  Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with
square-summable coefficients. We study composition operators
$\mathscr{C}_\varphi$ on $\mathscr{H}^2$ which are generated by symbols of the
form $\varphi(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq
1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet
series of $\varphi$, then the operator $\mathscr{C}_\varphi$ admits an
associated orthogonal decomposition. Under sparseness assumptions on
$\mathbb{P}$ we use this to asymptotically estimate the approximation numbers
of $\mathscr{C}_\varphi$. Furthermore, in the case that $\varphi$ is supported
on a single prime number, we affirmatively settle the problem of describing the
compactness of $\mathscr{C}_\varphi$ in terms of the ordinary Nevanlinna
counting function. We give detailed applications of our results to affine
symbols and to angle maps.
