It takes time to read

Let $k$ be a positive integer. Bernard and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernard starts by writing the smallest perfect square with $k+1$ digits. Every time Bernard writes a number, Silvia erases the last $k$ digits of it. Bernard then writes the next perfect square and Silvia erases the last $k$ digits of it and the process continues until the last 2 numbers that remain on the board differ by at least 2. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k=1$ , then the numbers that Bernard writes are $16,25,36,49,64$ and the numbers showing on the board after Silvia erases are $1,2,3,4,6$ and thus $f(1)=5$ . Compute the sum of $f(2)+f(4)+f(6)+\cdots+f(2016)$ .