Let $A_k$ be a random variable which represents the number of distinct integers seen after sampling $k$ independently and uniformly at random from the range $1, \dots, n$. Let $B_k$ be a random variable which represents the number of distinct integers seen after sampling $k$ independently and uniformly at random from the range $1, \dots, n^2$.
We know that $E(A_k) = n(1-(1-1/n)^k)$ and $E(B_k) = n^2(1-(1-1/n^2)^k)$. We also know that for $k \ll n $ both grow almost linearly and hence almost identically in $k$ but for $k \geq n$ they behave completely differently.
What is the asymptotic growth of the expected value of the ratio of $A_k$ and $B_k$? That is I am looking for an asymptotic expression for $E(B_k/A_k)$ assuming $n$ is large and especially a formula that captures the difference between the small and large $k$ phenomena.
My intuition is that $E(B_k/A_k)$ looks like $E(k/A_k)$ when $k < n\log{n}$ and like $E(B_k/n)$ otherwise. For large $n$ we also have that $E(A_k) \sim n(1-e^{-k/n})$ and $E(B_k) \sim n^2(1-e^{-k/n^2})$ it seems.