Properties of terms of OEIS A342810

TL;DR

This paper investigates the properties of sequence A342810 from OEIS, providing proofs about its terms based on prime factorization and modular equations involving powers of 10.

Contribution

It offers new theoretical insights into the structure of sequence A342810, linking its terms to prime factors and solutions of specific modular equations.

Findings

Terms with form 3^m * y have prime factors related to solutions of 10^n ≡ 1 (mod n)

Sequences with form 3^m * p * q generate infinite related terms

Proved structural properties of sequence terms based on prime factorization

Abstract

The OEIS sequence A342810 contains the numbers that divide the smallest number that has the sum of their digits. It is proved that if a term $x$ has the form $3^m \times y$ where $m \geq 2$, then all prime factors of $y$ are prime divisors of solutions to $10^n \equiv 1 \; ( \bmod n)$. It is also proved that if a term $x$ has the form $3^m \times p \times q$ where $m \geq 2$, where $p$ is a prime divisor of a solution to $10^n \equiv 1 \; ( \bmod n)$ and where $q$ is the product of all other factors of the prime factorisation of $x$, then all numbers $3^m \times p^i \times q$ are also terms of the sequence A342810 for any integer $i$.