Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910262979723264 |
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| author | Povolotsky, Alexander R |
| author_facet | Povolotsky, Alexander R |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_08256 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$ Povolotsky, Alexander R Number Theory 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. |
| title | Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.08256 |