The sequence $13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, \dots,$ (sequence A348300 in the OEIS) shows the largest digital sum the square of an $n$-digit (decimal) number has.
Is this sequence strictly increasing?
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asked Oct 10, 2021 at 22:47
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2$\begingroup$ Did you check that for small $n$, say, $n<10^9$? $\endgroup$– markvsCommented Oct 11, 2021 at 0:14
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7$\begingroup$ for the analogues sequence in base 2, namely $1, 2, 3, 5, 6, 8, 9, 13, 13, 15, 16, 18, 20, 22, 24, 25, 27, ...$ it is not strictly increasing, since $13, 13$ appears. $\endgroup$– Moritz FirschingCommented Oct 11, 2021 at 8:52
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5$\begingroup$ @markvs: What do you mean exactly? For $n=10^9$ one would have to compute the squares of about $10^{10^9-1}$ numbers. If you meant $n=9$ then the OP did do that. $\endgroup$– Yaakov BaruchCommented Oct 11, 2021 at 19:21
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2$\begingroup$ @markvs Can you please elaborate on how exactly one might computationally check the case $n=10^9$ in a reasonable amount of time? $\endgroup$– Timothy ChowCommented Oct 12, 2021 at 12:28
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2$\begingroup$ @markvs Your question was whether the OP checked $n< 10^9$ and I do not expect you to know whether the OP did in fact check $n < 10^9$. But saying that $n \approx 10^9$ is "small" implies that checking $n < 10^9$ is a feasible computation. When Yaakov Baruch asked for confirmation and pointed out that $n \approx 10^9$ does not appear to be a feasible computation, you affirmed that you did mean $n < 10^9$, and by saying, "You do not need squares of all numbers" you further implied that you had in mind some nontrivial way of making the computation feasible. I am just asking you for more details. $\endgroup$– Timothy ChowCommented Oct 12, 2021 at 14:41
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1 Answer
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After 10 + days calculating, I got the following sequence : {13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, 193, 211, 229, 244, 262, 277, 297, 310, 331, 343, 360, 378, 396, 406, 423, 436, 454, 469, 487, 517}
The conclusion seems correct at least for 1 <= n <= 31.
a (n) appears to be approximately equal to 16.5*n.
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2$\begingroup$ I disagree with Alex. This is not a full answer, but clearly this is a partial answer that adds information towards the question. Quoting an answer in Meta: "Half an answer is better than no answer at all. Posting the progress you have made may provide the piece missing in someone else's partial answer. An answer doesn't have to be complete, just helpful." $\endgroup$ Commented Apr 16 at 8:52
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$\begingroup$ @JukkaKohonen: I have suggested to the moderators that this post should be converted into a comment, not that it should be deleted. (Compare this post with Moritz Firsching's comment, which says something similar but for base $2$.) $\endgroup$– Alex M.Commented Apr 16 at 10:19
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$\begingroup$ Is your sequence exact, or lower bounds? On math.se edited later on, you say 469 and 484 are lower bounds. $\endgroup$ Commented Apr 18 at 7:09
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$\begingroup$ @Claude Chaunier Perhaps “ lower bounds” can more accurately describe the sequence.Thanks. $\endgroup$– MrexcelCommented Apr 18 at 9:15