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:

maxNumber = 324
len isSum[] maxNumber
maxSquare = floor sqrt maxNumber
# 
proc flagSum currSum sqPos . .
   nextSum = currSum + sqPos * sqPos
   if nextSum <= maxNumber
      isSum[nextSum] = 1
      for i = sqPos + 1 to maxSquare
         flagSum nextSum i
      .
   .
.
for i = 1 to maxSquare
   flagSum 0 i
.
for i = 1 to maxNumber
   if isSum[i] = 0
      write i & " "
   .
.