  For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The
asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th
prime $p_n$ have been studied intensively in analytic number theory.
Surprisingly, we find that $\pi(x)$ and $p_n$ have many combinatorial
properties which should not be ignored. In this paper we pose 60 open problems
on combinatorial properties of primes (including connections between primes and
partition functions) for further research. For example, we conjecture that for
any integer $n>1$ one of the $n$ numbers $\pi(n),\pi(2n),...,\pi(n^2)$ is
prime; we also conjecture that for any integer $n>6$ there exists a prime $p<n$
such that $pn$ is a primitive root modulo $p_n$. One of our conjectures
involving the partition function $p(n)$ states that for any prime $p$ there is
a primitive root $g<p$ modulo $p$ with $g\in\{p(n):\ n=1,2,3,...\}$.
