A special sequence and primorial numbers

TL;DR

The paper studies a gcd-based recursive permutation f_a on natural numbers, showing the maps are bijections that decompose into finite cycles and that all f_a fall into two EI-equivalence classes (identity vs f_3-like). It develops turning-point theory, proves explicit results for f_3 including f_3(2k+1)=2k and primes as records, and unveils a primorial-translation and almost periodic structure in f. It also characterizes when f_a is eventually the identity through the set A and explores densities of records κ and primes among records, with numerous conjectures linking to primorials, twin primes, and sieve-like phenomena.

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.

Figures (4)

• Figure 1: Twin primes distribution into cycles
• Figure 2: Discrete Derivative of f
• Figure 3: $\frac{\# \{r|r\ \ prime r<n\} }{\# \{r| r<n \} }\ln n$
• Figure 4: Number of primes within the records

Theorems & Definitions (37)

• Proposition 1.1
• Proposition 1.2
• Definition 1
• Definition 2
• Theorem 2.1
• proof
• Remark 1
• Remark 2
• Definition 3
• Corollary 2.2