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| Autor principal: | |
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| Formato: | Recurso digital |
| Idioma: | inglês |
| Publicado em: |
Zenodo
2025
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| Assuntos: | |
| Acesso em linha: | https://doi.org/10.5281/zenodo.16945277 |
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Sumário:
- <p>We investigate two families of extensions of the well-known Syracuse sequence. In the standard case, starting from an initial integer $v_0 > 0$, the sequence $(v_n)$ is defined by the recurrence: $v_{n+1} = (3v_n + 1)/2$ if $v_n$ is odd, and $v_n/2$ if $v_n$ is even.</p> <p>For the generalizations studied here, we denote the sequence by $(V_n)$ to clearly distinguish it from the classical case. In all cases, the rule applied to even integers remains $V_{n+1} = V_n/2$. We consider two types of extensions:</p> <p>- $3n + b$ extensions: for a fixed odd integer $b$, the transformation becomes $V_{n+1} = (3V_n + b)/2$ when $V_n$ is odd;<br>- The $5n + 1$ extension: the rule becomes $V_{n+1} = (5V_n + 1)/2$ when $V_n$ is odd.</p> <p>Building upon results established in the classical case (when $b = 1$), we show that:<br>- for every odd integer $b$, the $3n + b$ extension admits only finitely many cycles, all of which can be detected by testing initial values $V_0 \leq 2^{f(b)} \cdot |b|$. Moreover, no sequence diverges. In particular, when $|b| < 2^{10}$, the bound $f(b) = 48$ suffices;<br> - in contrast, the $5n + 1$ extension admits at least one divergent orbit.</p>