  We compute the equivariant K-theoretic Donaldson--Thomas invariants of
$[\mathbb{C}^2/\mu_r]\times \mathbb{C}$ using factorization and rigidity
techniques. For this, we develop a generalization of Okounkov's factorization
technique that applies to Hilbert schemes of points on orbifolds. We show that
the (twisted) virtual structure sheaves of Hilbert schemes of points on
orbifolds satisfy the desired factorization property. We prove that the
generating series of Euler characteristics of such factorizable systems are the
plethystic exponential of a simpler generating series. For
$[\mathbb{C}^2/\mu_r]\times \mathbb{C}$, the computation is then completed by a
rigidity argument, involving an equivariant modification of Young's
combinatorial computation of the corresponding numerical Donaldson-Thomas
invariants.
