How to show that every integer greater than $23$ is the sum of two squareful numbers? I checked up to $50000$. The argument I used in this answer to a similar problem doesn't work because it would rely on most numbers being squareful which isn't the case.
Every large enough $n$ can be represented as $n=4x+9y$, where $x$ and $y$ are non-negative integers. Here large enough means $\ge (4-1)(9-1)$.
For completeness, we give a proof. The four numbers $24$, $25$, $26$, and $27$ are so representable. And every integer $\ge 28$ is of the shape $4k+t$, where $t$ is one of these four numbers, and $k$ is a positive integer.