user
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
assistant
Try it online!

**Spoiler:** *(highlight to read)*  
Once the longest run of consecutive generated sums is longer the the next square, every number after can be generated by adding the next square to every number in the run. Find the *new* longest run, add the *next* square, etc.

```
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];
}

```

Output:

```
2 3 6 7 8 11 12 15 18 19 22 23 24 27 28 31 32 33 43 44 47 48 60 67 72 76 92 96 108 112 128
```