Practical numbers among the binomial coefficients

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 (nk), 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 (2nn) 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.