A highly composite number (HCN) is a positive integer having more divisors than any smaller positive integer (sequence A002182 in OEIS).

Please prove the following:

1. There is an infinite number of highly composite numbers.
2. For any highly composite number (n= p₁^c₁*p₂^c₂* p₃^c₃*...p_k^c_k) the k given prime numbers p_i must be precisely the first k prime numbers ( i.e. 2, 3, 5,7,...).
3. The sequence of exponents c_k must be non-increasing.
4. Only in two special cases (which?) the last exponent c_k is greater than 1.
Rem: Although number 1 does not exactly comply with my definition it is considered an HC number.

(In reply to First three and part of 4. by Charlie)

It may have something to do with the relative size of consecutive primes.

The gap from 2 to 3 is 50% of 2 and the gap from 3 to 5 is 67% of 3.  After that they are all under 50% and decrease quickly.  So for the HCN's we want to concentrate on 2's and 3's.

As the primes get bigger the relative gaps get smaller so for large HCN's it is better to have more primes with exponent 1 than to increase the other exponents.

It's a thought anyway.  I haven't taken more time to explore though.

Posted by Jer on 2012-10-26 16:22:26