An identity for a class of arithmetical functions of two variables

For a positive integer r, let r∗ denote the quotient of r by its largest squarefree divisor (1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1)   or   0according as r∗|n or not where C(n,r) is the well known Ramanujan's sum. In this paper, using a different method, we generalize (∗) to a wide class of arithmetical functions of 2 variables and deduce as special cases (∗) and similar formulae for several generalizations of Ramanujan''s sum