发表机构
National University of Science and Technology Politehnica Bucharest; Academy of Romanian Scientists（布加勒斯特理工大学; 罗马尼亚科学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过二元结构分解和theta函数求值，解释两块奇数分拆函数a(n)的递推关系，并利用生成函数因式分解建立模任意奇素数的同余式，获得模3、5、13的无限族加权同余。
AI 中文摘要
两块奇数分拆函数 $a(n)$ 是对恰好具有两个不同部分大小、且每个部分大小出现奇数次的分拆的带符号计数。通过根据两个部分大小的 $2$-进赋值来分离相应的表示，我们获得了一个带符号分解，并由两个经典 theta 函数求值补全，这解释了结构关系 $a(2^km)=a(m)+(2^{k-1}-1)\sigma(m)$，其中 $m$ 为奇数，$k\ge1$，$\sigma(m)$ 为除数函数。结合先前建立的 $a(n)$ 的算术公式，该关系给出了沿二元等差数列 $2^k(4n+3)$ 的同余式，以及族 $a\\!\left(2^\alpha3^\beta(12n+11)\right)\equiv0\pmod3$ 和 $a\\!\left(2^\alpha(18n+15)\right)\equiv0\pmod3$。我们还利用涉及 $\operatorname{pod}(n)$（奇数部分互异且偶数部分不受限制的分拆数）的 $a(n)$ 生成函数的因式分解，建立了相关辅助级数模任意奇素数的统一约化。由此得到的涉及 $\operatorname{pod}(n)$ 的 theta 加权和与 $a(n)$ 同余，因此具有涉及除数函数和与模 $4$ 非主 Dirichlet 特征相关的特征除数和的显式求值。作为特例，我们获得了模 $3$、$5$ 和 $13$ 的无限族加权同余式。
英文摘要
The two-block odd partition function \(a(n)\) is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the \(2\)-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation \(a(2^km)=a(m)+(2^{k-1}-1)σ(m)\), where \(m\) is odd, \(k\ge1\), and \(σ(m)\) is the sum-of-divisors function. Combined with a previously established arithmetic formula for \(a(n)\), this relation yields congruences along the dyadic progressions \(2^k(4n+3)\), as well as the families \(a\!\left(2^\alpha3^β(12n+11)\right)\equiv0\pmod3\) and \(a\!\left(2^α(18n+15)\right)\equiv0\pmod3\). We also use a factorization of the generating function of \(a(n)\) involving \(\operatorname{pod}(n)\), the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving \(\operatorname{pod}(n)\) are congruent to \(a(n)\) and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo \(4\). As special cases, we obtain infinite families of weighted congruences modulo \(3\), \(5\), and \(13\).
CommentsSubmitted to Quaestiones Mathematicae on 30 August 2026
