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Exploring HCNs (Posted on 2012-10-26) Difficulty: 3 of 5
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* 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    
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Part 1 | Comment 3 of 5 |

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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