  We determine the asymptotic density $\delta_k$ of the set of ordered
$k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime
power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$,
$1\le i\le k$, and deduce asymptotic formulae with error terms regarding this
problem and analogous ones. We give numerical approximations of the constants
$\delta_k$ and improve the error term of formula (1.2) due to {\sc E. Cohen}.
  We point out that our treatment, based on certain inversion functions, is
applicable also in case $k=1$ in order to establish asymptotic formulae with
error terms regarding the densities of subsets of $\N$ with additional
multiplicative properties. These generalize an often cited result of {\sc G. J.
Rieger}.
