On certain generalizations of the Smarandache function

positive integer. This arithmetical function is connected to the number of divisors of n, and other important number theoretic functions (see e.g. [6], [7], [9], [10]). A very natural generalization is the following one: Let f: N * ~ N * be an arithmetical function which satisfies the following property: (Pd For each n E N * there exists at least a k E N * such that nlf(k). Let Fj: N * ~ N- defined by Fj(n) = min{k EN: nlf(k)}. (1) Since every subset of natural numbers is well-ordered, the definition (1) is correct, and clearly Fj ( n) ~ 1 for all n E N-. Examples. 1) Let id ( k) = k for all k ~ 1. Then clearly (Pd is satisfied, an