Coprime partitions and Jordan totient functions

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 p_k′(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 p_k′(n) can be always expressed as a C-linear combination of them.