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

TL;DR

This paper explores a frequency-ordered ratio sequence derived from the factorization of $p_{m-1}+1$, revealing natural structural patterns and proposing a heuristic model based on prime distribution in arithmetic progressions.

Contribution

It introduces a novel sequence based on prime factorization patterns, explains its emergence from a known sequence, and develops a heuristic asymptotic model supported by numerical analysis.

Findings

The sequence of $R_m$ values exhibits a specific frequency ordering.

The frequency ordering reflects dominant prime factorization families.

A heuristic model explains the observed frequency distribution using classical prime distribution results.

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, $m \in \mathrm{A223881}$), we obtain a multiset of values $R_m$. Sorting the distinct $R_m$ by decreasing frequency yields a new sequence beginning \[ 2,3,4,8,6,12,10,14,15,18,20,24,\dots. \] 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.