Let $\phi_{n}(x)$ be the $n$th cyclotomic polynomial. What are the restrictions to $n$ (if any) to have $\phi_{n}(x)$ divides $\phi_{2n}(x)$ (where division is in $\mathbb{Z}[x]$)?Or is it true that $\frac{\phi_{2n}(x)}{\phi_{n}(x)}\in\mathbb{Z}[x]$ for all integers $n$?

When is a primitive *n*th root of unity also a primitive 2*n*th root of unity? Please note that the answer is never, and this can also be seen by unique factorisation. 


Those polynomials are irreducible in $\mathbb Z[X]$ and have different degree... see http://en.wikipedia.org/wiki/Cyclotomic_polynomial 


The restrictions are $n$ nonnegative with $n \le 0$. Another characterization is $n=2n.$ I mention that mainly for the humor value. The OEIS comments:
I suppose one could equally well just not define it, a number of sources don't. $\Phi_{2n}(x)$ is $\Phi_{n}(x)$ for odd $n$ and $\Phi_n(x^2)$ for even $n$. That would argue that $\Phi_0$ should be $1$ if it is defined. 

