  In 2015, S. Hong and C. Wang proved that none of the elementary symmetric
functions of $1,1/3,\ldots,1/(2n-1)$ is an integer when $n\geq 2$. In 2017, Kh.
Pilehrood, T. Pilehrood and R. Tauraso proved that the multiple harmonic sums
$H_n(s_1,\ldots,s_r)$ are never integers with exceptions of $H_1(s_1)=1$ and
$H_3(1,1)=1$. They also proved that the multiple harmonic star sums are never
integers when $n\geq 2$. In this paper, we consider the odd multiple harmonic
sums and the odd multiple harmonic star sums and show that none of these sums
is an integer with exception of the trivial case. Besides, we give evaluations
of the odd (alternating) multiple harmonic sums with depth one.
