KöMaL 2011 B4360

Let $S$($n$) denote the sum of the digits for any positive integer $n$. For what integers $k$greater than 1 is there a positive real number $c_k$ such that $S$($kn$)$\ge c_k ^. S$($n$) for all positive integers $n$? (Suggested by P. Erben, Budapest)