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Math Help - Divisors

  1. #1
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    Divisors

    Let tau(n) = # of pos odd divisors of n

    Let sigma(n) = sum of pos. odd divisors of n

    Note that tau and sigma are both multiplicative functions.

    1.) Evaluate the following:

    tau(12)
    tau(16)
    sigma(12)
    sigma(16)

    2.) Let n = p_(1)^(a_1)*p_(2)^(a_2)*p_(3)^(a_3)* ... *p_(k)^(a_k)

    a.) Determine a formula for tau(n) in terms of the p's and a's.
    b.) Determine a formula for sigma in terms of the p's and a's
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  2. #2
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    ..
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    ..
    I believe this is the formula for reg tau(n) and reg sigma(n). I need to find it for the ODD divisors of n (number/sum, respectively for tau/sigma)
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  4. #4
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    Quote Originally Posted by flash101 View Post
    I believe this is the formula for reg tau(n) and reg sigma(n). I need to find it for the ODD divisors of n (number/sum, respectively for tau/sigma)
    I see.

    Factorize n>1,

    n=p_1^a_1 * p_2^a_2 * ... * p_n^a_n

    Say, p_1=2, the even part of the number in factorization.

    You can choose 1 choice for a_1 because you want them all odd, hence a_1=0 is only possibility.

    You can choose a_2+1 for second and so on.

    Thus,

    (a_2+1)(a_3+1)...(a_n+1)

    Same thing as all but without first factor.
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