All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
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    
No Rating

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Part 3Daniel2012-10-28 07:09:31
Part 2Daniel2012-10-28 07:02:11
Part 1Daniel2012-10-28 06:58:03
An idea for 4.Jer2012-10-26 16:22:26
Some ThoughtsFirst three and part of 4.Charlie2012-10-26 12:58:26
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information