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The StormerVerlet 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 heatbath is included through the fluctuationdissipation, the nature of the conservation requirements change, and the premise for the algorithm is put into question. We present a simple rederivation of the StormerVerlet 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 within the usual stability limit. The method, which is as simple as conventional Verlet schemes, is numerically tested on both lowdimensional nonlinear systems as well as more 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. Host: Turab Lookman 