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14102Variation of perfect numbers

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  • Jack Brennen
    Dec 1, 2003
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      I've been looking into a couple of minor variations on the "perfect
      number"... Of course, a perfect number is equal to the sum of its
      proper factors.

      What if we exclude its prime factors? What numbers are equal to the
      sum of their non-prime proper factors? (Note that the factor one
      is included, since 1 is not prime.) I can find five:


      Note that all of these except for 72872313094554244192 are of a
      special form:

      q = 2^p-1
      r = 2^(2*p)-2^(p+1)-1

      N = 2^(p-1)*q*r

      where p,q,r are all prime. In this case, we get solutions for

      That leaves the "oddball" number:

      72872313094554244192 == 2^5 * 109 * 151 * 65837 * 2101546957

      Which seems to be just a stroke of random luck.

      Perhaps even more interesting, what numbers are equal to the sum
      of their composite proper factors? Same problem as above,
      except that the factor 1 is not counted. Despite searching
      long and hard, I haven't found a single example. I can't think
      of any reason why such numbers shouldn't exist, nor why they
      should be so scarce.

      Has anybody ever heard of other research on these two types of
      numbers? I will continue my search for now, but if I'm treading
      on already explored ground, I would love to hear about it. A
      Google search turned up nothing of interest. In particular,
      the number 131080256 doesn't show any hits, despite its rather
      "rare" nature.
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