  If a sequence $(a_n)$ of non-negative real numbers has ``best possible''
distribution in arithmetic progressions, Bombieri showed that one can deduce an
asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$.
By constructing appropriate sequences, we show that any weakening of the
well-distribution property is not sufficient to deduce the same conclusion.
