Polynomials with divisors of every degree

We consider polynomials of the form tn−1tn−1 and determine when members of this family have a divisor of every degree in Z[t]Z[t]. With F(x)F(x) defined to be the number of such integers n⩽xn⩽x, we prove the existence of two positive constants c1c1 and c2c2 such thatc1xlogx⩽F(x)⩽c2xlogx.