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A $2$-Regular Sequence That Counts The Divisors of $n^2 + 1$

Anton Shakov

TL;DR

This work identifies a $2$-regular sequence $s(n)$ whose value-multiplicities encode the divisor counts of $m^2+1$ by proving $\#\{n:s(n)=m\}=\tau(m^2+1)$ and deriving a generating function $\sum_{m\ge0} \tau(m^2+1)x^m=\sum_{n\ge1} x^{s(n)}$. A binary integer-pair tree with left/right maps $L(d,m)=(d,m+d)$ and $R(d,m)=(((m+d)^2+1)/d, m+(m^2+1)/d)$ together with an involution provides a combinatorial proof of the counting identity and ties to the $2$-regular recurrence structure of $s(n)$. The paper also investigates further properties, including a row-sum recurrence $r_n=5r_{n-1}-2r_{n-2}$, an explicit row-average formula, a primality criterion for $n^2+1$ via level sets $igl\{m:s(m)=n\bigr\}$, and a Fibonacci-path connection yielding $s(a(n))=F_n$. Overall, the results embed the divisor function $\tau(m^2+1)$ within a $2$-regular framework and reveal structural links to Fibonacci sequences and symmetric tree constructions.

Abstract

We introduce the $2$-regular integer sequence A383066 $= (s(n))_{n \geq 1}$, which begins $0, 1, 1, 2, 3, 3, 2, \ldots$. We prove that the number of occurrences of an integer $m \geq 0$ in this sequence is equal to $τ(m^2+1)$, the number of divisors of $m^2 + 1$. Using this fact, we give a generating function for $τ(m^2+1)$. We also discuss other interesting properties of $s(n)$, including its relationship to the Fibonacci sequence.

A $2$-Regular Sequence That Counts The Divisors of $n^2 + 1$

TL;DR

This work identifies a -regular sequence whose value-multiplicities encode the divisor counts of by proving and deriving a generating function . A binary integer-pair tree with left/right maps and together with an involution provides a combinatorial proof of the counting identity and ties to the -regular recurrence structure of . The paper also investigates further properties, including a row-sum recurrence , an explicit row-average formula, a primality criterion for via level sets , and a Fibonacci-path connection yielding . Overall, the results embed the divisor function within a -regular framework and reveal structural links to Fibonacci sequences and symmetric tree constructions.

Abstract

We introduce the -regular integer sequence A383066 , which begins . We prove that the number of occurrences of an integer in this sequence is equal to , the number of divisors of . Using this fact, we give a generating function for . We also discuss other interesting properties of , including its relationship to the Fibonacci sequence.
Paper Structure (3 sections, 8 theorems, 12 equations, 9 figures)

This paper contains 3 sections, 8 theorems, 12 equations, 9 figures.

Table of Contents

  1. Introduction and proof of the main result
  2. Other properties of the sequence
  3. Acknowledgments

Key Result

Theorem 4

We have where $\tau$ is the usual divisor counting function and $s(n)_{n\geq1}$ is a $2$-regular sequence defined recursively by with initial conditions $s(1) = 0,\ s(2)=1,\ s(3) = 1$.

Figures (9)

  • Figure 1: Integer pair tree.
  • Figure 2: Second component tree.
  • Figure 3: Three generations of second pair components.
  • Figure 4: Linear dependencies between second pair components.
  • Figure 5: Dependencies re-indexed in terms of $s(n)$.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Definition 1
  • Example 2
  • Example 3
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 8 more