Is there any computable function $f(n)$, which given any integer $n$ has been proven to return either $0$ or $1$ in finite time, and for which the statement "$f(1), f(2), f(3),\ldots$ contains arbitrarily large sequences of $0$'s" has been proved to be undecidable in PA or ZFC?
If not, is there any proof of the existence or non-existence of such a function?
Edit: Is there one which is also morally undecidable?