27729 is a member of OEIS A053061: a(n) is the decimal concatenation of n and n^2 where n=27.
The members from 27729 to 40000 are:
27729, 28784, 29841, 30900, 31961, 321024, 331089, 341156, 351225, 361296, 371369, 381444, 391521
27729 is a member of OEIS A048390: digits d in decimal expansion of n replaced with d^3. Here d=39.
27729 is the second number in a run of FOUR consecutive positive negabinary-Niven numbers. The run begins with 27728.
27729 is a katadrome in base 13 : c810. This is because the digits of the number are in strictly decreasing order.
27729 is a xenodrome in base 7 : 143562. Notice that all the digits in this number base (except 0) are present. There are of course 6! = 720 such numbers and they can be generated using the following flexible algorithm: permalink. Here is a permalink to numbers in base 7 that contain all the digits from 0 to 6 exactly once: permalink.
27729 is a Duffinian number.
27729 is a modest number since divided by 729 gives 27 as remainder.
27729 is a Harshad (or Niven) number since it is a multiple of its sum of digits (27). Note also that it is the second in a string of four consecutive numbers that are negabinary Niven numbers. See blog post Negabinary Niven Numbers.
27729 can be rendered as a digit equation as follows:
( 2 + 7 / 7 ) ^ 2 = 9