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Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08 In an introductory post on ...

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$. For example $f(6)=6+3+2=11$, $f(5)=5$. Note that $x$ is a fixed point for ...