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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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 Part 3 Comment 5 of 5 |

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.

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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