Is this polynomial familiar? - III
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?