TITLE: Can the sum of two distinct factorizations of a number be equal?
QUESTION [13 upvotes]: Given two distinct factorizations of a positive integer with the same number of factors (not necessarily prime or all distinct), must the sums of the respective sets of factors also be distinct?  This question arises frequently in puzzles of the KenKen or Killer Sudoku type. I have found no obvious counter examples searching by hand. For the purpose at hand, the numbers being factored may be limited to less than 1000, say.

REPLY [2 votes]: I thought it might be fun to write a Mathematica code to find instances of same-sum factorizations. Using the following code, you can find integers such that num different sets of len different same-length factorizations of an integer less than max have the same sum. It omits multiples of lower cases on the grounds that those are boring.
The smallest integers with increasingly large sets of different same-sum factorizations are:72, 432, 3456, 5760, 7200, 12096, 17280, 21600, ...
The smallest integers with pairs of increasingly large sets of different same-sum factorizations are: 144, 720, 2160, 5040, 8640, 10080, 14400, 25920, 30240, ...
The smallest integers with triples of increasingly large sets of different same-sum factorizations are: 144, 1440, 2880, 7200, 8640, 15120, 17280, 30240, 30240, ...
30240 is an interesting case with 6 different same-sum factorization 9-sets, 4 10-sets, an 11-set and a 12-set.
factorizationList[n_Integer?PrimeQ] := {{n}}
factorizationList[n_Integer] := 
  (factorizationList[n] = 
    Union[Append[Join@@
      (#/{L__List}:>Sort/@Flatten[Outer[Join,L,1],1]&/@
        Map[factorizationList,{#,n/#}&/@
           #[[-Ceiling[Length[#]/2];;-2]]&[Divisors[n]],{2}]),{n}]])
Block[{num=2,len=10,max=25000},
  DeleteDuplicates[DeleteCases[
    Table[{n,Select[GatherBy[factorizationList[n],
      {Total[#],Length[#]}&],
      Length[#]>=num&]},
      {n,max}],
  {_, L_ /; Length[L] < num}],IntegerQ[#2[[1]]/#1[[1]]]&]]

Output:
{{21600,{{{2,3,9,20,20},{2,3,10,15,24},{2,3,12,12,25},{2,5,5,18,24},
          {2,5,8,9,30},{2,6,6,10,30},{3,3,8,10,30},{3,4,4,18,25},
          {3,4,5,12,30},{3,5,5,9,32}}}}}