Let
denote the number of positive regular integers
less than or equal to
and let
(
) be the multidimensional generalization of the arithmetic function
. We study the behaviour of the sequence
. We also investigate the average orders of the functions
,
and
. Here the functions
,
,
generalize the Dedekind function, the sum of the divisors of
and the sum of the unitary divisors of
, respectively. Finally, we give the extremal orders of some compositions involving the functions mentioned previously and the functions
and
which generalize
, the Euler function and the unitary function corresponding to
.