Adrian Puzzlinger has gone missing again. Even worse, for puzzle lovers, the latest puzzle under construction is a mess. The only shred of order in the chaos of Puzzlinger's puzzler parlor is this puzzling chart, which, predictably, puzzles the perpetually puzzled puzzle police.

                                  _______                               _______
               4 _______        9|   | 1 |          _______          25|   | 1 |
               2|   | 2 |      18|   | 2 |       32|   | 2 |         50|   | 2 |
               3| 1 | 3 |      36| 3 | 4 |       48| 4 | 3 |         75| 5 | 3 |
               5|   | 5 |      45|   | 5 |       80|   | 5 |     14 100|   | 4 |
           7 ___|___|___|     ___|___|___|      ___|___|___|        ___|___|___|
           4|   | 2 |      27|   | 1 |      256|   | 2 |        125|   | 1 |
           9| 1 | 3 |     432| 3 | 4 |      576| 4 | 3 |       1125| 5 | 3 |
          25|   | 5 |     675|   | 5 |     1600|   | 5 |    23 2000|   | 4 |
      10 ___|___|___|     ___|___|___|      ___|___|___|        ___|___|___|
       8|   | 2 |      81|   | 1 |     2048|   | 2 |        625|   | 1 |
      27| 1 | 3 |    5184| 3 | 4 |     6912| 4 | 3 |      16875| 5 | 3 |
     125|   | 5 |   10125|   | 5 |    32000|   | 5 |   32 40000|   | 4 |
  13 ___|___|___|     ___|___|___|      ___|___|___|        ___|___|___|
  16|   | 2 |     243|   | 1 |    16384|   | 2 |       3125|   | 1 |
  81| 1 | 3 |   62208| 3 | 4 |    82944| 4 | 3 |     253125| 5 | 3 |
 625|   | 5 |  151875|   | 5 |   640000|   | 5 |  41 800000|   | 4 |
    |___|___|        |___|___|         |___|___|           |___|___|

What kind of puzzle was Adrian making?

As this kind of puzzle has assumed different aliases, the authorities need a full description of how Adrian used this chart, not just a name.

Other missing-puzzler case
The lists of Adrian Puzzlinger