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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