In this paper, we present a generalization of the super-twisting algorithm for perturbed chains of integrators of arbitrary order. This higher order super-twisting (HOST) controller is homogeneous with respect to a family of dilations and is continuous. It is built as a dynamic controller (with respect to the state variable of the chain of integrators) and the convergence analysis is performed by the use of a homogeneous strict Lyapunov function which is explicitly constructed. The effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four, first pure then perturbed, where we compare the performances of two HOST controllers.