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

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