The jth divisor function \(d_j\), which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function\(c_j\), which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor function\(c_j^{(r)}\), for \(r\ge 0\), whose definition is motivated by the sum-over divisors recurrence for \(d_j\). We give an overview of properties of \(d_j\), \(c_j\) and \(c_j^{(r)}\), specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers.

中文翻译：

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非平凡除数函数的一些性质和应用

第j个除数函数\（d_j \）是一个非常著名的乘法算术函数，该函数将一个正整数的有序分解分解为j个正整数因子。但是，将正整数的有序因式分解计数为j个因子（每个因子大于或等于2 ）的非乘法第j个非平凡除数函数\（c_j \）很少被研究。此外，我们认为相关的除数函数\（^ C_J {（R）} \） ，对\（R \ GE 0 \） ，其定义是由总和，超过除数复发动机\（D_J \）。我们将概述\（d_j \），\（c_j \）和\（c_j ^ {（r）} \）的属性，特别是关于它们的Dirichlet级数和生成函数以及用二项式系数和和超几何系列。注意三种类型的除数函数之间的一般不等式，然后观察它们的比率如何表示为二项式系数和和超几何级数，并为其中的一些找到明确的Dirichlet级数和Euler积。作为非平凡和相关除数函数的说明性应用，我们展示了如何使用它们来计算Ollerenshaw和Brée所考虑类型的主可逆平方矩阵以及整数和与距离系统。