关于包含分拆函数的某一数列的值域与遗漏值
The range and omitted values of a certain sequence involving the partition function
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中文总结 AI 辅助
研究分拆函数p(n)与n之差构成数列的值域，给出组合解释，证明其稀疏性（计数函数阶为(log x)^2，自然密度为零），并扩展到Durfee正方形约束的分拆。
中文摘要 AI 辅助
设\\(p(n)\\)表示普通分拆函数。受关于欧拉函数及其互补计数函数的类似问题的启发，我们研究由分拆导出的数列\\(p(n)-n\\)的值域。我们给出该数列的组合解释，并研究其取到的正整数与遗漏的正整数。我们获得关于相邻取到值之间间隔的精确与渐近信息，并证明该值域异常稀疏：其计数函数的阶为\\((\log x)^2\\)，因此该值域的自然密度为零。我们还将讨论扩展到其Durfee正方形边长至少为固定正整数的分拆。
英文摘要
Let \(p(n)\) denote the ordinary partition function. Motivated by analogous questions concerning Euler's totient function and its complementary counting function, we study the range of the partition-derived sequence \(p(n)-n\). We give combinatorial interpretations of this sequence and investigate both the attained and omitted positive integers. We obtain exact and asymptotic information about the gaps between consecutive attained values and show that the range is remarkably sparse: its counting function has order \((\log x)^2\), and consequently the range has natural density zero. We also extend the discussion to partitions whose Durfee square has side at least a fixed positive integer.
发表机构
- SRM University-AP（SRM大学安得拉邦校区）
- Sister Nivedita University（西瓦·尼维迪塔大学）
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