Sum of factors from consecutive sequence $24$ can be written as $2\times 3\times 4$, $2\times 2\times 6$, $1\times 4\times 6$ or $1\times 3\times 8$.  
The sums of those triples are $2+3+4=9$, $2+2+6=10$, $1+4+6=11$, $1+3+8=12$.
That is a consecutive string $9$, $10$, $11$, $12$ of length $4$.
Are there other numbers, instead of $24$, that factor into triples, where the triples' sums form a consecutive string much longer than that - as long as you like?
EDIT : The number 1441440 is made from triples that sum to any number from 343 to 377.
 A: If you're willing to let the factorizations be large (but all the same length), then powers of 2 can produce very long strings of consecutive integers through a 22/14 interchange. For instance,
$$64 = 2\times2\times2\times2\times2\times2 \to 12$$
$$64 = 1\times2\times2\times2\times2\times4 \to 13$$
$$64 = 1\times1\times2\times2\times4\times4 \to 14$$
$$64 = 1\times1\times1\times4\times4\times4 \to 15$$
... and by taking larger powers of 2, I can clearly make arbitrarily long strings. In fact, the above can be extended a few more times by interchanging 44/28, which lets us increment by 2:
$$64 = 1\times1\times2\times2\times2\times8 \to 16$$
$$64 = 1\times1\times1\times2\times4\times8 \to 17$$
If this feels "cheaty" with all the extra factors of 1, you can just build chains like
$$1024 = (2,2,4,4,4,4) \to 20$$
$$1024 = (1,4,4,4,4,4) \to 21$$
$$1024 = (2,2,4,4,2,8) \to 22$$
$$1024 = (1,4,4,4,2,8) \to 23$$
$$1024 = (2,2,2,8,2,8) \to 24$$
$$1024 = (1,4,2,8,2,8) \to 25$$
This just uses the two "exchanges" above, and only ever has a single "1". (And clearly generalizes to build very long strings).
