Universal sums of generalized heptagonal numbers

TL;DR

This paper characterizes which sums of generalized heptagonal numbers are universal, providing a finite criterion to determine universality based on representing integers up to a certain bound.

Contribution

It establishes a finite bound for universality of sums of generalized heptagonal numbers, extending classical results to a broader class of figurate number sums.

Findings

Identifies a finite bound for universality of these sums

Provides a classification criterion for universal sums

Extends classical polygonal number results to generalized heptagonal numbers

Abstract

In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such sums which are universal, i.e., those that represent every positive integer. We prove an explicit finite bound such that a given sum is universal if and only if it represents positive integer up to the given bound.