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 Exploring HCNs (Posted on 2012-10-26)
A highly composite number (HCN) is a positive integer having more divisors than any smaller positive integer (sequence A002182 in OEIS).

1. There is an infinite number of highly composite numbers.
2. For any highly composite number (n= p1c1*p2c2* p3c3*...pkck) 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.

 No Solution Yet Submitted by Ady TZIDON No Rating

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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

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