On the Density of Happy Numbers

TL;DR

This paper investigates the density of happy numbers using probabilistic methods, establishing bounds on their upper and lower densities and showing that certain generalizations lack an asymptotic density.

Contribution

It introduces probabilistic techniques to bound happy number densities and demonstrates the non-existence of asymptotic density for some generalizations.

Findings

Upper density > 0.18577

Lower density < 0.1138

Asymptotic density does not exist for some generalizations

Abstract

The happy function $H: \mathbb{N} \rightarrow \mathbb{N}$ sends a positive integer to the sum of the squares of its digits. A number $x$ is said to be happy if the sequence $\{H^n(x)\}^\infty_{n=1}$ eventually reaches one. A basic open question regarding happy numbers is what bounds on the density can be proved. This paper uses probabilistic methods to reduce this problem to experimentally finding suitably large intervals containing a high (or low) density of happy numbers as a subset. Specifically we show that $\bar{d} > .18577$ and $\underline{d} < .1138$. We also prove that the asymptotic density does not exist for several generalizations of happy numbers.