# Large prime factors in a sequence of consecutive numbers

When trying to solve a number theory problem I came across this other problem which sounds interesting. Let $$n$$ be a positive integer, and consider $$n$$ consecutive positive integers $$a_1, \ldots, a_n$$ that are at most $$n^2$$.

What is an upper bound for the number of integers in this kind of list that have a prime factor greater than $$n$$?

What's interesting is that for any such prime factor, it appears only once as a factor in the list, and there can only be at most $$n$$ such primes. I'm guessing that $$n$$ is too large an upper bound and cannot be reached, i.e. there is always at least one number with prime factors all less than or equal to $$n$$.

I don't have any results besides checking some lists of numbers, and I don't really know how to approach this. Any ideas?

At least one of the numbers will be divisible by $$n$$. It cannot have a prime factor greater than $$n$$ because then it would be at least $$n(n+1)$$, so at most $$n-1$$ of them can have a prime factor larger than $$n$$. An example that meets this is $$n=5$$ and the numbers $$19-23$$ where only $$20$$ does not have a prime factor greater than $$5$$.
• Silly me, I should have thought of that. Now I guess the next natural question is whether the bound $n-1$ can be achieved infinitely often. – Paul Cusson Jan 18 at 4:05