A special sequence and primorial numbers

TL;DR

This paper introduces a class of recursively defined functions based on gcd algorithms, revealing their permutation structure, cycle decomposition, and connection to primorial numbers, with implications for number theory.

Contribution

It characterizes these functions as permutations with specific cycle structures, showing they form only two equivalence classes related to the identity and primorial-based maps.

Findings

Functions are permutations with infinitely many cycles.

They split into two equivalence classes: identity and primorial-related maps.

The primorial maps have almost periodic derivatives.

Abstract

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.