Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares

Theorem
The largest odd positive integer that cannot be expressed as the sum of exactly $4$ (distinct) non-zero square numbers all of which are coprime is $157$.

The full sequence of such odd positive integers which cannot be so expressed is:


 * $1, \ldots, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 67, 69, 73, 77, 83, 89, 97, 101, 103, 115, 157$

where the sequence contains all odd integers between $1$ and $37$.

This sequence appears not to be documented on the.

Proof
The condition on the summands to be distinct is so as to disallow expressions such as:
 * $67 = 64 + 1 + 1 + 1 = 8^2 + 1^2 + 1^2 + 1^2$

as technically $1$ is coprime with itself.

It is noted that the smallest odd integer that can be expressed as the sum of exactly $4$ (distinct) coprime non-zero square numbers is $39$:


 * $39 = 1 + 4 + 9 + 25 = 1^2 + 2^2 + 3^2 + 5^2$

since the smallest coprime positive integers are $1, 2, 3$ and $5$.

Then we have:

It remains to be demonstrated that the above sequence is indeed complete.