Consecutive integers with no large prime factors 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.
 A: You are asking for consecutive runs of smooth  numbers.  I do not have the keyboard to spell Stormer with a stroke over the o, but http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem has information for you.  Unfortunately, I do not know of bounds for the largest pair of consecutive smooth numbers, but perhaps you can find out and report back here.  I will say that I suspect a sequence of k such numbers will not exist once you reach numbers the size of A.
Gerhard "Ask Me About System Design" Paseman, 2012.08.03 
A: Yuta is correct, Størmer's method, or preferably, D.H. Lehmer's 1963 refinement of that method, applies to a finite set of primes.  If the set contains $k$ primes, then you have $2^k-1$ Pell equations to solve (Størmer's original method involved solving $3^k$ Pell's.

I have had a particular interest in this subject for some years, and have provided most of the high-end values at OEIS, see http://oeis.org/A002072.  I intend to raise some questions arising from this work in a new posting here at mathoverflow.
But to Yuta's original question, the only way I can think of to identify smooth numbers in a particular range is to use a smart sieving algorithm. But if the interval is really large, and the number of primes is also very large, this may not be practical.
By the way, in Lehmer's paper, "Størmer"  appears as "Störmer", I have yet to determine which is correct! I suspect Lehmer was probably correct.
