Numbers $n$ such that the digit sum of $n^2$ is a square
Solution 1:
As Djalal Ounadjela outlined in the comments, there is probably no such maximal k, we can expect to find an n for any k at an order of magnitude n ~ 10^m with approximately $ m/\log_{10}(m) \sim k $.
PS: http://oeis.org/A061910 lists numbers n for which the sum of digits of n^2 is a square.