I am looking for a clear mathematical way to describe the product of the minimum set of primes and their powers that are contained within a sequence of consecutive integers.
For example, in the sequence: $2,3,...,10$, this product is $2^33^25^17^1= 2520$
For the sequence $20,21,22,23,24,25$, this would be $2^33^15^27^1(11)^123^1$
Let $x$ be the first integer in the sequence and $n$ be the number of consecutive integers.
Let $v_p(x,n)$ be the highest power of $p$ that divides $x+i$ where $0 \le i < n$
Would this be a clear way of describing this product:
$$\prod\limits_{p | \frac{(x+n-1)!}{(x-1)!}} p^{v_p(x,n)}$$
I ask because I am trying to analyze an intuition I have that:
$$\frac{(x+n-1)!}{(x-1)!} \div \prod\limits_{p | \frac{(x+n-1)!}{(x-1)!}} p^{v_p(x,n)} \le (n-1)! $$
I have a question about this relationship and don't want to waste anyone's time if I can't state the relationship clearly.
Are you looking for a way to describe the product using mathematical notation ? If so, as far as I can tell there is nothing wrong with your representation.
Also note that you can write your product as :
$\frac{(x+n-1)!}{(x-1)!} \div \prod\limits_{p | \frac{(x+n-1)!}{(x-1)!}} p^{v_p(x,n)} \le \Gamma (n)$