On Positive Integers Represented as Arithmetic Series

TL;DR

This paper investigates how positive integers can be expressed as sums of arithmetic sequences, focusing on sums of consecutive odd, even, or general positive integers, revealing fundamental additive and multiplicative relationships.

Contribution

It introduces a systematic study of representing positive integers as sums of specific arithmetic sequences, highlighting their fundamental properties and connections.

Findings

Characterization of integers as sums of successive odd or even numbers

Analysis of integers as sums of consecutive positive integers

Identification of relationships between additive representations and multiplicative properties

Abstract

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In particular, we shall be concerned with the representation of positive integers as arithmetic series of the simplest kind, i.e., either as sums of successive odd positive numbers, or as sums of successive even positive numbers (both treated as Problem 1), or as sums of consecutive positive integers (treated as Problem 2).