Practical numbers among the binomial coefficients

Abstract A practical number is a positive integer n such that every positive integer less than n can be written as a sum of distinct divisors of n. We prove that most of the binomial coefficients are practical numbers. Precisely, letting f ( n ) denote the number of binomial coefficients ( n k ) , with 0 ≤ k ≤ n , that are not practical numbers, we show that f ( n ) n 1 − ( log ⁡ 2 − δ ) / log ⁡ log ⁡ n for all integers n ∈ [ 3 , x ] , but at most O γ ( x 1 − ( δ − γ ) / log ⁡ log ⁡ x ) exceptions, for all x ≥ 3 and 0 γ δ log ⁡ 2 . Furthermore, we prove that the central binomial coefficient ( 2 n n ) is a practical number for all positive integers n ≤ x but at most O ( x 0.88097 ) exceptions. We also pose some questions on this topic.