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 .

Key Words: arithmetical function, composition, regular integers , average orders, extremal orders.

2010 Mathematics Subject Classification: Primary 11A25,

Secondary 11N37