The function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ given by $f(x) = \|x\|^{2-n}$, where $\|~\|$ denotes the Euclidean norm, is harmonic. This is just a simple computation.
My question is: why should we expect this function to be harmonic? In other words, if someone asked you whether this function is harmonic or not, what kind of reasoning could lead you to suspect that this is the case, before actually computing its laplacian?
I'm just trying to understand the intuition behind harmonic functions. Since this can be a little vague, I'm willing to accept alternative proofs of this fact as an answer, even if they are overkill.