# Is this polynomial familiar? - III

Algebra Level 5

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?

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