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The Hofstadter consecutive-sum sequence omits infinitely many positive integers

Quanyu Tang

Abstract

Let $(a_n)_{n\ge 1}$ be the greedy self-generating sequence defined by $a_1=1$, $a_2=2$, and, for $k\ge 3$, by taking $a_k$ to be the least integer greater than $a_{k-1}$ that can be written as a sum of at least two consecutive earlier terms. Hofstadter asked about the asymptotic behavior of this sequence. In this paper we prove that $$ n+ω(1)\le a_n \ll n^{4175/2506+o(1)}. $$ In particular, $(a_n)_{n\ge1}$ omits infinitely many positive integers, thereby settling a conjecture from the OEIS entry A005243.

The Hofstadter consecutive-sum sequence omits infinitely many positive integers

Abstract

Let be the greedy self-generating sequence defined by , , and, for , by taking to be the least integer greater than that can be written as a sum of at least two consecutive earlier terms. Hofstadter asked about the asymptotic behavior of this sequence. In this paper we prove that In particular, omits infinitely many positive integers, thereby settling a conjecture from the OEIS entry A005243.
Paper Structure (13 sections, 18 theorems, 87 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A qualitative lower bound and infinitely many omissions
  4. Monotonicity and eventual linearity
  5. Consecutive-block decompositions beyond the linear tail
  6. Powers of two force a quadratic exponential equation
  7. Completion of the proof
  8. A quantitative upper bound for the classical sequence
  9. Greedy enumeration of the representable integers
  10. From consecutive sums to a convex difference set
  11. Bloom's convex difference-set bound and the basic recursion
  12. Iteration and the final exponent
  13. Concluding remarks

Key Result

Theorem 1.4

For every integer $s\ge 2$ and every seed $\mathbf{u}$ of length $s$ consisting of positive integers, the sequence $(b_n^{\mathbf{u}})_{n\ge1}$ is eventually nondecreasing; indeed, it is nondecreasing for all $n\ge s$. Moreover, $(b_n^{\mathbf{u}})_{n\ge1}$ is unbounded.

Figures (1)

  • Figure 1: Numerical data for sequence \ref{['eq:hof']}: the top panel shows $b_n=a_n-n$ (on a logarithmic $n$-axis), and the remaining panels plot the ratios $b_n/n^{1/k}$ for $k=5,4,3,2$ over $1\le n\le 30000$.

Theorems & Definitions (37)

  • Conjecture 1.2: OEIS-A005243
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2: SchinzelTijdeman1976
  • Corollary 2.3
  • proof
  • ...and 27 more