This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(2) = 2 because 2^4 = 16 and 1+6 = 7>= 2
Select[Range[0,50],Total[IntegerDigits[#^4]]>=#&] (* Harvey P. Dale, Jan 20 2020 *)
a(2) = 2 because 2^5 = 32 and 3+2 = 5>= 2
Select[Range[0,50],Total[IntegerDigits[#^5]]>=#&] (* Harvey P. Dale, Jul 03 2021 *)
a(2) = 2 because 2^6 = 64 and 6+4 = 10>= 2
a(4) = 4 because 4^3 = 64 and 6+4 = 10>= 4
Select[Range[300],Total[IntegerDigits[#^3]]>=#&] (* Harvey P. Dale, Aug 27 2013 *)
a(2) = 17. We have 17^2 = 289 and 2+8+9 >= 17. No number > 17 has this property.
(* the constant 20 may need to be higher for larger n *) Table[Select[Range[20*n], Total[IntegerDigits[#^n]] >= # &][[-1]], {n, 100}] (* T. D. Noe, Oct 21 2011 *)