I did experiments, and it looks like the density of those $n$ for which the ratio is an integer approaches a limit, and the limit is approximately $0.2187.$ Of this writing I went up to $1000000$ - (with *Mathematica*); the convergence seems reasonably rapid, however. Note that the density is **not** the same as the density of square-free integers - I am not sure if there is any sort of heuristic that predicts what it should be.

However, another experiment shows that the density of square-free numbers amongst sums of the first $n$ primes is about $6/\pi^2,$ as expected. There are two things at play here: the sum of the first $n$ primes is square free, and also its largest prime divisor is no bigger than the $n$-th prime. Now, the sum of the first $n$ primes is of order of $\frac{n^2}{2 \log n},$ and the expected value of the largest prime divisors of numbers smaller than $N$ is $\frac{\pi^2}{12} \frac{n^2}{\log n},$ not sure what the distribution looks like. Assuming the events of square-freeness and "reasonable smoothness" are independent should give a heuristic asymptotic.

**UPDATE** Now up to $10^7,$ and the probability of integrality is slowly declining (now $0.2174$). So, not sure what to think.

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