APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
more than 20 bytes golfed off with dzaima's help at the APL orchard.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n to, cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?
APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
more than 20 bytes golfed off with dzaima's help at the APL orchard.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n to cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?
APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
more than 20 bytes golfed off with dzaima's help at the APL orchard.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n, cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?
APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
more than 20 bytes golfed off with dzaima's help at the APL orchard.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n to cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?
APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n to cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?
APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
more than 20 bytes golfed off with dzaima's help at the APL orchard.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n to cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?
APL (dzaima/APL), 15 bytes
{4≤⍵⍧∘.+⍨3*⍨⍳⍵}
Same method as all the other answers. Only works till 4104 due to brute forcing.
The output appears with a ⊂ next to it, which has been patched in the latest version of the interpreter.
Explanation
{4≤⍵⍧∘.+⍨3*⍨⍳⍵} ⍵ → input
3*⍨⍳⍵ range 1 to n to cubed
∘.+⍨ dot product, addition with itself
⍵⍧ number of times ⍵ appears in the sums
4≤ is 4 less than or equal to that?