Observe that $16,383 = 2^{14} - 1 = (2^7 + 1)(2^7 - 1) = 129 \cdot 127 = 3 \cdot 43 \cdot 127$. Because all three of these factors are prime, the greatest prime number that is a divisor of $16,383$ is $127$, and the requested sum of digits is $1 + 2 + 7 = 10$.