A naïve solution would iterate over positive integers and check their luck until the $K^{\text{th}}$ lucky integer is found. Such a solution is worth $20\mathrm{\%}$ of points.

A better solution is possible if a pattern is noticed in lucky numbers.

Specifically, if we substitute the digit $4$ with $0$, and $7$ with $1$, the lucky numbers correspond to binary number notation, with an additional quirk that leading zeros are allowed.

A solution iteratively generating the first $K$ lucky numbers using this observation has a complexity of $\mathcal{O}(K)$ and is worth $50\mathrm{\%}$ of points.

Another speedup can be made by observing the number of lucky numbers with a given length. There are $2^N$ lucky numbers with length $N$ (since each of $N$ positions can contain one of two digits).

Knowing this, we can determine the length $L$ of the required lucky number and the index $P$ of that number among all lucky numbers of length $L$. Then, it is sufficient to output the number $P-1$ in binary with the appropriate number of leading zeros, substituting digits $0$ and $1$ with $4$ and $7$, respectively.