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

Theorem
The following positive integers cannot be expressed as the sum of distinct non-pythagorean primes:
 * $1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24, 25, 27, 28, 32, 35, 36, 39, 48, 51, 55$

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