On a variant of the happy numbers and their generalizations

TL;DR

This paper studies a variant of happy numbers, defining the sequence, presenting results and conjectures about its convergence to 1 within three steps, and providing computational tools and new generalizations.

Contribution

It introduces a new variant of happy numbers, proves key properties, and offers computational methods and conjectures for future research.

Findings

Sequence likely converges to 1 within 3 steps

Provided Wolfram code for sequence computation

Proposed new conjectures and generalizations

Abstract

This paper investigates a variant of the famous "happy numbers" sequence, given by A351327 on the oeis. First of all we'll define this integer sequence, and then we'll show some important results about it; in particular we conjectured that if $k$ is a term of the sequence, then it converges to 1 in a number of steps less or equal to 3. Furthermore it will be possible to find some codes written in Wolfram language in order to compute large terms of the sequence and to support our hypothesis. At the end we'll explore some new conjectures and generalizations about this kind of integer sequence.