Odd Perfect Numbers?

Discussion in 'Undergraduate Math' started by Pedhuts, Dec 3, 2010.

  1. Pedhuts

    Pedhuts Guest

    Is there some characterization of the sum of divisors for an odd square number?
     
    Pedhuts, Dec 3, 2010
    #1
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  2. The sum-of-divisors is multiplicative, and sigma(p^k) = 1 + p + \cdots
    + p^k = (p^{k+1}-1}/(p-1).

    So the sum of divisiors of an odd square number is always of the form

    (p_1^{2k_1+1}-1)(p_2^{2k_2+1} -1) * ... * (p_m^(2k_m+1))/(p_1-1)
    (p_2-1)...(p_m-1)

    or

    (1+p_1+...+p_1^{2k_1})(1 + p_2 + ... + p_2^{2k_2})...(1 + p_m + ... +
    p_m^{2k_m}).

    with p_1<p_2<...<p_m odd primes, k_1,k_2,...,k_m positive integers.
    And any such number is the sum of the divisors of an odd square.
     
    Arturo Magidin, Dec 4, 2010
    #2
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