Is there any relation between n(A)and n(B)under following condition

Question

$A=\{1,2,3,4,.....,n\}$, $n(A)=n$, $B=\{(x,y)| x \text{ and } y \text{ are co-prime }, x,y \in A\}$ ,$B \subseteq A \times A$. Is there any relation between $n(A)$ and $n(B)$? If yes what is the relation?

Answer 1

In terms of Euler's Totient function, $n(B)$ is just $$2\sum_1^{n(A)}\phi(k)-1.$$

I don't believe there is any neater form than this.