Numbers not Sum of Distinct Squares

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Theorem

The positive integers which are not the sum of $1$ or more distinct squares are:

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

This sequence is A001422 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Sources

  • 1948: R. SpragueÜber Zerlegungen in ungleiche Quadratzahlen (Math. Z. Vol. 51: 289 – 290)