Build a chain of sets $(A_{k})_{k \in \mathbb{N}}$

Question

Build a chain of sets $(A_{k})_{k \in \mathbb{N}}$ ($A_{k} \subseteq A_{k+1} \forall k$), that

if $k$ is even, then $A_{k}$ is recursive set

if $k$ is not even, then $A_{k}$ is non-computable set

Any help greatly appreciated. Thanks

Answer 1

Let $p_0,p_1,\ldots$ enumerate the primes. Let $S\subset\mathbb{N}$ be a fixed non-computable set.

Start with $A_0=\emptyset$.

Then given $A_k$ for even $k$, let $A_{k+1}=A_k\cup\{p_{k/2}^s:s\in S\}$ and $A_{k+2}=A_k\cup\{p_{k/2}^n:n\in\mathbb{N}\}$.