Is a function looking for subsequences of digits of $\pi$ computable?
Note that $f$ can be the constant $1$ even if $\pi$ is not a normal number. (In French we say if $f$ is constant that $\pi$ is a nombre univers. I don't know the corresponding term in English)

For what it's worth: it could be, in the following way:

Proving $f$ is computable would not necessarily imply the resolution of the open question whether $f$ is constant or not. For example you can build $g$ that is computable but such that the constantness of $g$ is equivalent to Goldbach's conjecture.

Of course that does not even begin to answer your question, but it's likely open to me.