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 with greatest common divisor $1$ 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 statement of the result was taken from the paper cited below by and others:
 * The largest odd integer not expressible as a sum of $4$ distinct non-zero squares with greatest common divisor $1$ is $157$.

It is noted that the smallest odd integer that can be expressed as the sum of $4$ distinct non-zero squares with greatest common divisor $1$ is $39$:


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

since the smallest set of $4$ integers the sum of whose squares is odd is $\set {1, 2, 3, 5}$.

Then we have:

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

The original article by states:


 * $3$. Genau dann is $n \in N$ nicht Summe von vier verschiedenen positiven (im Falle $n \not \equiv 0 \bmod 8$ auch teilerfremden) Quadraten, wenn entwieder
 * $n = 4^h a \quad$ mit $h \ge 0$ und $a \in \set {1, 2, \ldots, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103}$
 * oder
 * ''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$

That is, in English:
 * '' $3$. $n \in N$ is not the sum of four different positive squares (in the case where $n \not \equiv 0 \bmod 8$ also non-prime) if and only if:
 * ''$n = 4^h a \quad$ where $h \ge 0$ and $a \in \set {1, 2, \ldots, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103}$
 * or
 * ''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$