Results 1 to 4 of 4

Math Help - Sum of square free divisors of integer

  1. #1
    Member
    Joined
    Nov 2010
    Posts
    95

    Sum of square free divisors of integer

    Hi, is there any forumula to find the sum of square free (Square-free integer - Wikipedia, the free encyclopedia) divisors of an integer?
    One way is to get the divisors then check for each if it is square free and take the sum. However is there any formula for calculating it?
    I think mobius (Möbius function - Wikipedia, the free encyclopedia) function and Euler-Totient(Euler's totient function - Wikipedia, the free encyclopedia) functions can be used here but am not sure how?
    E.g for n = 9 the divisors are 1,3,9 out of which the square free divisors are 1,3 hence sum = 4.
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2010
    Posts
    993
    Thanks
    244

    Re: Sum of square free divisors of integer

    Isn't that just the sum of divisors of the radical of n? The radical of n is the product of the distinct prime divisors of n. For example, for n=504=2^3\cdot3^2\cdot7, \text{rad}(504)=2\cdot3\cdot7=42. The divisors of n are 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504 of which 1, 2, 3, 6, 7, 14, 21, and 42 are square free. This is exactly the set of divisors of 42.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2010
    Posts
    95

    Re: Sum of square free divisors of integer

    thanks for the reply but is there any more other efficient way?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Mar 2010
    Posts
    993
    Thanks
    244

    Re: Sum of square free divisors of integer

    It would be the product of p+1 for all primes p dividing n. So in the case of 504, (2+1)(3+1)(7+1)=96.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Determining if n is a square from number of divisors?
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: August 21st 2009, 08:38 AM
  2. Square free numbers
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: June 14th 2009, 03:30 AM
  3. gaps between square free
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 23rd 2008, 07:19 PM
  4. square free
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: July 12th 2008, 07:30 PM
  5. Square free problem
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: January 30th 2008, 03:49 PM

Search Tags


/mathhelpforum @mathhelpforum