A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven.
Williams, E. Simonton
"Further Generalizations of Happy Numbers,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 24:
2, Article 11.
Available at: https://scholar.rose-hulman.edu/rhumj/vol24/iss2/11