enigmatic puzzle - The mess of Adrian Puzzlinger

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

Answer

With credit to Dan Russel for identifying the puzzle involved, Adrian appears to be

identifying the numbers which, if put in the top left hand corner of a multiplicative, staircase-shaped cage within a 5x5 KenKen puzzle, would mean that just two distinct numbers are used in that cage. By staircase-shaped cages, I mean one of the following (or their reflections/rotations):

 __ __
|   __|
|__|

    __ __ 
 __|   __|
|   __|
|__|

       __ __ 

    __|   __|
 __|   __|
|   __|
|__|  
          __ __    
       __|   __|
    __|   __|
 __|   __|
|   __|
|__|

Each row of the chart refers to one of the above shapes (and the layout of the chart points to the staircase shapes).

For example, the second row, second column(/diagonal) of the chart), 27 432 675 | 3 | 1 4 5 means that

if we see 27, 432 or 675 as the multiplicative total of a cage like the second one I drew above, then we know that there are three 3's along the main diagonal. Additionally if the number is, say 675, then we know that there are two 5's in the other cells.

Notably, the missing 1's 2's and 4's in the chart are where

it would be impossible to tell whether there are two 2's or a 1 and a 4.

The additional numbers, 4 7 10 13 and 14 23 32 41 are

the minimum/maximum sums of the above cages and are put next to their respective products. eg 41 is from five 5's along the main diagonal and four 4's in the other cells (41=5*5+4*4) and is put next to 800000 (=5^5*4^4).