Abstract

We generalize the concept of happy number as follows. Let e = (e_{0}, e_{1}, ....) be a sequence with e_{0} = 2 and e_{i} = {1, 2} for i > 0. Define S_{e} : ℤ^{+} → ℤ^{+} by

S_{e} (∑_{i=0}^{n} a_{i}10^{i}) = ∑_{i=0}^{n} a_{i}^{e_{i}}.

If S_{e}^{k}(a) = 1 for some k ∈ ℤ^{+}, then we say that a is a semihappy number or, more precisely, an e-semihappy number. In this paper, we determine fixed points and cycles of the functions S_{e} and discuss heights of semihappy numbers. We also prove that for each choice of e, there exist arbitrarily long finite sequences of consecutive e-semihappy numbers.