n^2 - n - 1 is not one of the cyclotomic polynomials (but it looks very similar to one of them).
For integers n >= 3, x = n^2 - n - 1 is a sequence of prime and composite numbers. The first 10 elements of this sequence are 5, 11, 19, 29, 41, 55, 71, 89, 109, 131. There is nothing unusual about 9 of the first 10 elements in this sequence being prime - in the long run the number of primes among elements of this sequence decays logarithmically, just like the primes do among the full set of positive integers.
However, the interesting thing I noticed by looking at only the composite values of x, ASYMPTOTICALLY JUST ONE HALF OF ALL PRIMES ARE FACTORS OF COMPOSITE ELEMENTS OF THIS SEQUENCE. Prime numbers that never divide any of the composite values of x include: 2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, ... and this list is infinite.
For the asymptotic behaviour I checked values of n up to 1'000'003 at intervals of powers of 10 (plus 3): so at 4, 13, 103, 1'003, 10'003, 100'003, and 1'000'003. I'm currently checking 10'000'003. That's a relatively small n to make an asymptotic generalization, but I conjecture that the ratio converges non-monotonically to 1/2 as n increases, with an accuracy of about 1 part in sqrt(n).
It would be interesting, and perhaps exploitable for improving the efficiency of sieves, if this kind of relationship is found in other polynomials.
For the curious..., except for the number 2, all of these non-divisors end with the digit 3 or 7, and this holds as far as I have seen.