The only such $n$ less than $5.7\cdot10^9$ are all even: $2, 38040, 51888, 236644, 260880, 3097024, 5283852, 5667312, 11777472, 46120848, 52981252, 69128640, 121352208, 330364848, 485906400, 662736600, 769422720, 1111869360, 1267978320, 1272335760, 1426817904, 1807128528, 2107406448, 2381691312, 2452404544, 2691587568, 3758996016, 4403660352, 5139308592.$

This doesn't seem to work for $n = 2$. Do you mean to have $\displaystyle \frac1n\sum_{k=0}^n(-1)^k\binom{n}{k}\log^{n-k}(n)\int_0^\infty e^{-x}\log^k(x)\,dx$?

Oh sorry, I thought this was a sum over both $k$ and $j$.

Looking at the first 20 values of $N$, it looks like the sum equals $\frac{4}{45} (n-2) (n-1) (n^2 + 3n + 2)$.

And by the way, I made a couple edits since posting -- I don't know if you saw them -- a conjectured closed form and an approximate distribution fit.

Thanks. Yes I was surprised by that histogram too. Originally I thought there might have been a bug in my code. I reran my code without the intersection stopping condition and the histogram came out uniform. An example of that is in the linked notebook.

@MarkS I did indeed try and nothing fruitful came out for the entire distribution. But the tail does seem to have some regularity. I have code that can run $10^11$ simulations in the order of hours. And this weekend I’ll have access to a machine with many cores and I’m hoping to run $10^12$. After this I plan to post an answer detailing the data.