>Versatile Numbers are numbers that have more factors than any smaller >number. >Twelve is a versatile number because 12 has 6 factors and no number less >than >12 has that many factors. These are the first few versatile numbers: 2, >4, >6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 840, 1260, 1680, 2520, 5040, >...
I have also been fooling around with this set of numbers, and, strangly enough, I also labeled them "versatile numbers" :) But my definition of a versatile number is different from yours:
Let f(n)=the sum of the factors of n V(n) --the versatility of n-- = f(n)/n.
I believe both definitions lead to the same set of numbers, because your list is equivalent to mine, at least for the first 40 terms, which is all I have calculated... Prime numbers have V(n)=(n+1)/n, and the 43rd versatile number (the largest one I've found--122522400), has V(n) slightly above 5.
It's quite maddenning to try and search for patterns in the versatile numbers; every time you think you've spotted one, it turns out not to work. One interesting pattern, though, is that the first two versatile numbers are powers of 2, then a factor of 3 is added, then later a factor of 5, 7, 11, etc.