Difference between 2$\pi f$ and $\omega$ in Fourier transform
You might be mixing two concepts, pertaining to (following Robert Bristow-Johnson) 


the "analog context" (more formally, the "continuous-time case"),
the "digital context" (more formally the "discrete-time case").


The first concept corresponds to the continuous Fourier transform, for which you can use 
a form of  normalized frequency cycles per second or Hertz ($e^{-j2\pi f}$), while the second is the angular frequency in radians per second ($e^{-j\omega}$).

The second one, with integer index $n$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes  $e^{-j2\pi f n}$ with $f\in \mathbb{R}$ (for instance) or $e^{-j\omega n}$ with $\omega\in \mathbb{R}$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.