Is there some way to use Euler's Totient function to represent the amount of odd composites that share a factor with integer $n$ less than $n$? For instance,
$n-\phi(n)$ represents the total amount of composites less than $n$ that share a factor with $n$. How might I reduce this to only the odd composites?
If $n$ is even, then no even number is relatively prime to $n,$ so the $n-\phi(n)$ numbers not-relatively-prime-to-n include the $n/2$ even numbers, so in that case, your number is: $$n/2 - \phi(n).$$
If $n$ is odd, then the number of even numbers not-relatively-prime to $n$ is the same as the number of number $\leq \lfloor \frac{n}{2}\rfloor$ not-relatively-prime to $n.$ Now the number of integers relatively prime to $n$ and less than or equal to $n/2$ is $\Phi(n, \lfloor \frac{n}{2}\rfloor),$ where $\Phi(n, x)$ is defined (see this question) as $$\Phi(n,x)=x-\sum_i\lfloor{x/{p_i}}\rfloor+\sum_{i \lt j}\lfloor x/{p_ip_j}\rfloor-\cdots+(-1)^{k}\lfloor x/{n}\rfloor,$$ and thus the number of even numbers not-relatively-prime-to-n equals $$E(n) = \lfloor \frac{n}{2}\rfloor - \Phi(n, \lfloor \frac{n}{2}\rfloor).$$ So, the number of odd non-relativel-prime integers is $$n-\phi(n) - E(n).$$ I am not sure if you can simplify this further.