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Let $k$ be a positive integer. Consider the following condition:

There exists a polynomial $f(n)$ with rational coefficients such that for all positive integers $n<k$, $f(n)$ is the $n^{th}$ digit after the decimal representation of $\pi$. For example, $f(1)=1, f(2)=4, f(3)=1, ...$

How many positive integers $k$ exist such that the above condition is false?