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Let $\pi(n)$ denote the prime-counting function and let$$f(n)=\left|\left\lfloor\log n-\lfloor\logn\rfloor-0.1\right\rfloor\right|\left\lfloor\frac{\left\lfloor n/\lfloor\logn-1\rfloor\right\rfloor\lfloor\log n-1\rfloor}{n}\right\rfloor\text{.}$$ Inthis paper we prove that if $n$ is an integer $\ge 60184$ and $f(n)=0$, then$\pi(n)$ does not divide $n$. We also show that if $n\ge 60184$ and $\pi(n)$divides $n$, then $f(n)=1$. In addition, we prove that if $n\ge 60184$ and$n/\pi(n)$ is an integer, then $n$ is a multiple of $\lfloor\log n-1\rfloor$located in the interval $[e^{\lfloor\log n-1\rfloor+1},e^{\lfloor\logn-1\rfloor+1.1}]$. This allows us to show that if $c$ is any fixed integer $\ge12$, then in the interval $[e^c,e^{c+0.1}]$ there is always an integer $n$ suchthat $\pi(n)$ divides $n$.  Let $S$ denote the sequence of integers generated by the function$d(n)=n/\pi(n)$ (where $n\in\mathbb{Z}$ and $n>1$) and let $S_k$ denote the$k$th term of sequence $S$. Here we ask the question whether there areinfinitely many positive integers $k$ such that $S_k=S_{k+1}$.

When $\pi(n)$ does not divide $n$