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

Question

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.

Answer 1

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.