If we take a prime $p$, how large does $p$ have to be so that $p-1$ has at least $n$ factors between $f_1$ and $f_2$? (Note that the factors can be prime or composite) Note that I'm looking more for an asymptotic analysis or formula, not an algorithm to solve for exact values. In other words, how large does $p_2$ have to be so that we have a prime $p$ such that $p<p_2$ so that $p-1$ has at least $n$ factors between $f_1$ and $f_2$.
For example, If we take $p=31$, then $p-1=30$. For $f_1=4$ and $f_2=12$, we know that 5, 6, and 10 are all factors of 30, so we find 3 factors between 4 and 12. In this case we were given $p$, $f_1$, and $f_2$. My question is: given $f_1$, $f_2$, and $n$, find $p$.
Let $Y={p_1}^{k_1}{p_2}^{k_2}...$ , we know the number of divisor of a number, is the product of the values 1 more than the powers (i.e letting d(Y) mean the number of divisors of Y,$d(Y)=(k_1+1)(k_2+1)...$). If $\frac{Y}{f_2-f_1}$ is an integer x, then we can say ( via the pigeonhole principle, and possible variants), that at least one gap of the same length where $\lceil d(Y)/x \rceil$ of it's divisors lie (a weighted version of the principle may be better but I couldn't find a link to the weighted version I wanted). The problem then becomes, what is the form of p-1, such that, we can create n divisors in the latter part.