  After the work of Bordell\`{e}s, Dai, Heyman, Pan and Shparlinki (2018) and
Heyman (2019), several authors studied the averages of arithmetic functions
over the sequence $[x/n]$ and the integers of the form $[x/n]$. In this paper,
we give three remarks on this topic. Firstly, we improve the result of Wu and
Yu (2022) on the distribution of the integers of the form $[x/n]$ in arithmetic
progressions by using a variant of Dirichlet's hyperbola method. Secondly, we
prove an asymptotic formula for the number of primitive lattice points with
coordinates of the form $[x/n]$, for which we introduce a certain averaging
trick. Thirdly, we study a certain "multiplicative" analog of the Titchmarsh
divisor problem. We derive asymptotic formulas for such "multiplicative"
Titchmarsh divisor problems for "small" arithmetic functions and the Euler
totient function with the von Mangoldt function. However, it turns out that the
average of the Euler totient function over the $[x/p]$-sequence seems rather
difficult and we propose a hypothetical asymptotic formula for this average.
