Subbalancing Numbers

Abstract

A natural number $n$ is called balancing number (with balancer $r$)if it satisfies the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r).$ However, if for some pair of natural numbers $(n,r)$, $1+2+\cdots+(n-1) < (n+1)+(n+2)+\cdots+(n+r)$ and equality is achieved after adding a natural number $D$ to the left hand side then we call $n$ a $D$-subbalancing number with $D$-subbalaner number $r$. In this paper, such numbers are studied for certain values of $D$.