Augmented generalized happy functions

TL;DR

This paper investigates the properties of augmented generalized happy functions, proving the existence of arbitrarily long sequences of integers attracted to cycles, with distinctions based on whether the base is even or odd.

Contribution

It establishes new results on the attraction properties of augmented generalized happy functions for all bases and non-negative constants, including the existence of long sequences of attracted integers.

Findings

For even bases, arbitrarily long sequences of consecutive attracted integers exist.

For odd bases, arbitrarily long sequences of two consecutive attracted integers exist.

The results generalize previous work on happy functions to augmented and generalized cases.

Abstract

An augmented happy function, $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base-$b$ digits and a non-negative integer $c$. A positive integer $u$ is in a cycle of $S_{[c,b]}$ if, for some positive integer $k$, $S_{[c,b]}^k(u) = u$ and for positive integers $v$ and $w$, $v$ is $w$-attracted for $S_{[c,b]}$ if, for some non-negative integer $\ell$, $S_{[c,b]}^\ell(v) = w$. In this paper, we prove that for each $c\geq 0$ and $b \geq 2$, and for any $u$ in a cycle of $S_{[c,b]}$, (1) if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted integers and (2) if $b$ is odd, then there exist arbitrarily long sequences of 2-consecutive $u$-attracted integers.