Elated Numbers

Abstract For a base $$b \ge 2$$ , 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.