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

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 {{OEISLink}}.

Proof:
The statement of the result was taken from the paper cited below by {{AuthorRef|Paul T. Bateman}} and others:
:''The largest odd integer not expressible as a sum of $4$ distinct non-zero squares with greatest common divisor $1$ is $157$.''
By a brute force exercise, we assemble all the sets of $4$ distinct non-zero integers and calculate the sum of their squares such that those sums are less than $157$.
It is noted in passing that all such sets have a greatest common divisor of $1$.
Thus we have:
{{begin-eqn}}
{{eqn | l = 39
      | r = 1 + 4 + 9 + 25
      | rr= = 1^2 + 2^2 + 3^2 + 5^2
      | c = 
}}
{{eqn | l = 51
      | r = 1 + 9 + 16 + 25
      | rr= = 1^2 + 3^2 + 4^2 + 5^2
      | c = 
}}
{{eqn | l = 57
      | r = 1 + 4 + 16 + 36
      | rr= = 1^2 + 2^2 + 4^2 + 6^2
      | c = 
}}
{{eqn | l = 63
      | r = 1 + 4 + 9 + 49
      | rr= = 1^2 + 2^2 + 3^2 + 7^2
      | c = 
}}
{{eqn | l = 65
      | r = 4 + 9 + 16 + 36
      | rr= = 2^2 + 3^2 + 4^2 + 6^2
      | c = 
}}
{{eqn | l = 71
      | r = 1 + 9 + 25 + 36
      | rr= = 1^2 + 3^2 + 5^2 + 6^2
      | c = 
}}
{{eqn | l = 75
      | r = 1 + 9 + 16 + 49
      | rr= = 1^2 + 3^2 + 4^2 + 7^2
      | c = 
}}
{{eqn | l = 79
      | r = 1 + 2 + 25 + 49
      | rr= = 1^2 + 2^2 + 5^2 + 7^2
      | c = 
}}
{{eqn | l = 81
      | r = 4 + 16 + 25 + 36
      | rr= = 2^2 + 4^2 + 5^2 + 6^2
      | c = 
}}
{{eqn | l = 85
      | r = 1 + 4 + 16 + 64
      | rr= = 1^2 + 2^2 + 4^2 + 8^2
      | c = 
}}
{{eqn | l = 87
      | r = 4 + 9 + 25 + 49
      | rr= = 2^2 + 3^2 + 5^2 + 7^2
      | c = 
}}
{{eqn | l = 91
      | r = 1 + 16 + 25 + 49
      | rr= = 1^2 + 4^2 + 5^2 + 7^2
      | c = 
}}
{{eqn | l = 93
      | r = 4 + 9 + 16 + 64
      | rr= = 2^2 + 3^2 + 4^2 + 8^2
      | c = 
}}
{{eqn | l = 95
      | r = 1 + 9 + 36 + 49
      | rr= = 1^2 + 3^2 + 6^2 + 7^2
      | c = 
}}
{{eqn | l = 99
      | r = 1 + 9 + 25 + 64
      | rr= = 1^2 + 3^2 + 5^2 + 8^2
      | c = 
}}
{{eqn | r = 9 + 16 + 25 + 49
      | rr= = 3^2 + 4^2 + 5^2 + 7^2
      | c = 
}}
{{eqn | l = 105
      | r = 4 + 16 + 36 + 47
      | rr= = 2^2 + 4^2 + 6^2 + 7^2
      | c = 
}}
{{eqn | l = 107
      | r = 1 + 9 + 16 + 81
      | rr= = 1^2 + 3^2 + 4^2 + 9^2
      | c = 
}}
{{eqn | l = 109
      | r = 4 + 16 + 25 + 64
      | rr= = 2^2 + 4^2 + 5^2 + 8^2
      | c = 
}}
{{eqn | l = 111
      | r = 1 + 25 + 36 + 49
      | rr= = 1^2 + 5^2 + 6^2 + 7^2
      | c = 
}}
{{eqn | r = 1 + 4 + 25 + 81
      | rr= = 1^2 + 2^2 + 5^2 + 9^2
      | c = 
}}
{{eqn | l = 113
      | r = 4 + 9 + 36 + 64
      | rr= = 2^2 + 3^2 + 6^2 + 8^2
      | c = 
}}
{{eqn | l = 117
      | r = 1 + 16 + 36 + 64
      | rr= = 1^2 + 4^2 + 6^2 + 8^2
      | c = 
}}
{{eqn | l = 119
      | r = 4 + 9 + 25 + 81
      | rr= = 2^2 + 3^2 + 5^2 + 9^2
      | c = 
}}
{{eqn | r = 9 + 25 + 36 + 49
      | rr= = 3^2 + 5^2 + 6^2 + 7^2
      | c = 
}}
{{eqn | l = 121
      | r = 1 + 4 + 16 + 100
      | rr= = 1^2 + 2^2 + 4^2 + 10^2
      | c = 
}}
{{eqn | l = 123
      | r = 1 + 9 + 49 + 64
      | rr= = 1^2 + 3^2 + 7^2 + 8^2
      | c = 
}}
{{eqn | r = 1 + 16 + 25 + 81
      | rr= = 1^2 + 4^2 + 5^2 + 9^2
      | c = 
}}
{{eqn | l = 125
      | r = 9 + 16 + 36 + 64
      | rr= = 3^2 + 4^2 + 6^2 + 8^2
      | c = 
}}
{{eqn | l = 127
      | r = 1 + 9 + 36 + 81
      | rr= = 1^2 + 3^2 + 6^2 + 9^2
      | c = 
}}
{{eqn | l = 129
      | r = 4 + 9 + 16 + 100
      | rr= = 2^2 + 3^2 + 4^2 + 10^2
      | c = 
}}
{{eqn | r = 4 + 25 + 36 + 64
      | rr= = 2^2 + 5^2 + 6^2 + 8^2
      | c = 
}}
{{eqn | l = 131
      | r = 9 + 16 + 25 + 81
      | rr= = 3^2 + 4^2 + 5^2 + 9^2
      | c = 
}}
{{eqn | l = 133
      | r = 4 + 16 + 49 + 64
      | rr= = 2^2 + 4^2 + 7^2 + 8^2
      | c = 
}}
{{eqn | l = 135
      | r = 1 + 4 + 9 + 121
      | rr= = 1^2 + 2^2 + 3^2 + 11^2
      | c = 
}}
{{eqn | r = 1 + 4 + 49 + 81
      | rr= = 1^2 + 2^2 + 7^2 + 9^2
      | c = 
}}
{{eqn | r = 1 + 9 + 25 + 100
      | rr= = 1^2 + 3^2 + 5^2 + 10^2
      | c = 
}}
{{eqn | l = 137
      | r = 4 + 16 + 36 + 81
      | rr= = 2^2 + 4^2 + 6^2 + 9^2
      | c = 
}}
{{eqn | l = 139
      | r = 1 + 25 + 49 + 64
      | rr= = 1^2 + 5^2 + 7^2 + 8^2
      | c = 
}}
{{eqn | l = 141
      | r = 16 + 25 + 36 + 64
      | rr= = 4^2 + 5^2 + 6^2 + 8^2
      | c = 
}}
{{eqn | l = 143
      | r = 1 + 25 + 36 + 81
      | rr= = 1^2 + 5^2 + 6^2 + 9^2
      | c = 
}}
{{eqn | r = 4 + 9 + 49 + 81
      | rr= = 2^2 + 3^2 + 7^2 + 9^2
      | c = 
}}
{{eqn | l = 145
      | r = 4 + 16 + 25 + 100
      | rr= = 2^2 + 4^2 + 5^2 + 10^2
      | c = 
}}
{{eqn | l = 147
      | r = 1 + 16 + 49 + 81
      | rr= = 1^2 + 4^2 + 7^2 + 9^2
      | c = 
}}
{{eqn | r = 1 + 9 + 16 + 121
      | rr= = 1^2 + 3^2 + 4^2 + 11^2
      | c = 
}}
{{eqn | r = 9 + 25 + 49 + 64
      | rr= = 3^2 + 5^2 + 7^2 + 8^2
      | c = 
}}
{{eqn | l = 149
      | r = 1 + 4 + 25 + 121
      | rr= = 1^2 + 2^2 + 5^2 + 11^2
      | c = 
}}
{{eqn | r = 4 + 9 + 36 + 100
      | rr= = 2^2 + 3^2 + 6^2 + 10^2
      | c = 
}}
{{eqn | l = 151
      | r = 9 + 25 + 36 + 81
      | rr= = 3^2 + 5^2 + 6^2 + 9^2
      | c = 
}}
{{eqn | l = 153
      | r = 4 + 36 + 49 + 64
      | rr= = 2^2 + 6^2 + 7^2 + 8^2
      | c = 
}}
{{eqn | l = 155
      | r = 9 + 16 + 49 + 81
      | rr= = 3^2 + 4^2 + 7^2 + 9^2
      | c = 
}}
{{end-eqn}}
It remains to be demonstrated that the above sequence is indeed complete.
The original article by {{AuthorRef|Franz Halter-Koch}} states:
:{{SmallCaps|Satz}} $3$. ''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:
:''{{SmallCaps|Statement}} $3$. $n \in N$ is not the sum of four different positive squares (in the case where $n \not \equiv 0 \bmod 8$ also non-prime) {{iff}}:
::''$n = 4^h a \quad$ 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''
::''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$
{{ProofWanted}}