On the Erd\H{o}s-Sloane and Shifted Sloane Persistence Problems

TL;DR

This paper explores two variations of the persistence problem related to digit sequences, examining their connections to conjectures on digit distribution and providing insights into their mathematical properties.

Contribution

It introduces and analyzes the shifted and nonzero persistence problems, establishing links to conjectures on digit equidistribution and generalizing previous results.

Findings

Connections between persistence problems and digit distribution conjectures

Insights into the structure of shifted and nonzero persistence sequences

Potential avenues for future research in digit sequence analysis

Abstract

In this paper, we investigate two variations on the so-called persistence problem of Sloane: the shifted version, which was introduced by Wagstaff; and the nonzero version, proposed by Erd\H{o}s. We explore connections between these problems and a recent conjecture of de Faria and Tresser regarding equidistribution of the digits of some integer sequences and some of its natural generalizations.