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We then alter the function to contain the set of primes rather than the set of real numbers. This continued fraction function also has a similar form with coefficients $2,6,48,594,10520,\cdots$, which are twice OEIS:A258301.  \section{Main}  Define the function \begin{equation}  f(x)=\frac{1}{x+\frac{2}{x+\frac{3}{x+\cdots}}} = \underset{k=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{k}{x}  \end{equation}  evaluating this function until convergence for 16 decimal places, gives something that looks like $1/x$, however, after diagnosing the coefficients of the Laurent series, by subtracting likely integer terms we very easily find \begin{equation}  f(x)=\frac{1}{x}-\frac{2}{x^3}+\frac{10}{x^5}-\frac{74}{x^7}+\frac{706}{x^9}-\cdots, \lim_{x\to \infty}f(x)=\frac{1}{x}-\frac{2}{x^3}+\frac{10}{x^5}-\frac{74}{x^7}+\frac{706}{x^9}-\cdots,  \end{equation}  with a search on OEIS giving sequence A000698 which we believe to be plausible.  We can easily imagine a similar function