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For the sake of convenience, let's define an integer to be a "supersquare" if:

  • The number itself is a positive square number
  • Each digit of the number is a positive square (1, 4, 9)
  • The sum of digits is a square

For example, 144 is a supersquare because:

  • $144=12^{2}$
  • Each digit is a positive square (1, 4 and 4)
  • $1+4+4=9=3^{2}$

Here are the first few supersquares that I could find: 1, 4, 9, 144, 441, 44944

Now I'd like to ask:

  1. Are there infinite supersquares?
  2. If we allow "almost supersquares" to have exactly one 0, for example, $9941409=3153^2$, are there infinite almost supersquares?

(allowing for two 0s is trivial, because multiplying a supersquare by powers of 100 will give the required integers, e.g. 4494400, 449440000, etc.)

Unfortunately, motivation is out of my own personal curiosity and because of that I have little idea as to how to approach the problem. However, a quick check with Python for integers below $10^{14}$ has found 44944 to be the highest supersquare so far, and 4410449411449 to be the highest almost supersquare.

Edit: After searching through integers below $10^{18}$ the highest supersquare is still 44944, but here is the sequence of almost supersquares:

9941409, 141111419904, 941911011441, 1119444409444, 1144944940441, 4410449411449, 4991441999419044, 49041994144141441, 141114911949411904, ...

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1 Answer 1

up vote 2 down vote accepted

OEIS A061269 says

Next term, if it exists, is > 90000000000 - Larry Reeves (larryr(AT)acm.org), May 11 2001

Your search seems to go a bit further.

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Thanks for the find, I forgot to check OEIS first. There doesn't seem to be much information on it though, although the related sequences are interesting. –  Sp3000 Oct 13 '11 at 12:27

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