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Thread: Sigma Function Help

  1. December 6th 2007, 07:39 PM #1
    suissa
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    Sigma Function Help

    Sigma Function

    σ: ℕ ->ℕ

    n
    Σ k = σ (n)
    k|n

    1. Is the Sigma Fctn an injection?
    2. Is it a surjection?
    3. Can you find a formula for the sigma fctn which allows to calculate the sum of divisors of n based on the decomposition of n into a product of prime numbers?

    Thanks guys!
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  2. December 6th 2007, 08:07 PM #2
    ThePerfectHacker
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    Quote Originally Posted by suissa View Post
    n
    Σ k = σ (n)
    k|n

    1. Is the Sigma Fctn an injection?
    2. Is it a surjection?
    3. Can you find a formula for the sigma fctn which allows to calculate the sum of divisors of n based on the decomposition of n into a product of prime numbers?
    1) Consider n=15,23.
    2) Can it be that \sigma (n) = 2?
    3) If n=p_1^{a_1}...p_n^{a_n} then \sigma (n) = \frac{p_1^{a_1}-1}{p_1-1}\cdot ... \cdot \frac{p_n^{a_n}-1}{p_n-1}.
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  3. December 6th 2007, 08:17 PM #3
    badgerigar
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    Sorry please ignore this, the perfect hacker already answered in more detail and clarity

    1. Is the Sigma Fctn an injection?
    No. Look for a counter-example, it shouldn't take too long.

    2. Is it a surjection?
    No. look for another counter-example

    3. Can you find a formula for the sigma fctn which allows to calculate the sum of divisors of n based on the decomposition of n into a product of prime numbers?
    Yes.
    If n = \prod_{i=0}^ma_i^{k_i} where all a are prime,
    then \sigma(n) = \prod_{i=0}^m\sum_{j_i=0}^{k_i}a_i^{j_i}
    This can be simplified using the formula for geometric series
    Last edited by badgerigar; December 6th 2007 at 08:19 PM. Reason: Perfect hacker beat me to it
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  4. December 6th 2007, 08:25 PM #4
    suissa
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    I'm sorry I don't understand 1 and 2.
    Thanks a lot for the reply though!
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  5. December 6th 2007, 08:46 PM #5
    suissa
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    Oh I understand it.
    Thanks PerfectHacker and badgerigar
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