Problem Statement

Integer squares are the set of integers multiplied by themselves: 1 x 1 = 1, 2 × 2 = 4, 3 × 3 = 9, etc. ( 1, 4, 9, 16 ... ) Most positive integers can be generated as the sum of 1 or more distinct integer squares. Many can be generated in multiple ways: The number of positive integers that cannot be generated by any combination of distinct squares is in fact finite:

Find and show here, on this page, every positive integer than cannot be generated as the sum of distinct squares. Do not use magic numbers or pre-determined limits. Justify your answer mathematically.

Let's start with the solution:

task=: {{
  r=. 'invalid'
  lim=. 1
  whilst. -.r -: have do.
    have=. r
    lim=. lim+1
    r=. (i.*:lim) -. (#:i.2^lim)+/ .**:i.lim
  end.
}}


   task''
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