Hi,
it depends on the "implementation" of the FFT, by means using a scaling factor or not. The picture below is from "Uderstanding Digital Signal Processing" by Richard G. Lyons and addresses this issue.
View attachment 169320
At the modaterators (e.g.
@KlausST or
@FvM ), I hope it is fine to post the picture of two page of a book (not available on google books). To be hornest it's long time ago I have read and agreed to the board rules. If it is not in agreement with the board rules, please delete the picture above.
BR