Sequences of Consecutive Happy Numbers in Negative Bases

TL;DR

This paper investigates the properties of happy numbers in negative bases, establishing specific congruence conditions for -2 and -3 bases and demonstrating the existence of arbitrarily long sequences of consecutive happy numbers in certain negative bases.

Contribution

It characterizes negative-base happy numbers and proves the existence of arbitrarily long sequences of consecutive happy numbers in specific negative bases.

Findings

-2-happy numbers are exactly those congruent to 1 mod 3.

-3-happy numbers are precisely the odd integers.

Existence of arbitrarily long sequences of consecutive happy numbers in certain negative bases.

Abstract

For $b\leq -2$ and $e \geq 2$, let $S_{e,b}:\mathbb{Z}\to\mathbb{Z}_{\geq 0}$ be the function taking an integer to the sum of the $e$-powers of the digits of its base $b$ expansion. An integer $a$ is a $b$-happy number if there exists $k\in\mathbb{Z}^+$ such that $S_{2,b}^k(a) = 1$. We prove that an integer is $-2$-happy if and only if it is congruent to 1 modulo 3 and that it is $-3$-happy if and only if it is odd. Defining a $d$-sequence to be an arithmetic sequence with constant difference $d$ and setting $d = \gcd(2,b - 1)$, we prove that if $b \leq -3$ odd or $b \in \{-4,-6,-8,-10\}$, there exist arbitrarily long finite sequences of $d$-consecutive $b$-happy numbers.