How to simulate a signal of a particular frequency according to the number of points in a period of the signal?
I suppose that the $n$ points (hereafter, the chunk) precisely divide the base period (which you don't know yet)  evenly. You are lucky. You can verify that by concatenating  the $n$  discrete samples  (or possibly $n-1$ consecutive ones)
by an integer number $K$ of (periodic) repetitions (copy-paste, several times, hereafter the chain), with $Kn\ge m$ (or $K(n-1)\ge m$. It should be smooth  visually. The FFT of the chunk and the chain should be superposable (with appropriate scaling in time and amplitude).
Then, crop the chain of $Kn$ samples to $m$.
Then, using a non-parametric (discrete Fourier transform, FFT), you can estimate the spectrum of the (underlying) continuous data, and find the minimum Nyquist sampling frequency, which may differ from the original one (the initial sampling).