Currently, I try to improve the solution $$466 369 719 316$$ which is the first of $13$ consecutive non-squarefree numbers. I wonder whether this is the smallest solution.
@MartinHopf You have written history ! I currently search for a large prime : $\phi_{5207}(10^6+n)$. Chances should be good because small factors are impossible, nevertheless, I Passed $n=1800$ without success.
Only PRP of this cyclotomic polynomial I know is Phi(5207,1365)
I only found some very small prime factors of generalized fermat numbers and a solution for a problem a user proposed (but probably not the smallest). I had neither luck with finding a large prime nor with finding significant factors.
I could partially answer why $\phi_n(n-21)$ is a good prime generator. The prime factors upto $11$ occur very very rare. Try to work it out yourself.