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 $\gamma(t)$ ?



Smallest integer not divisible by integers in a finite set
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.


