joshuazelinsky

Hmm, thinking about this in terms of lattices raises a question: Let n be a positive integer. Call b a maximal highly composite divisor of n if 1) b is highly composite, 2) b|n and 3) If b|a and a|n and a is highly composite then a=b. How many maximal highly composite divisors can a number have? The obvious attempt above to make one with 2 maximal highly composite divisors is to take the lcm of 27720 and 45360. But this doesn't work because their lcm is 498960, which is itself highly composite. There are, however much smaller examples. For example, 72 has two maximal highly composite divisors 36 and 48. My guess is that for any k there exists an n with k maximal highly composite divisors. But I don't see any immediate way to prove this.