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.