Seeking upper bound on largest prime factor of colossally abundant numbers I recently came across an older post where the OP referred to papers giving upper bounds on the size of the largest prime factor in superabundant and colossally abundant numbers. I asked for links to those papers in the comments. Am still hoping for a response, however maybe it's a good stand-alone question.
Does anyone know of results on the upper bound on the size of the largest prime factor of colossally abundant numbers? It seems that the answer will roughly be slightly less than log(n) but I'd be grateful if there's some known explicit results out there. Thanks.
Edit
The poster of the original question has answered this question. Thanks Ahmad! Here is the original post.
 A: The main paper i rely on in my question was "A Class of Equivalent Problems Related to the Riemann Hypothesis" by Sadegh Nazardonyavi in which in proposition 3.33 he proves that $ \sqrt{p} < \ln n -\theta(p) < \sqrt{3p}$ for large enough super abundant (SA) number $n$ and $p$ being the largest prime divisor of $n=2^k \cdots p$.
Other important papers are for Erdos, Nicolas on highly composite numbers.
Also one can show from PNT that any counter-example for RH must have these properties :
$n =2^k \cdots p$ a multiplication of primes in ascending order and the powers in descending order.
And that $\ln n \approx \theta(p) \approx p$.
I did it 2 years ago and never looked back to it, so i don't have any papers on my pc nor my own proof for the statements above but you should find them in Sadegh Nazardonyavi,Erdos works.
I tried to make them sharp as a way to attack RH, but i couldn't improve on the bounds you should get from PNT.
A: The largest possible factor of a number will be $\lfloor\sqrt{x}\space\rfloor$ but these numbers have many factors by definition so the largest factor will be much less and one might think it would be the $x^{th}$ root of the number where x is the number of digits but this proves to be far too aggressive with larger values of n.
Your conjecture appears correct. For instance, the largest prime factor of $224403121196654400$ is $37$ and
log $224403121196654400\approx 40.$ A discussion of this and other 'abundant numbers is here.
