Exponentiation Sequence

05AB1E, 12 11 10 bytes

Supports the sequence up to 2^95 = 39614081257132168796771975168

₃ÝoŒʒg≠}Jù

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Explanation

₃Ý            # push range [0 ... 95]
  o           # raise 2 to the power of each
   Π         # get a list of all sublists
    ʒ         # filter, keep elements that satisfy:
     g        # length
      ≠       # false (not equal to 1)
       }      # end filter
        J     # join each
         ù    # keep numbers of length matching the input

Saved 1 byte thanks to Erik the Outgolfer
Saved 1 byte thanks to Riley

Pyth, 22 21 20 17 bytes

fqQlTjLkt#.:^L2yT

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Explanation

fqQlTjLkt#.:^L2yT
            ^L2yT  Get the powers of 2 up to 2^20
        t#.:       Get all consecutive sequences of at least 2
     jLk           Concatenate each
fqQlT              Get the ones whose length is the input

Jelly,  19 18  16 bytes

^{There may be a shorter solution now that we may use any cut-off (not just 2048), although this change to the specification has allowed a one byte save from this implementation by moving to a cut-off of 32768.
--yep...}

-2 bytes thanks to Erik the Outgolfer (use of V to allow implicit right argument of the filter and tightening)
--yes it is very similar to his inefficient one now; go upvote his!

⁴Ḷ2*Ẇṫ17VDL$⁼¥Ðf

A monadic link taking a number and returning a list of numbers.

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How?

⁴Ḷ2*Ẇṫ17VDL$⁼¥Ðf - Link: number, n        e.g. 3
⁴                - literal sixteen             16
 Ḷ               - lowered range               [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
  2              - literal two                 2
   *             - exponentiate                [1,2,4,8,16,32,...,32768]
    Ẇ            - all sublists                [[1],[2],...,[1,2],[2,4],...,[1,2,4],...]
      17         - literal seventeen           17
     ṫ           - tail from index             [[1,2],[2,4],...,[1,2,4],...]]
        V        - evaluate as Jelly code      [12,24,...,124,...]
              Ðf - filter keep:
             ¥   -   last two links as a dyad
           $     -     last two links as a monad:
         D       -       decimal list (of entry) (i.e. 816 -> [8,1,6] or 24 -> [2,4])
          L      -       length                  (i.e. 816 -> 3, or 24 -> 2)
            ⁼    -   equals (n)                  (i.e. 816 -> 1, or 24 -> 0)
                 - ...resulting in             [816, 124, 248]