The property discovered by Pal Erdos alluded to in the above writeup is an easy consequence of the definition. In fact, we can show that any very composite number is of the form
N = 2k13k25k3...
Indeed, for any indices a < b, we have that
N' = 2k13k2...pakb...pbka...
(i.e. what you get by switching the powers if pa and pb in N)
has exactly the same number of factor
s as does N. It follows from the definition that (if N is very composite) N ≤ N', so ka
, as required.