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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).

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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 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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