Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes

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Theorem

The following positive integers cannot be expressed as the sum of distinct non-pythagorean primes:

$1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24, 25, 27, 28, 32, 35, 36, 39, 48, 51, 55$

This sequence is A048262 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


All positive integers greater than $55$ can be so expressed.


Proof


Sources