Sequences of Consecutive Happy Numbers in Negative Bases

Abstract

For $b\le -2$ and $e\ge 2$, let $S_{e,b}:\mathbb{Z}\to {\mathbb{Z}}_{\ge 0}$ 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 $b\le -3$ odd or $b\in \{-4,-6,-8,-10\}$, there exist arbitrarily long finite sequences of $d$-consecutive $b$-happy numbers.

8 pages, 1 table