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The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows: Given an ...

Given a strictly positive initial rational number $x_0$ in $\mathbb Q_>$ we define a sequence $x_0,x_1,\ldots$ recursively by setting $x_{n+1}=x_n+1/S(x_n)$ for $S(x)=a_0+a_1+\ldots+a_k$ where $[...