This question is in reference to this answer to a question I asked.
It can be proved that numbers that have most factors than the numbers before it have the degrees of their prime factors arranged in the descending order-:
$N = 2^p\cdot 3^q\cdot5^r\cdot\ldots \text{ so on}$
where
$p \ge q \ge r$
I want to know why this is the case.
What is the intuition behind it?
What I can know is that the lowest prime factors will occur more and more as I move towards $\infty+$. But it is not clear to me why this property is followed by the number that has the maximum factors.
For example:
Take $2^2\cdot3^3$ and $2^3\cdot3^4$ they have a total number of factors of $12$ and $20$, yet the second number does not follow the above-mentioned property.
Also if someone can point me to a proof of this, it would be great.
As Wojowu mentioned, the number of divisors only depends on the exponents in the prime factorization, to be more precise, if $$N=p_1^{a_1}\cdots p_n^{a_n}$$ then the number of divisors is $$(a_1+1)\cdots (a_n+1)$$ Hence changing the order of the exponents does not change the number of divisors. But the smallest number we have with a given set of exponents is $2^{a_1}\cdot 3^{a_2}\cdot 5^{a_3}\cdots p_n^{a_n}$ with $a_1\ge a_2\ge \cdots \ge a_n$ if $p_n$ is the $n$-th prime.