*greater than*10^12. This can be accomplished by brute-forcing all possible ways of creating a valid factorization using seven or fewer digits. I generated no solutions for one to four digits. For five digits, one gets the 4 solutions of the known 1-balanced factorizations (that started it all):

26487 = 3^5 * 109

28651 = 7 * 4093

61054 = 2 * 7^3 * 89

65821 = 7 * 9403

For six digits, there are again no solutions. For seven digits, there are 4 solutions:

1495476527089 = 83^2 * 601^3

3392164558027 = 7^9 * 84061

8789650571264 = 2^31 * 4093

9418623046875 = 3^9 * 5^10 * 7^2

These then are (necessarily) the

*largest*2-balanced factorization integers! I'm going to run the program again for eight digits. This will generate all 12-digit 2-balanced factorization integers and may do so (I'm hoping) in less time than my exhaustive forward search. In addition, there's a chance that it may find a k > 2 solution, since nothing in the reverse search precludes it.