Fabricate Frequently Factored Functions
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!uΣz:NCNmΠN


Try it online! Or showing the stats for the first 100 n in the first 5000 terms

Explanation

        mΠ     Map factorial over 
          N    the Natural numbers
      CN       Cut into lists of lengths corresponding to the natural numbers
   z:N         ZipWith append the natural numbers and ^
  Σ            Concatenate into one list
 u             Remove duplicates
!              Index


Proof (Without the u)

For triangle numbers : f(½n*(n+1))=n

Otherwise : f(n)=k! for some k.

As the density of triangular numbers is zero, this has the same probability as the sequence of factorials: 1