# [Jelly], 9 bytes

    Żp`×ÄSƊ€i

[Try it online!][TIO-kgcfa5cs] or try the [test suite]() (some larger inputs excluded for time reasons)

[Jelly]: https://github.com/DennisMitchell/jelly
[TIO-kgcfa5cs]: https://tio.run/##ASEA3v9qZWxsef8Kw4fGrP/Fu3Bgw5fDhFPGiuKCrGn///8xMDg "Jelly – Try It Online"

Returns a non-zero integer for Loeschian numbers and \$0\$ for non-Loeschian numbers

May be a little cheeky to post, as the only reason this is shorter than Leaky's Jelly answer is due to newer features, but it's different enough from the existing Jelly answers.

## How it works

    Żp`×ÄSƊ€i - Main link. Takes k on the left
    Ż         - Yield [0, 1, 2, ..., k]
      `       - Use this as both left and right argument for:
     p        -   Cartesian power
          Ɗ€  - Over each pair, [i, j], run the previous three commands:
        Ä     -   Cumulative sum; [i, i+j]
       ×      -   Multiply; [i×i, j×(i+j)]
         S    -   Sum; i×i+j×(i+j) = i×i + i×j + j×j
                This yields all Loeschian numbers below 3k²
            i - Index of k in this list, or 0 if not found