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Re: Convergence of reciprocal odd perfect numbers

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  • djbroadhurst
    ... The really tough question is whether the sum of the reciprocals of the sociable numbers is finite. One might guess so, from the opinions in
    Message 1 of 7 , Aug 24, 2011
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > The asymptotic number of perfect numbers less than x was proven in
      >
      > MR0090600 (19,837d)
      > Hornfeck, Bernhard; Wirsing, Eduard
      > Über die Häufigkeit vollkommener Zahlen. (German)
      > Math. Ann. 133 (1957), 431-438.
      >
      > to be less than x^eps for all eps>0.
      > From this it trivially follows that the sum of the reciprocals
      > of the perfect numbers is finite.
      >
      > I regard Carl's result in
      >
      > MR0618552 (82m:10012)
      > Pomerance, Carl
      > On the distribution of amicable numbers. II.
      > J. Reine Angew. Math. 325 (1981), 183-188.
      >
      > as "stronger" since it proves the finiteness of
      > of the sum of the reciprocals of the amicable numbers.

      The really tough question is whether the sum of the
      reciprocals of the sociable numbers is finite.

      One might guess so, from the opinions in
      http://www.math.dartmouth.edu/~carlp/sociable.pdf
      but there seems little hope of a proof,
      given that we do not know whether an integer as small
      as 276 is sociable.

      David
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