New recurrences for divisor sum functions and triangular numbers

TL;DR

This paper introduces new recurrence relations for divisor sum functions and triangular numbers, expanding on Euler's classical work and inspired by Ewell's 1977 findings.

Contribution

It presents novel recurrence formulas for divisor sum functions and triangular numbers, building on historical mathematical theorems.

Findings

Derived new recurrence relations for divisor sum functions

Established recurrences for triangular numbers

Extended Euler's classical results with modern proofs

Abstract

Euler discovered recurrence for divisor sum functions as a consequence of the pentagonal numbers theorem. With similar idea and also motivated by Ewell's work in 1977, we prove new recurrences for certain divisor sum functions and triangular numbers.