plusquamperfect squares

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A perfect riddle:

For some perfect squares, when you remove the last digit, you get another perfect square. The first five perfect squares are 16, 49, 169, 256 and 361. What are the next three ones? Is there a more than perfect square other than 169 such that removing the last two digits returns a perfect square?

Writing an R code for plusquamperfect squares is straightforward and returns the following first 20 values

 [1]         16         49        169        256        361       1444
 [7]       3249      18496      64009     237169     364816     519841
[13]    2079364    4678569   26666896   92294449  341991049  526060096
[19]  749609641 2998438564

Adding the second constraint does not return a solution other than 169.

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