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    $\begingroup$ Well, the ratio is supposed to tend to $1$ in the limit, so if it got all the way up to $1{.}76$, it has to start falling back down at some point. $\endgroup$ Commented Dec 2, 2021 at 16:35
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    $\begingroup$ @EmilJeřábek Hard to disagree. Still, I find this kind of behavior somehow surprising. Besides, the curve is quite smooth so maybe somebody can guess some more precise statement about asymptotics... $\endgroup$ Commented Dec 2, 2021 at 19:50
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    $\begingroup$ The correct constant is actually $2 \pi / \sqrt{3}$ rather than $\pi \sqrt{2/3}$. That brings the peak down to about 1.24 (but doesn't change the surprising shape). There is some similar discussion at oeis.org/search?q=A000607 of the numerics. $\endgroup$ Commented Dec 3, 2021 at 15:12
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    $\begingroup$ I guess what is happening is lower order terms are something like $\exp(c \sqrt{n / \log n})$ for smaller constants $c$, and it takes a while before these are insignificant compared to the main term. $\endgroup$ Commented Dec 3, 2021 at 15:15
  • $\begingroup$ @SeanEberhard In fact the proposed asymptotic for A000607 has an opposite property. They supply the list up to 50000 and the ratio there grows with increasing speed, crossing 1 at $n=13194$. $\endgroup$ Commented Dec 4, 2021 at 5:47