Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it and adds $1$ to the remaining number. Bernardo then writes the next perfect square, Silvia performs her operation, and this process continues until the last two numbers that remain on the board differ by at least 2. Let $f(k)$ be the smallest positive integer not written on the board. What is the sum of the digits of $f(2)+f(4)+f(6)+\dots +f(2016)$?