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The Stormer-Verlet type numerical integrators for simulating equations of motion are well known and widely used in many contemporary scientific contexts. We review why the simple "Verlet" algorithm, which is a direct second order finite difference approximation to a second order differential, is so desirable for initial value problems with conservation properties. For Langevin dynamics, where coupling to a heat-bath is included through a dissipation- fluctuation balance in the equation of motion, the nature of the conservation requirements change, and the premise for the algorithm is put into question. We present a simple re-derivation of the Stormer-Verlet algorithm, including linear friction with associated stochastic noise. We analytically demonstrate that the new algorithm correctly reproduces diffusive behavior of a particle in a flat potential, and, for a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency, and time step for both underdamped and overdamped dynamics within the usual stability limit of the Verlet algorithm. The method, which is as simple as the conventional Verlet scheme, is numerically tested on both low-dimensional nonlinear systems as well as complex systems with many degrees of freedom. Finally, we discuss the opportunities and benefits of proper thermodynamic properties of the numerical integrator for simultaneous accuracy and efficiency in, e.g., Molecular Dynamics simulations.