We show that while the number of coprime compositions of a positive integer n into k parts can be expressed as a $\mathbb{Q}$-linear combination of the Jordan totient functions, this is never possible for the coprime partitions of n into k parts. We also show that the number $p_k^{\prime }(n)$ of coprime partitions of n into k parts can be expressed as a $\mathbb{C}$-linear combination of the Jordan totient functions, for n sufficiently large, if and only if $k\in \{2,3\}$ and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that $p_k^{\prime }(n)$ can be always expressed as a $\mathbb{C}$-linear combination of them.

中文翻译：

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互质分区和 Jordan 函数

我们表明，虽然正整数n到k部分的互质组合数可以表示为$\mathbb{问}$-Jordan totient 函数的线性组合，这对于n到k部分的互质分区是不可能的。我们还证明了数$p_{ķ}^{\text{'}}(n)$n到k部分的互质分区可以表示为$\mathbb{C}$-Jordan tient 函数的线性组合，对于n足够大，当且仅当$ķ\in \{2,3\}$并以独特的方式。最后，我们介绍了 Jordan totient 函数的一些推广，并证明了$p_{ķ}^{\text{'}}(n)$总是可以表示为$\mathbb{C}$-它们的线性组合。