Smallest integer not divisible by integers in a finite set Hello all, if $a_1,a_2, \ldots a_t$ are $t$ integers $\geq 2$, the set
$G(a_1,a_2, \ldots a_t)=\lbrace N \geq 1 |$ In any sequence of $N$ consecutive
integers there is at least one not divisible by any of $a_1,a_2, \ldots a_t\rbrace$
is nonempty (it contains $a_1a_2 \ldots a_t$) so it has a minimal element
which we denote by $g(a_1,a_2, \ldots a_t)$.
Question 1 : Is there a uniform bound $\gamma (t)$, depending
only on $t$, such that $\gamma (t) \geq g(a_1,a_2, \ldots a_t)$ for any
$a_1,a_2, \ldots a_t$ ? For example, we may take $\gamma(2)=4$.  
Question 2 : If $\gamma$ is well-defined, 
are any asymptotics known about $\gamma(t)$ ?
 A: Given an integer $n$, the Jacobsthal function $g(n)$ is the least integer, so that among any $g(n)$ consecutive integers $a,a+1,\dots,a+g(n)-1$ there is at least one that is coprime to $n$. Let $\nu(n)$ count the distinct prime factors of $n$. You can define $$C(r)=\max_{\nu(n)=r} g(n)$$ and as Jonas Meyer points out in the comments this is precisely $C(t)=\gamma (t)$ (i.e. it is enough to consider when all $a_i$ are prime).
For the bounds $$\frac{c_1t (\log t)^2 \log \log \log t}{(\log\log t)^2}\le C(t)\le c_2 t^{c_3}$$
see the paper "On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal"" by Erdos. I don't know if there are better bounds.
A: Thomas Hagedorn has a short survey on results related to the Jacobsthal function, as well as recent computations for a_i being the first t primes for t up to 50 . It is at 
 http://www.tcnj.edu/~hagedorn/papers/JacobPaper.pdf .
In his section 1, Hagedorn cites a result of Iwaniec which gives an asymptotic upper bound of order O(t log(t))^2, and he cites a more explicit upper bound that was given by Stevens as 2t^(2 + 2elog(t)).  (He also cites a lower bound by Pintz which is a mild improvement on the Erdos lower bound.) I am working on replacing the bound in Stevens' result by something asymptotically smaller (involving log(log(tlog(t))).  I will post it as an answer to 
Erik Westzynthius's cool upper bound argument: update?
when I am confident it is valid.
UPDATE 2011.02.25  I have posted an improvement of Stevens's result as an answer to the linked question above.  I welcome a review of it.END UPDATE 2011.02.25
Gerhard "Ask Me About System Design" Paseman, 2011.02.13
