I've conjectured a general form of the sum of entries in the Cayley table for $(\mathbb Z\_n, \times)$, though I'm not sure how to prove it:
$$\frac{n^2(n-1)}{2} - \frac{n}{2} \sum\_{i=1}^{n-1} \gcd (i,n)$$
seems to work for small values of $n$. Is this right, and how can it be proven?