

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.

     1 == 1
     5 == 4 + 1
    25 == 16 + 9
    77 == 36 + 25 + 16
   103 == 49 + 25 + 16 + 9 + 4

Many can be generated in multiple ways:

    90 == 36 + 25 + 16 + 9 + 4 == 64 + 16 + 9 + 1 == 49 + 25 + 16 == 64 + 25 + 1 == 81 + 9
   130 == 64 + 36 + 16 + 9 + 4 + 1 == 49 + 36 + 25 + 16 + 4 == 100 + 16 + 9 + 4 + 1 == 81 + 36 + 9 + 4 == 64 + 49 + 16 + 1 == 100 + 25 + 4 + 1 == 81 + 49 == 121 + 9    

The number of positive integers that cannot be generated by any combination of distinct squares is in fact finite:

   2, 3, 6, 7, etc.



Task
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.



See also
 OEIS: A001422 Numbers which are not the sum of distinct squares


my @squares = ^∞ .map: *²; # Infinite series of squares
 
for 1..∞ -> $sq {          # for every combination of all squares
    my @sums = @squares[^$sq].combinations».sum.unique.sort;
    my @run;
    for @sums {
        @run.push($_) and next unless @run.elems;
        if $_ == @run.tail + 1 { @run.push: $_ } else { last if @run.elems > @squares[$sq]; @run = () }
    }
    put grep * ∉ @sums, 1..@run.tail and last if @run.elems > @squares[$sq];
}