Arithmetic progressions of integers that are relatively prime to their digital sums

TL;DR

This paper explores the maximum possible lengths of arithmetic progressions composed of integers that are coprime to their digit sums in a given base, advancing understanding of their distribution and properties.

Contribution

It introduces the concept of b-anti-Niven numbers and investigates the maximum lengths of their arithmetic progressions, providing new insights into their structure.

Findings

Determined bounds for the lengths of such progressions.

Identified conditions under which long progressions exist.

Enhanced understanding of the distribution of b-anti-Niven numbers.

Abstract

For an integer $b\geq 2$, we call a positive integer $b$-anti-Niven if it is relatively prime to the sum of the digits in its base-$b$ representation. In this article, we investigate the maximum lengths of arithmetic progressions of $b$-anti-Niven numbers.