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.
see this wikipedia
let d(n)= # of divisors of n
if n=p1^c1*p2^c2*...*pk^ck then
thus the number of divisors depends only on the exponents of the primes and not the primes itself. Thus, for any number n for which the prime divisors are not the first k primes, you can reduce n without changing d(n) by changing the primes to the first k primes and thus this n is minimal.
Posted by Daniel
on 2012-10-28 07:02:11