  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.
