The infinitely admissable sequence
Brachylog, 44 bytes

-:[1]r:[1]cit.
t&lt;gX=:?rc.'(l:2eLg:.rz:%adlL)


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Predicate 0 (main predicate)

-:[1]r:[1]cit.
-:[1]r:[1]c     create array [[1], input-1, 1]
           i    iterate: [1] as input, iterate (input-1) times, predicate 1
            t.  last element is the output


Predicate 1 (auxiliary predicate)

t&lt;gX=:?rc.'(l:2eLg:.rz:%adlL)
t&lt;                              last element of input &lt; Temp
  gX                            [Temp] = X
   X=                           Assign a value to X, which assigns
                                a value to Temp
   X :?rc.                      [Input,X] concatenated is output
         .'(                )   The following about the output cannot
                                be proven:
            l:2eL                   L is a number between 2 and length
                                    of output, inclusive
                Lg:.rz:%a           every element of output, taken modulo L,
                         dlL        after removing duplicates, has length L