let $X: \mathbb{N}^2 \to \mathbb{N}$
Let $X(a ,b)$ be the number of unique ways we can write $a$ as the sum of $b$ many numbers, where each of the $b$ numbers are co-prime to $a$. Where $a$ $\in \mathbb{N}$ and $b$ $\in \mathbb{N}$
Example:
$X(a ,2) = |\{(x, y): x + y = a$, $gcd(a, x) = gcd(a, y) = 1\}|$
I can easily show that $X(a, 2) = \frac{\phi(a)}{2}$, where $\phi$ is Euler's totient function, when $a > 2$.
Proof:
Let $\Phi_{a} = \{k: gcd(a, k) = 1, k \in \mathbb{N}\}$
let $k \in \mathbb{N}$, then it is easy to see $\forall k < a$, $\exists n \in \mathbb{N}$ such that $a = k + n$
Specifically, this is $n = a - k$
Now if we only consider $k \in \Phi_{a}$ we can see that $n \in \Phi_{a}$. Why is this so? Here is why:
Let $k \in \Phi_{a}$ i.e. $gcd(a, k) = 1$, now if we assume $n \notin \Phi_{a}$ i.e. $gcd(a, n) \neq 1$ it implies that $\exists m \in \mathbb{N}$ such that $m | a$ and $m | n$. But this means that $m | k$ (because $n = a - k$) which contradicts $gcd(a, k) = 1$. Thus our assumption was wrong and therefore $n \in \Phi_{a}$.
Hence it follows that, $\forall x \in \Phi_{a}$ $\exists y \in \Phi_{a}$ such that $x + y = a$. We also know that $|\Phi_{a}| = \phi(a)$ thus once we pair our numbers together we have exactly $\frac{\phi(a)}{2}$ unique pairs $(x, y)$. Here unique means if we have counted the pair $(x, y)$ then we do not count the pair $(y, x)$ as we consider them to be the same pair.
End of proof
Now to my actual question: Can we find the value of $X(a, b)$ in terms of $a$ and $b$ when $b > 2$? So far I have only defined the trivial cases:
$X(a ,b) = \begin{cases} 0, & b \gt a\ &OR& b + 1\equiv a\equiv 0\pmod 2 &OR& b = 1, a\gt 1\\ 1, & b = a\\ \frac{\phi(a)}{2},& b = 2, a \gt 2\\ ?,& Otherwise\end{cases}$
Just so that my question is clear, for $b = 3$ we have:
$X(a ,3) = |\{(x, y, z): x + y + z = a$, $gcd(a, x) = gcd(a, y) = gcd(a, z) = 1\}|$
This question is purely out of interest, thanks in advance for any answers.