Reduce the number of frequencies after FFT
Apply a sliding window function of 500 samples length to the signal, for example a Hamming window. Choose a suitable overlap between windows e.g. 75%. Calculate the spectrum of each window and then average over all windows. Then the frequency bins lie exactly at the natural numbers as you wanted.

But beware: each bin does not only represent one exact frequency, but the energy in a frequency band, in this case  $[f-0.5,f+0.5]$Hz