lower density(maybe asymptotic)?

Well I did try Buchstab's identity together with usual induction argument. I got no result.

Thanks

Answer 1

Let $A$ be the set of integers $n$ which are divisible by a prime number $p>n^\theta$, $\frac{1}{2}<\theta<1$. Then for a prime number $q<x^{1/3}$ we have \begin{eqnarray*} \#\{n\leq x, n\in A, q|n\} & = & \underset{p\neq q}{\sum_{p\leq x}}\underset{q|m}{\sum_{m\leq\min(p^{(1-\theta)/\theta}, x/p)}}1 + \mathcal{O}(q^2)\\ & = & \sum_{p\leq x}\left\lfloor\frac{\min(p^{(1-\theta)/\theta}, x/p)}{q}\right\rfloor+ \mathcal{O}(q^2)\\ & = & \sum_{x^\theta\leq p\leq x}\frac{x}{pq} + \mathcal{O}\left(\frac{x}{\log x}\right). \end{eqnarray*} Hence $A$ is well distributed in residue classes, but does not contain smooth numbers.