For which positive integers $n$ is the following property satisfied: $$\prod_{k|n}k\text{ divides } \prod_{k|n}(k+1)$$
For $n$ up to $15$, only $n=1$ and $n=2$ work. If $n>2$ is prime then $n$ does not divide $2(n+1)$. For other $n$, it looks like the greatest prime divisor of $n$ will appear with greater exponent in $\prod_{k|n}k$ than in $\prod_{k|n}(k+1)$.