Fixed points of augmented generalized happy functions II: Oases and mirages

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 ε 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.