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In olympiad teaching period, we have a session that students must try to design a good problem for others. Many times we arrive to good questions but sometimes there are some challenges. In one of our exam we gave this problem to our students:

Find the 5-digit numbers that the summation of the digits of their square is maximum? In the problem designing session, we got this problem (the notations are corrected and redefined. Also, I give here the generalized form of question):

Let $D_n^2(k)$ denotes the total number of $n$ digits number such that the summation of the digits of its square is less than or equal to $k$. What is the behaviour of $D_n^2(k)$?

I think this general form is so hard. Is the question famous or previously studied somewhere? Is there any approximation for the values of $D_n^2(k)$ based on $n$ and $k$?

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    $\begingroup$ You might start by looking at the links at OEIS sequence A004159. $\endgroup$
    – Robert Israel
    Commented Mar 6, 2020 at 2:58
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    $\begingroup$ This is loosely related to an early proposed (by Von Neumann) pseudo-random number generator the Middle square method, which would square an $n$ digit number (producing a $2n$ digit number), extract the middle $n$ digits, then iterate. Some cryptanalysis of this could theoretically be based on understanding the digits of the square of a number, but I'm unfamiliar with it. $\endgroup$
    – Mark Schultz-Wu
    Commented Mar 6, 2020 at 7:19
  • $\begingroup$ Dear Mark, I am familiar with that method which you mentioned. But, I can not see how we can use it for this question! $\endgroup$
    – Shahrooz
    Commented Mar 6, 2020 at 7:30
  • $\begingroup$ Dear Israel, I did not find something there. Am I did wrong search? $\endgroup$
    – Shahrooz
    Commented Mar 6, 2020 at 7:34
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    $\begingroup$ Digits of squares are very difficult to control. That is why it is easy to formulate such problems and hard to solve them. For example an old unsolved (as far as I know) problem asks if there are squares which are not powers of 10 and whose digits are 1 and 0. $\endgroup$
    – user6976
    Commented Mar 6, 2020 at 14:49

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