\(k\)-gap balancing numbers (Q2341950)

A natural number \(n\) is called a balancing number with balancer \(r\) if \[ 1 + 2 + \ldots + (n - 1) = (n + 1) + (n + 2) + \ldots + (n + r). \] On other hand \(n\) is called a cobalancing number with cobalancer \(r\) if \[ 1 + 2 + \ldots + n = (n + 1) + (n + 2) + \ldots + (n + r). \] Several papers in this area are presently available. In a very recent paper, the authors [Fibonacci Q. 51, No. 3, 239--248 (2013; Zbl 1350.11038)] introduced the concept of gap balancing number by deleting more than one consecutive numbers from the list of the first \(m\) natural numbers, so that the sum of the numbers to the left of these deleted numbers is equal to the sum to the right. In the same paper, they defined \(k\)-gap balancing numbers (which they call \(g_k\)-balancing numbers) differently for odd and even \(k\) and studied the case \(k = 2\) completely. These definitions are the following: Definition 1. Let \(k\) be an odd natural number. We call a natural number \(n\) a \(k\)-gap balancing number (or \(g_k\)-balancing number) if \[ 1 + 2 + \ldots + \left(n-\frac{k+1}{2}\right) = \left(n+\frac{k+1}{2}\right) + \left(n+\frac{k+3}{2}\right) + \cdots + (n + r). \] for some natural number \(r\), which we call a \(k\)-gap balancer(or a \(g_k\)-balancer) corresponding to \(n\). Definition 2 Let \(k\) be even. If \[ 1 + 2 + \ldots + \left(n-\frac{k}{2}\right) = \left(n+\frac{k}{2}+1\right) + \left(n+\frac{k}{2}+2\right) + \cdots + (n + r). \] for some natural numbers \(n\) and \(r\) then we call \(2n + 1\) a \(k\)-gap balancing number (or \(g_k\)-balancing number) and \(r\) a \(k\)-gap balancer (or a \(g_k\)-balancer) corresponding to this \(k\)-gap balancing number. In this paper, the authors provide a detailed discussion on \(g_k\)-balancing numbers for \(k = 3, 4, 5\). Furthermore this paper deals with the study of \(k\)-gap balancing numbers for arbitrary \(k\). For \(k \geq 2\), the \(k\)-gap balancing numbers partition into two or more disjoint classes. It is always possible to explore two classes of \(k\)-gap balancing numbers for all \(k \geq 2\) and a third class for some specified \(k\). However, they have failed to present all classes for arbitrary \(k\).