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Can we find the distribution and/or expected value of $$S=\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$$

where $r_n$ is a uniformly distributed random integer, $r_n \in [1,n]$ and $\text{lcm}$ is the least common multiple?

Or maybe an estimate is possible?

Some boundaries are easy to see. For example:

$$1

For two trials with $10000$ runs each and summing up to $n=20000$ I obtained the following distributions:

enter image description here

It seems as if there are two peaks.

For 20000 trials we obtain the following distribution:

enter image description here

For 100000 trials:

enter image description here

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    Cool question! I can compute the distribution of $gcd(n, X)$ for a uniform random $X \in [n]$, but it's not clear to me how to understand the distribution of $lcm(n, X) = n X / gcd(n, X).$ – 2017-02-19
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    This paper -- https://arxiv.org/pdf/1305.0536.pdf -- might be useful. The authors describe the limiting behavior of the distribution of least common multiple of some number of uniformly chosen integers in [0,n]. I guess you want to take the lcm of $n$ with a random integer... but maybe some of the same techniques can be used to compute it. – 2017-02-26
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    @JRichey, thank you. This might be more useful for my other question, which has a bounty set rn. So if you want to make it into an answer... – 2017-02-26

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