$P$ is a powerful number if it can be represented as $A^2B^3$ Where $A,B\in\Bbb{N}$. It has been proven by Heath-Brown that if $M\in\Bbb{N}$ and is sufficiently large then it can always be written as the sum of three powerful numbers. In other words the list of positive whole numbers that cannot be written as the sum of three powerful numbers is finite.
It is known that the numbers that cannot be written as the sum of three squares iff they are of the following form $4^k(8n+7)$ where $n,k\in\Bbb{N}$. This is a super-set of numbers that cannot be written as the sum of three powerful numbers.
Here is the OIES link to the sequence https://oeis.org/A056828.