2018 Fermat Problem 25

For each positive digit D and positive integer k, we use the symbol D_{(k)} to represent the positive integer having exactly k digits, each of which is equal to D. For example, 2_{(1)} = 2 and 3_{(4)} = 3333. There are N quadruples (P, Q, R, k) with P, Q and R positive digits, k a positive integer with k \le 2018, and P_{(2k)} - Q_{(k)} = \left(R_{(k)}\right)^2. The sum of the digits of N is

\textbf{(A)}\ 10\quad \textbf{(B)}\ 9\quad \textbf{(C)}\ 11\quad \textbf{(D)}\ 12\quad \textbf{(E)}\ 13

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Full credit to this problem is given to the CEMC, you may view all Fermat contests here.