ABSTRACT. An augmented generalized happy function, S [c,b] maps a positive integer to the sum of the squares of its base-b digits and a non-negative integer c. A positive integer u is in a cycle of S [c,b] if, for some positive integer k, S k [c,b] (u) = u and for positive integers v and w, v is w-attracted for S [c,b] if, for some non-negative integer ℓ, S ℓ [c,b] (v) = w. In this paper, we prove that for each c ≥ 0 and b ≥ 2, and for any u in a cycle of S [c,b] , (1) if b is even, then there exist arbitrarily long sequences of consecutive u-attracted integers and (2) if b is odd, then there exist arbitrarily long sequences of 2-consecutive u-attracted integ
An augmented generalized happy function S [c,b] maps a positive integer to the sum of the squares of its base b digits plus c. For b ≥ 2 and k ∈ Z + , a k-desert base b is a set of k consecutive non-negative integers c for each of which S [c,b] has no fixed points. In this paper, we examine a complementary notion, a k-oasis base b, which we define to be a set of k consecutive non-negative integers c for each of which S [c,b] has a fixed point. In particular, after proving some basic properties of oases base b, we compute bounds on the lengths of oases base b and compute the minimal examples of maximal length oases base b for small values of b.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.