For a given arithmetical function f : N → N, let F : N → N be defined by F(n) = min{m ≥ 1 : n|f(m)}, if this exists. Such functions, introduced in [4], will be called as the f-minimum functions. If f satisfies the property a ≤ b → f(a)|f(b), we shall prove that F(ab) = max{F(a), F(b)} for (a, b) = 1. For a more restrictive class of functions, we will determine F(n) where n is an even perfect number.