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:
- Are there infinite supersquares?
- 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, ...