Abstract
We prove that a sequence $X$ is integer-valued random ($\mathsf{IVR}$) if and only if for every cl-tt reduction $\Phi$, there are only finitely many oracles that compute $X$ via $\Phi$. A well-known property of integer-valued supermartingales is that under any cone, there is a subcone where the values are constant. We extend this property and prove that for a special integer-valued martingale $M$, which starts with initial capital $1$ and always bets $1$ on bit “1” if possible, and for any integer valued supermartingale $f$, under any cone there is a subcone such that $f$ actually simulates $M$. Using this property, we prove that there exists a non-$\mathsf{IVR}$ sequence that is cl-computable by only countably many oracles, as a complement to the first result.