The greatest expanded number expanded by summing of powers of its digits (Q2788847)

Given integers \(p\geq 2\) and \(q\geq 1\) for an integer \(x\) let \(S_{q,p}(x)\) be the sum of the \(q\)th powers of the digits of \(x\) in base \(p\). The authors say that \(x\) is expanded if \(x\leq S_{q,p}(x)\). As the title suggests the paper deals with the largest expanded number \(x\), given \(p\) and \(q\), which the authors denote by \(M(S_{q,p})\). This quantity has been studied by \textit{H. G. Grundman} and \textit{E. A. Teeple} who found \(M(S_{5,p})\) for \(2\leq p\leq 10\) (see [Rocky Mt. J. Math. 38, No. 4, 1139--1146 (2008; Zbl 1229.11018)]). In the paper, the authors give some bounds of \(M(S_{q,p})\) as well as an algorithm to find it. The proofs are elementary.