There's no need for an iterator in many such situations, especially as in this one where you're dealing with consecutive integers and individual functions (power-of-two and IntegerDigits) are listable. To obtain the desired first digits, you may simply do this:

      First /@ IntegerDigits[Range[20]^2]
 (*  {1, 4, 9, 1, 2, 3, 4, 6, 8, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4}  *)

If you want a table, you may use, for example:

      {Range[20], First /@ IntegerDigits[Range[20]^2]} // TableForm

Identical but extended with //Dataset or //Framed or //Dataset//Framed

TableForm[Table[{n, n^2, n^3}, {n, 20}], 
  TableHeadings -> {None, {"n", "n^2", "n^3"}}
  , TableAlignments -> {Right, Center}
  , TableSpacing -> {1.5, 2}
  ] // Framed

Yes Hans, thank you. How about:

Make a list for numbers n up to 20 in which each element is a column of the values of n, n^2 and n^3