Suppose we’re dealing with programs that output zeros and ones. Consider all such programs (in a fixed language) having at most n instructions. For each program in this finite set, look at how many ones the program produces when we run it on the empty argument. Call this number one(n).

Consider the set $ \{\langle a,b\rangle :one(a)=b\}$ where $ \langle \rangle$ is the Cantor pairing function. Assuming that this set is decidable (there is an oracle telling us the answer), how to prove that $ K=\{x:\phi_x(x)\downarrow\}$ is decidable? (i.e. how to show that $ K$ is decidable using that set above as an oracle)