Repunit Factors

Each of the factors that is this format follows a certain form. This form can be described as $a$ groups of $b$ digits, with $c$ ones in each of these groups.

We can place a few restrictions on these variables, and then look for all of the possibilities of this form. (Example: One factor of $R_{20}$ is $00110011001100110011$ : 5 groups of 4 digits, 2 ones in each group.)

For all factors, $ab$ must be a factor of 20 (total digits cannot exceed 20, and the number of digits in a group cannot be larger than the group). $b$ must be divisible by $c$ .

$b=c$ if and only if $b = 1$ , and $a$ cannot be 1 unless $b$ is also 1 (these two restrictions will prevent us from overcounting the case where $R_k$ is a factor).

Factors of 20 are: $\{1,2,4,5,10,20\}$

Factors of these factors (excluding itself) are $\{1,1,[2,1],1,[5,2,1],[10,5,4,2,1]\}$

Factors of these numbers (excluding itself, unless it is a 1) are: $\{1,1,[1,1],1,[1,1,1],[(5,2,1),1,(2,1),1,1]\}$

This totals to $\boxed{16}$ numbers.

Challenge: There is at least one factor of $R_{20}$ that consists of only the digits $0$ and $1$ that is not composed of only copies of a single ministring . Can you find them? Can you figure out for what values of $n$ does $R_n$ have these special factors?