Notes on Divisibility of Catalan Numbers

TL;DR

This paper studies the divisibility properties of Catalan numbers and their sum-of-divisors function, revealing new conditions and asymptotic behaviors that deepen understanding of their arithmetic structure.

Contribution

It establishes new criteria for prime factors of Catalan numbers and analyzes the growth of their sum-of-divisors function using advanced number-theoretic methods.

Findings

Catalan numbers often have prime factors of the form 6k-1 under certain conditions

Derived sufficient criteria for divisibility of the sum-of-divisors function of Catalan numbers

Provided asymptotic estimates for the growth of (C_n) using de Bruijn's theorem

Abstract

We investigate the divisibility properties of \sigma(C_n), the sum-of-divisors function applied to Catalan numbers, in relation to other number-theoretic functions. We establish conditions under which C_n has prime factors of the form 6k-1, derive sufficient criteria for divisibility of \sigma(C_n), and explore asymptotic estimates for the growth of \sigma(C_n) using de Bruijn's theorem. These results provide new insights into the arithmetic structure of Catalan numbers.