We know that in $\mathbb Q[x]$ that two cyclotomic polynomials $\Phi_n(x),\Phi_m(x)$ are relatively prime when $m\neq n$. This means that there are integer polynomials $a(x),b(x)$ and a positive integer $D$ such that:
$$a(x)\Phi_n(x)+b(x)\Phi_m(x)=D$$
The smallest such $D$ can be thought of as some kind of integer $GCD$ of the polynomials. I'm wondering if there is a good general formula for this?
Lets write $\langle m,n\rangle$ to be this $D$.
If $m=1$ then $$\langle 1,n\rangle=\Phi_n(1)=\begin{cases}p&n=p^k\\1&\text{otherwise}\end{cases}$$
If $m,n>1$ and $\gcd(m,n)=1$ then we easily get that $\langle m,n\rangle = 1$, by the same argument that $\gcd(x^n-1,x^m-1)=x^{\gcd(m,n)}-1$, since the proof of that is entirely in $\mathbb Z[x]$.
When $m=p,n=2p$ with $p$ prime, you get $\langle m,n\rangle = 2$, since we get $\Phi_{p}(x)(1-x)+\Phi_{2p}(x)(1+x)=2$.
A related question is, given distinct integers $m,n$, what is the smallest $D$ such that for all integers $k$:
$$\gcd(\Phi_m(k),\Phi_n(k))\mid D$$
In general, this $D\mid\langle m,n\rangle$ but is not necessarily equal. For example, when $m=p,n=2p$ where $p$ is an odd prime, then $\Phi_m(k)$ is never even, so $D=1$ when $\langle m,n\rangle = 2$.