This repository contains the Jupyter notebook \texttt{scatter\_Rk\_density.ipynb}, which computes and visualizes the normalized iterative divisor-sum ratios:\[R_k(n) = \frac{\sigma^{(k)}(n)}{n},\]where $\sigma(n)$ is the sum-of-divisors function and $\sigma^{(k)}(n)$ denotes its $k$-fold iterate. The notebook implements an \emph{efficient sieve-based algorithm} to calculate $\sigma(n)$ and its iterates for large ranges of $n$, making it suitable for high-resolution empirical studies. It produces a \emph{scatter plot} of $R_k(n)$ versus $n$ and saves the figure in \textbf{PDF format} for direct inclusion in \LaTeX{} documents. \subsection*{Key Features}\begin{itemize} \item Efficient computation of $\sigma(n)$ using a \textbf{sieve-based algorithm}. \item Iterative computation of $\sigma^{(k)}(n)$ for arbitrary $k \ge 1$. \item Visualization of $R_k(n)$ up to large $n$. \item Scatter plot highlighting \emph{measure density} and \emph{asymptotic behavior}. \item Generates a \textbf{publication-ready PDF} figure for research papers.\end{itemize} \subsection*{Applications}\begin{itemize} \item Empirical analysis of \emph{iterated arithmetic functions}. \item Investigation of \emph{measure density} and \emph{boundedness} properties. \item Numerical support for conjectures related to divisor-sum iterates (e.g., Schinzel’s conjecture).\end{itemize} \subsection*{Output}\begin{itemize} \item \texttt{scatter\_density\_Rk.pdf} — scatter plot visualization. \item Ready-to-use \LaTeX{} figure integration.\end{itemize} \subsection*{Requirements}\begin{itemize} \item Python 3.x \item \texttt{numpy} and \texttt{matplotlib} \item Optional: \texttt{sympy} (if using direct divisor enumeration)\end{itemize}