Suppose that $p\_1,...,p\_k$ are distinct prime numbers. 


Let $f(n,l)$ be equal to the number of elements from set $\{n+1,n+2,...,n+l\}$ that are divisible by some $p\_1,...,p\_k$. Is it true that
$$\sup\_{n,m,l \in \mathbb{N}} |f(n,l)-f(m,l)| = O(k).$$



The maximum difference between the number of elements in the two sets of equal length of consecutive numbers that divisible by some prime numbers
No. There are various results which give counterexamples.


For example, Rankin's construction of large prime gaps boils down to the fact that if $p\_1, \ldots, p\_k$ denote all prime numbers below $x$, then there exists some $n$ with $f\left(n, \frac{x\log x\log\log\log x}{(\log\log x)^2}\right)=0$.


If $n=x, \ell=x^2-x$, then $f(n, \ell)=\pi(x^2)+\mathcal{O}(x)\sim\frac{x^2}{2\log x}$, but the average taken over all $n$ is $\ell \prod\_{p\leq x}\left(1-\frac{1}{p}\right)\sim \frac{e^{-\gamma}x^2}{\log x}$.


