Generalized happy numbers (Q2765408)

Let \(a= \sum_{i=0}^n a_i10^i\) be the decimal representation of \(a\) and \(S_2(a)= \sum_{i=0}^n a_i^2\). Then \(a\) is called a happy number if, when \(S_2\) is applied to \(a\) iteratively, the resulting sequence of integers reaches 1. A first possible generalization is to take \(S_e(a)= \sum_{i=1}^n a_i^e\), and there are some results for \(e=3\). Another generalization uses different bases. For \(b\geq 2\) and \(a= \sum_{i=1}^n a_ib^i\) with \(0\leq a_i\leq b-1\), \(S_{e,b}= \sum_{i=1}^n a_i^e\). If the \(S_{e,b}\)-sequence reaches 1, \(a\) is called an \(e\)-power \(b\)-happy number, and also for these some results are given.