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.

again, see this link
http://en.wikipedia.org/wiki/Divisor_function
d(n)=# divisors of n
if n=p1^c1*p2^c2*...*pk^ck then
d(n)=(c1+1)*(c2+1)*...*(ck+1)

for this one, I'm assuming that it is meant for p1<p2<...<pk.

Proof by contradiction.
assume that n is an HCN and let n=p1^c1*...*pk^ck
also, let x=d(n).
if there exists an i,j such that pi<pj and ci<cj then n can be reduced without changing n by exchanging pi and pj.  Thus contradicting n being and HCN.  Thus the ci's must be non increasing.

Posted by Daniel on 2012-10-28 07:09:31