On an algorithm that generates an interesting maximal set P(n) of the naturals for any n greater than or equal to 2

TL;DR

This paper introduces an algorithm to generate the maximal set P(n) of natural numbers with a specific sum property, proving Q(n) is finite and enabling efficient computation of P(n) for any n ≥ 2.

Contribution

The paper presents a novel algorithm with worst-case O(1) complexity for generating the finite complement Q(n), thereby enabling the construction of the maximal set P(n) for any n ≥ 2.

Findings

Q(n) is finite for all n ≥ 2

An O(1) worst-case complexity algorithm is developed for Q(n)

P(n) can be constructed once Q(n) is known

Abstract

The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in P(n). Here largest is in the set theoretic sense and n is greater than or equal to 2. We call P(n) a maximal set obeying this property. For small n say 2 or 3, it is possible to develop P(n) intuitively but we strongly felt the necessity of an algorithm for any n greater than or equal to 2. Now P(n) shall invariably be a infinite set so we define another set Q(n) such that Q(n)=N-P(n), prove that Q(n) is finite and, since P(n) is automatically known if Q(n) is known, design an algorithm of worst case O(1) complexity which generates Q(n).