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.

1.  First, see this wikipedia
http://en.wikipedia.org/wiki/Divisor_function
for a number n=p^c  with p prime and c an integer greater than 0.  Then number of divisors is c+1.  Thus, for any integer x>1 we can find an integer n>=1 with x divisors.  And, because n is bounded below, there must be a minimum n.  Thus, for this x, n would be an HCN.  Thus the HCN's are infinite.

Posted by Daniel on 2012-10-28 06:58:03