Positive integer $m≤n$ such that $m$ has the largest number of distinct prime factors among all positive integers $\le n$

Question

How do I interpret the following mathematically:

Consider a positive integer $n$ such that $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_m^{\alpha_m}$ where $p_i$ are primes and $\alpha_i$ are positive integers.

I want to consider the positive integer $m\le n$ such that $m$ has the largest number of distinct prime factors among all positive integers less than or equal to $n$.

I explain it with the following example:

Consider $n=36$. Here my $m$ will be $30$ since $30=2\times 3\times 5$ and it has the largest number of distinct prime factors among all positive integers which are less than or equal to $n=36$.

But I cant express my view mathematically. How do I present my mathematical idea? Can someone please help me to write the above fact in a more precise and concise manner which can be easily understood by mathematicians.

Please help.

Answer 1

Find the largest positive integer $\ k\ $ with $\ p_k$#$\le n\ $

Then, there is a number not exceeding $\ n\ $ , namely $\ p_k$# , with $\ k\ $ distinct prime factors and a number with $\ k+1\ $ or more distinct prime factors must be at least $\ p_{k+1}$#$>n\ $

Hence , the desired number is $\ m=p_k$# , in other words, the largest primorial not exceeding $\ n\ $.