  In this paper, we study a class of functions defined recursively on the set
of natural numbers in terms of the greatest common divisor algorithm of two
numbers and requiring a minimality condition. These functions are permutations,
products of infinitely many cycles that depend on certain breaks in the natural
numbers involving the primes, and some special products of primes with a
density of approximately $29.4\%$. We show that these functions split into only
two equivalence classes (modulo the natural equivalence relation of eventually
identical maps): one is the class of the identity map and the other is
generated by a map whose discrete derivative is almost periodic with ``periods"
the primorial numbers.
