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Runge-Kutta methods solve equations of the form $$\dot y = \frac{\mathrm{d} y}{\mathrm{d} t} = F(t,y(t)),$$ where $y$ can be multidimensional. The first step is reconducting your equation to this form. Starting from  \begin{cases} \dot x_n(t) = \frac{p_n(t)}{m} = f(p_{n}(t))& \\ \dot p_n(t) = -k \left[\left(x_n(t)-x_{n-1}(t)\right) + \...
I'm not sure why you think your system of ODEs cannot be solved by using RK4, but I think the easiest way to do this is using DifferentialEquations.jl. Basically you have $2n$ unknowns and $2n$ ODEs, which should be good to solve as long as you have their initial conditions. I think you could look at an example for system of ODEs here from their official ...