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= p1c1*p2c2*
the k given prime numbers pi must be precisely the first k prime numbers ( i.e. 2, 3, 5,7,...).
3. The sequence of exponents ck must be non-increasing.
4. Only in two special cases (which?) the last exponent
ck 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
d(n)=# divisors of n
if n=p1^c1*p2^c2*...*pk^ck then
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