TL;DR: Is there a name for the largest value of $f$ for which $f|n$ and $n/f \ge f$? Or a name for the smallest value of $f$ for which $f|n$ and $n/f < f$?
Additional clarification. This question occurred to me while writing a very short piece of code to find the factors of a number and making a small optimisation to it. When you try to find the factors of a number $n$ you can simply take each element of the sequence $1, 2, 3, ..., n$ and check if $n$ is divisible by it. A first optimisation is to limit yourself to possible factors $f$ for which $f <= n/2$ (yielding only the proper divisors) and add $n$ itself to the set. A second optimisation is to stop at the first $f$ for which $n/f < f$. At that point the remainder of the factors can be found by dividing $n$ by each already found factor. I'm interested in knowing whether there's a name for this "pivot" element and its properties.
- For 12 the largest such $f$ is 3, since the remaining factors can be found by dividing 12 by 1, 2 and 3. The smallest "pivot" would be 4 as at that point $12/4 < 4$
- For any prime number $p$ the largest $f$ so that $f|p$ and $p/f \ge f$ is 1, and the smallest $f$ so that $f|p$ and $p/f < f$ is $p$.