Elated Numbers

For a base \(b \geq 2\), the \(b\)-elated function, \(E_{2,b}\), maps a positive integer written in base \(b\) to the product of its leading digit and the sum of the squares of its digits. A \(b\)-elated number is a positive integer that maps to \(1\) under iteration of \(E_{2,b}\). The height of a \(b\)-elated number is the number of iterations required to map it to \(1\). We determine the fixed points and cycles of \(E_{2,b}\) and prove a range of results concerning sequences of \(b\)-elated numbers and \(b\)-elated numbers of minimal heights. Although the \(b\)-elated function is closely related to the \(b\)-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work.