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

Home > Numbers
Primevens and primeodds (Posted on 2016-01-10) Difficulty: 4 of 5
Let a primeven be a positive integer that is the product of an even number of primes. Let a primeodd be a positive integer that is the product of an odd number of primes. Then, 1 is a primeven because it is the product of 0 primes. 2 is a primeodd because it is the product of 1 prime. 3 is a primeodd because it is the product of 1 prime. 4 is a primeven because it is the product of 2 primes. Here are the first 10 positive integers.
Number:   Factorization:      Number of primes:   Type:
1                             0                   primeven
2         2                   1                   primeodd
3         3                   1                   primeodd
4         2*2                 2                   primeven
5         5                   1                   primeodd
6         2*3                 2                   primeven
7         7                   1                   primeodd
8         2*2*2               3                   primeodd
9         3*3                 2                   primeven
10        2*5                 2                   primeven
Suppose the primevens and primeodds had a race. First, the primevens would be ahead because 1 is a primeven. Then, there would be a tie because 2 is a primeodd. Then, the primeodds would be ahead because 3 is a primeodd. Then, there would be a tie because 4 is a primeven. Here are the winners from 1 to 10.
Number:   Type:          Primevens:     Primeodds:     Winner:
1         primeven       1              0              primevens
2         primeodd       1              1              tie
3         primeodd       1              2              primeodds
4         primeven       2              2              tie
5         primeodd       2              3              primeodds
6         primeven       3              3              tie
7         primeodd       3              4              primeodds
8         primeodd       3              5              primeodds
9         primeven       4              5              primeodds
10        primeven       5              5              tie
The primevens were ahead at the start, but have not been ahead since then. Do the primevens ever become the winner again?

No Solution Yet Submitted by Math Man    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): some stats extended | Comment 4 of 8 |
(In reply to re: some stats extended by Charlie)

You just need one more power of 10.

https://en.wikipedia.org/wiki/Liouville_function

I doubt Haselgrove's proof is below D6.

  Posted by Jer on 2016-01-10 20:18:57

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 (3)
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