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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 3 Comment 5 of 5 |

again, see this link
http://en.wikipedia.org/wiki/Divisor_function
d(n)=# divisors of n
if n=p1^c1*p2^c2*...*pk^ck then
d(n)=(c1+1)*(c2+1)*...*(ck+1)

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
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