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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 2 | Comment 4 of 5 |

see this wikipedia
http://en.wikipedia.org/wiki/Divisor_function

let d(n)= # of divisors of n
if n=p1^c1*p2^c2*...*pk^ck then
d(n)=(c1+1)*(c2+1)*...*(ck+1)
thus the number of divisors depends only on the exponents of the primes and not the primes itself.  Thus, for any number n for which the prime divisors are not the first k primes, you can reduce n without changing d(n) by changing the primes to the first k primes and thus this n is minimal.

 Posted by Daniel on 2012-10-28 07:02:11

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