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arXiv · 2007.05972

Coprime partitions and Jordan totient functions

Daniela Bubboloni·Florian Luca
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Abstract

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 combinations of the Jordan totient functions, this is never possible for the coprime partitions of $n$ into $k$ parts. We also show that the number $p_k'(n)$ of coprime partitions of $n$ into $k$ parts can be expressed as a $\mathbb{C}$-linear combinations 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 $p_k'(n)$ can be always expressed as a $\mathbb{C}$-linear combinations of them.

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Daniela Bubboloni, Florian Luca. 2020-07-12. Coprime partitions and Jordan totient functions. https://arxiv.org/abs/2007.05972

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