very composite number (idea)

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 = 2^k₁3^k₂5^k₃...

where k₁≥k₂≥...

Indeed, for any indices a < b, we have that

N' = 2^k₁3^k₂...p_a^k_b...p_b^k_a...

(i.e. what you get by switching the powers if p_a and p_b in N) has exactly the same number of factors as does N. It follows from the definition that (if N is very composite) N ≤ N', so k_a ≥ k_b, as required.