Multiple harmonic sums and multiple harmonic star sums are (nearly)\n never integers

It is well known that the harmonic sum $H_n(1)=\\sum_{k=1}^n\\frac{1}{k}$ is\nnever an integer for $n>1$. In 1946, Erd\\H{o}s and Niven proved that the nested\nmultiple harmonic sum $H_n(\\{1\\}^r)=\\sum_{1\\le k_1<\\dots<k_r\\le\nn}\\frac{1}{k_1\\cdots k_r}$ can take integer values only for a finite number of\npositive integers $n$. In 2012, Chen and Tang refined this result by showing\nthat $H_n(\\{1\\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In\nthis paper, we consider the integrality problem for arbitrary multiple harmonic\nand multiple harmonic star sums and show that none of these sums is an integer\nwith some natural exceptions like those mentioned above.\n