Sequences of consecutive factoradic happy numbers

Abstract

Given a positive integer $n$, the factorial base representation of $n$ is given by $n={\sum }_{i=1}^k{a}_i\cdot i!$, where $a_k\ne 0$ and $0\le a_i\le i$ for all $1\le i\le k$. For $e\ge 1$, we define $S_{e,!}:{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$ by $S_{e,!}\left(0\right)=0$ and $S_{e,!}\left(n\right)={\sum }_{i=1}^n{a}_i^e$, for $n\ne 0$. For $\ell \ge 0$, we let $S_{e,!}^{\ell }\left(n\right)$ denote the $\ell$-th iteration of $S_{e,!}$, while $S_{e,!}^0\left(n\right)=n$. If $p\in {\mathbb{Z}}^{+}$ satisfies $S_{e,!}\left(p\right)=p$, then we say that $p$ is an $e$-power factoradic fixed point of $S_{e,!}$. Moreover, given $x\in {\mathbb{Z}}^{+}$, if $p$ is an $e$-power factoradic fixed point and if there exists $\ell \in {\mathbb{Z}}_{\ge 0}$ such that $S_{e,!}^{\ell }\left(x\right)=p$, 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. We prove that all positive integers are $1$-power factoradic happy and, for $2\le e\le 4$, we prove the existence of arbitrarily long sequences of $e$-power factoradic $p$-happy numbers. A curious result establishes that for any $e\ge 2$, 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 $2\le e\le 5$, for some values of $m$.