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We recently introduced the recursive divisor function $\kappa_x(n)$, arecursive analogue of the usual divisor function. Here we calculate itsDirichlet series, which is ${\zeta(s-x)}/(2 - \zeta(s))$. We show that$\kappa_x(n)$ is related to the ordinary divisor function by $\kappa_x *\sigma_y = \kappa_y * \sigma_x$, where * denotes the Dirichlet convolution.Using this, we derive several identities relating $\kappa_x$ and some standardarithmetic functions. We also clarify the relation between $\kappa_0$ and themuch-studied number of ordered factorizations $K(n)$, namely, $\kappa_0 = {\bf1} * K$.

Properties of the recursive divisor function and the number of ordered factorizations