Notes on Divisibility of Catalan Numbers

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.