A number is called lucky if it comprises only the digits $4$ and $7$. Your task is to return the k-th lucky number as a string for a given integer k. 

Constraints:

• $1 <=$ k $<= 10^9$

The essence of this solution lies in using bitwise manipulation to generate the k-th lucky number. The solution uses an approach by transforming the given integer k into a binary-like representation to determine the k-th lucky number. The key idea is to convert k into a string of digits composed of $4s$ and $7s$, the only digits considered lucky. This approach uses a combination of binary representation and simple string manipulation to construct the desired lucky number sequence.

Now, let’s walk through the steps of the solution:

1. We start by adjusting k for 1-based indexing. Since lucky numbers are usually counted starting from the first (i.e., 1-based indexing), we increment k by $1$. For example, if k $= 5$ ...