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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.

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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\ $.

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  • $\begingroup$ What is meant by #? $\endgroup$
    – Charlotte
    Commented Jun 30, 2020 at 10:08
  • $\begingroup$ Can you please provide more details $\endgroup$
    – Charlotte
    Commented Jun 30, 2020 at 10:08
  • $\begingroup$ $p$# stands for $\ 2\cdot 3\cdot 5\cdots p\ $.(called "primorial") $\endgroup$
    – Peter
    Commented Jun 30, 2020 at 10:08
  • $\begingroup$ It is easy to show that the primorial with the primes upto the $\ k\ $-th prime is the smallest positive integer with at least $\ k\ $ distinct prime factors. $\endgroup$
    – Peter
    Commented Jun 30, 2020 at 10:10
  • $\begingroup$ What does primorial mean? I am sorry but is it possible to give a more detailed answer $\endgroup$
    – Charlotte
    Commented Jun 30, 2020 at 10:13

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