Notes on Divisibility of Catalan Numbers
Volkan Yildiz
TL;DR
The paper investigates the divisibility properties of $\sigma(C_n)$ for Catalan numbers by combining elementary results on $\sigma(6k-1)$ with the prime-factor structure of $C_n$, and it establishes asymptotic estimates for prime-factor counts via de Bruijn-type results. It shows that for large $n$, $6 \mid \sigma(C_n)$ and that $\omega(C_n) \sim \frac{2n}{\log n}$, with about half of the prime factors congruent to $5$ mod $6$, i.e., of the form $6k-1$. Under Dirichlet distribution and the Hardy-Littlewood twin-prime conjecture, it predicts infinite twin-prime factors among the prime factors of $C_n$, with density approximately $\dfrac{C_2\,n}{(\log n)^2}$. Overall, the work links divisibility of $\sigma(C_n)$ to deep conjectures about prime distribution and twin primes, enriching the arithmetic understanding of Catalan numbers.
Abstract
We investigate the divisibility properties of σ(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 σ(C_n), and explore asymptotic estimates for the growth of σ(C_n) using de Bruijn's theorem. These results provide new insights into the arithmetic structure of Catalan numbers.