I give [here](https://mathoverflow.net/questions/132973/would-the-following-conjectures-imply-lim-inf-n-to-inftyp-nk-p-n-ok-lo) 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.



Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?
The [Hardy-Littlewood prime tuples conjecture](http://mathworld.wolfram.com/k-TupleConjecture.html) implies that $G\_k$ equals the smallest diameter of an admissible $(k+1)$-tuple. In particular, it implies that $G\_{10}=36$ (cf. [here](http://math.mit.edu/~primegaps/)).


Your conjecture implies for $k>3$ that
$$ G\_k>kH\_k+k-\frac{2k^2}{M\_k}>kH\_k+k-3.$$
In particular, it implies that $G\_{10}>36$.


So your conjecture contradicts a widely believed old conjecture.


**P.S.** I have not checked if the OP's first display yields an integral value for $G\_k$. If not, then it is false for the trivial reason that $G\_k$ is an integer.


