Binomial collisions and near collisions

A well known result of elementary number theory is that even though the partial sum H_n of the harmonic series increases to infinity, it is never an integer for n>1. Apparently the first published proof goes back to Leopold Theisinger in 1915, and, since then, it has been proposed as a challenging problem in several textbooks. In 1946, Erdos and Niven proved a stronger statement: there is only a finite number of integers n for which there is a positive integer r<=n such that the r-th elementary symmetric function of 1,1/2,...,1/n is an integer. In 2012, Chen and Tang refined this result and succeeded to show that the above sum is not an integer with the only two exceptions: either n=r=1 or n=3 and r=2. In this paper, we consider the integrality problem for sums which are not necessarily symmetric with respect to their variables: the multiple harmonic and multiple harmonic star sums.