How many pairwise non-similar rectangles are there in an 8x8 chessboard? 
How many pairwise non-similar rectangles are there in an $8 \times 8$ chessboard?

My answer is $22$ wherein I do trial and error.
I really cannot think of an equation or formula for this.
 A: Similar here means similar in the usual sense of Euclidean geometry. In particular: Two rectangles are similar if the ratios between the longer and the shorter (or equal) side lengths are equal. 
We are dealing here with rectangles having integer side lengths $1\leq a\leq b\leq 8$. Since, e.g., the rectangles $6\times8$ and $3\times 4$ are similar  we have to count the admissible pairs $(a,b)$ with ${\rm gcd}(a,b)=1$. Given $b$ there are $\phi(b)$ admissible values for $a$, where $\phi$ is Euler's totient function, i.e., $\phi(n)$ is the number of positive integers $\leq n$ that are coprime to $n$. Chasing the cases we obtain the following list:
$$\eqalign{
b=1:\qquad &a\in\{1\}\cr
b=2:\qquad &a\in\{1\}\cr
b=3:\qquad &a\in\{1,2\}\cr
b=4:\qquad &a\in\{1,3\}\cr
b=5:\qquad &a\in\{1,2,3,4\}\cr
b=6:\qquad &a\in\{1,5\}\cr
b=7:\qquad &a\in\{1,2,3,4,5,6\}\cr
b=8:\qquad &a\in\{1,3,5,7\}\cr}$$
It follows that there are $\sum_{k=1}^8\phi(k)=22$ different types. The summatory function of the totient function occurs at OEIS as A002088.
