Numbers which have a "compact" form...

[jb at brennen.net]

Consider the sequence of positive integers which have a "compact"
representation which uses fewer decimal digits than just writing
the number out normally.  You are allowed to use the following
symbols as well:

    ( )     grouping
     +      addition
     -      subtraction
     *      multiplication
     /      division
     ^      exponentiation

(Not sure if it matters, but assume that you can use the
 unary minus sign, as well...)

If I'm not mistaken, the sequence would begin:

  125, 128, 216, 243, 256, 343, 512, 625, 729, 1000,
  1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023,
  1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032,
  1033, ...

  (Note that 1015 to 1033 are all representable in the
   form 4^5-d or 4^5+d, where d is a single digit.
   1029 can also be written as 7^3*3.)

This sequence doesn't seem to be in the OEIS.

Does this sequence have density 1?  Or an even stronger
conjecture:  Are there only a finite number of positive
integers which are NOT in the sequence?  If so, can
anything be determined about the largest integer without
a "compact" representation?

  Jack