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arXiv · 1009.3985

Disquisitiones Arithmeticae and online sequence A108345

Paul Monsky
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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.

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Paul Monsky. 2010-09-21. Disquisitiones Arithmeticae and online sequence A108345. https://arxiv.org/abs/1009.3985

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