Disquisitiones Arithmeticae and online sequence A108345

TL;DR

This paper investigates the density of a specific subset of integers related to a power series in modular arithmetic, using classical number theory results and computational evidence to refine the understanding of its distribution.

Contribution

It applies Gauss's sums of three squares to analyze the density of a subset of sequence A108345, providing new bounds and computational insights.

Findings

Subset of B excluding integers congruent to 15 mod 16 has density 0.

Computational evidence suggests B has density approximately 1/32.

The study refines previous density bounds using classical number theory.

Abstract

Let g be the element that is the sum of x^(n^2) for n >= 0 of A=Z/2[[x]], and let B consist of all n for which the coefficient of x^n in 1/g is 1. (The elements of B are the entries 0, 1, 2, 3, 5, 7, 8, 9, 13, ... in A108345; see The On-Line Encyclopedia of Integer Sequences (OEIS).) Cooper, Eichhorn, and O'Bryant [1] have shown that the (upper) density of B is at most 1/4, and it is conjectured that B has density 0. This note uses results of Gauss on sums of 3 squares to show that the subset of B consisting of all n not congruent to 15 mod 16 has density 0. The final section gives some computer calculations, made by Kevin O'Bryant, indicating that, pace [1], B has density 1/32.