Can it be proven/disproven that there are highly composite numbers that prime-factorize into larger primes such as $9999991$?

Question

Of course, following the rules found by Ramanujan, such a highly composite number would need to factorize into all primes ascending up to 9999991 (with descending powers as the primes progress) so the highly composite number would be insanely large. However a highly composite number needs more factors than all other numbers before it, so surely any number with very large prime factors like 9999991 is automatically at a disadvantage? So is there a limit to the size of the largest prime factor of a highly composite number, or is it limitless? Is there even a way to know?

Answer 1

So, because there are an infinite number of primes, there are an infinite number of highly composite numbers. A highly composite number's prime factors are made up of the smallest consecutive primes with decreasing powers. For example, 5040 is a highly composite number because it has the factors 2⁴*3²*5*7. These primes are the smallest consecutive primes, and the powers they are raised to decrease (you can have 1 as a power repeating and it will still work as long as it never increases after you get to 1). Using these rules, we can take any number of consecutive primes and so long as the powers never increase we will have a highly composite number.

You are correct that highly composite numbers will be very large eventually, but there are an infinite number of them.