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 $m \geq 3$, 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.
From: Alexander Povolotsky R. [view email]
[v1]
Thu, 7 May 2026 20:41:46 UTC (3 KB)
[v2]
Tue, 26 May 2026 19:14:42 UTC (5 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。