  We proved recently (see \cite{lhgarasu}) the result on the title for odd
prime divisors of such an $n.$ The result implies for many $n's$, more
precisely, for an infinity of $n$'s with an arbitrary fixed number of prime
divisors, the inexistence of circulant Hadamard matrices, and the inexistence
of Barker sequences of length $n >13$. The proof used a result of Arasu. It
turns out that there is another, shorter proof, of the more general result that
includes the prime $p=2.$ This new proof is based on a result of Brock (see
\cite[Theorem 3.1]{brock}), and besides that, requires just the definition of
the Fourier transform. I noticed Brock's result in a preprint (see
\cite{winterhofetal}) of Winterhof et al. where it is used to study the
inexistence of related Butson-Hadamard matrices.
