I need answer of following Question for my study of an irrational number. (The raw problem is slightly different.)

Let $k$ be an arbitrary large positive integer, and let $A$ is a positive integer satisfying $Ce^{2^k}\le A\le Ce^{2^k}$ and have no prime factor larger than $2^k$.(Conventionally, $C$'s are certain positive constants.) Let $y_0$ be a positive integer which suffices $y_0< A$. We now think about $2^k$ products $$P_s=(y_0+As+1)(y_0+As+2)\cdots(y_0+As+k)\qquad (0\le s< 2^k).$$ Question is "Can we find some $s$'s such that $P_s$ has no prime factor larger than $2^k$?".

It is helpful not only answer for this question, but also introducing relating paper or research.(I can't discover relating research.)

If this question is nonsense or ridicurous, sorry for asking this question.

Sorry, I got some help which asserts some mistakes in my previous question.So probably, this question contains some mistakes. If you discover some of mistakes, it's helpful asserting that.