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= p_{1}^{c1}*p_{2}^{c2}*
p_{3}^{c3}*...p_{k}^{ck})
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.