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Coprime partitions and Jordan totient functions
Journal of Number Theory ( IF 0.6 ) Pub Date : 2021-07-28 , DOI: 10.1016/j.jnt.2021.05.017
Daniela Bubboloni 1 , Florian Luca 2, 3, 4
Affiliation  

We show that while the number of coprime compositions of a positive integer n into k parts can be expressed as a 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 pk(n) of coprime partitions of n into k parts can be expressed as a C-linear combination of the Jordan totient functions, for n sufficiently large, if and only if k{2,3} and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that pk(n) can be always expressed as a C-linear combination of them.



中文翻译:

互质分区和 Jordan 函数

我们表明,虽然正整数nk部分的互质组合数可以表示为-Jordan totient 函数的线性组合,这对于nk部分的互质分区是不可能的。我们还证明了数pķ'(n)nk部分的互质分区可以表示为C-Jordan tient 函数的线性组合,对于n足够大,当且仅当ķ{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pķ'(n)总是可以表示为C-它们的线性组合。

更新日期:2021-07-28
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