I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic number. It seems that whenever $k$ doesn't divide $G_{k}$ then we have $$\dfrac{G_{k}}{k}-(1+H_{k})=-\dfrac{k+\delta_{k}}{M_{k}}$$ where $M_{k}$ is the least common multiple of the first $k$ positive integers, and $\delta_{k}$ is the number of integers $m\lt k$ such that $M_{m}=M_{k}$.

My question is: has this already been conjectured? If so, could I get some references? Of course, I also welcome any insight.

Thanks in advance.