Sequences of consecutive factoradic happy numbers

Abstract: Given a positive integer $n$, the factorial base representation of $n$ is given by $n=\sum_{i=1}^ka_i\cdot i!$, where $a_k\neq 0$ and $0\leq a_i\leq i$ for all $1\leq i\leq k$. For $e\geq 1$, we define $$S_{e,!}:\mathbb{Z}<em>{\geq0}\to\mathbb{Z}</em>{\geq0}$$ by $S_{e,!}(0) = 0$ and $S_{e,!}(n)=\sum_{i=0}<sup>{n}a_i<sup>e$, for $n \neq 0$. For $\ell\geq 0$, we let $S_{e,!}<sup>\ell(n)$ denote the $\ell$-th iteration of $S_{e,!}$, while $S_{e,!}<sup>0(n)=n$. If $p\in\mathbb{Z}<sup>+$ satisfies $S_{e,!}(p)=p$, then we say that $p$ is an $e$-power factoradic fixed point of $S_{e,!}$. Moreover, given $x\in \mathbb{Z}<sup>+$, if $p$ is an $e$-power factoradic fixed point and if there exists $\ell\in \mathbb{Z}<em>{\geq 0}$ such that $S</em>{e,!}<sup>\ell(x)=p$, then we say that $x$ is an $e$-power factoradic $p$-happy number. Note an integer $n$ is said to be an $e$-power factoradic happy number if it is an $e$-power factoradic $1$-happy number. In this paper, we prove that all positive integers are $1$-power factoradic happy and, for $2\leq e\leq 4$, we prove the existence of arbitrarily long sequences of $e$-power factoradic $p$-happy numbers. A curious result establishes that for any $e\geq 2$ the $e$-power factoradic fixed points of $S_{e,!}$ that are greater than $1$, always appear in sets of consecutive pairs. Our last contribution, provides the smallest sequences of $m$ consecutive $e$-power factoradic happy numbers for $2\leq e\leq 5$, for some values of $m$.