It takes time to read

Discrete Mathematics Level pending

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 $$f(2)+f(4)+f(6)+\cdots+f(2016)$$.

×