Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$

Abstract

We investigate a ratio sequence derived from the factorization of $p_{m-1}+1$, where $p_n$ denotes the $n$th prime. For each $m\ge 3$, write $p_{m-1}+1=L_m{R}_m$ with $L_m$ the largest prime factor. Restricting to those $m$ for which $L_m>m$ (equivalently, $m\in \text{A223881}$), we obtain a multiset of values $R_m$. Since $p_{m-1}+1$ is even and $L_m>3$ is odd, all values of $R_m$ are strictly even. Sorting the distinct $R_m$ by decreasing frequency yields a new sequence beginning $2,6,4,8,10,12,14,16\dots$. This article explains how this construction arises naturally from the structure of A223881, why the ``family'' phenomenon appears in plots of $p_{m-1}+1$, and how the frequency ordering of $R_m$ captures the dominant families. Additionally, we propose a heuristic asymptotic model explaining the observed frequency ordering via classical results on primes in arithmetic progressions and support the model with numerical log-log analysis.