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Variation of perfect numbers

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  • Jack Brennen
    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
    Message 1 of 2 , 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:

      42
      1316
      131080256
      72872313094554244192
      37778715690312487141376

      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
      p=(2,3,7,19).

      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.
    • Edwin Clark
      I don t know of any work on the numbers you mention, but the function: sum of nonprime factors of n is listed in the OEIS: ID Number: A023890 URL:
      Message 2 of 2 , Dec 1, 2003
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        I don't know of any work on the numbers you mention, but the function:
        sum of nonprime factors of n is listed in the OEIS:

        ID Number: A023890
        URL: http://www.research.att.com/projects/OEIS?Anum=A023890
        Sequence: 1,1,1,5,1,7,1,13,10,11,1,23,1,15,16,29,1,34,1,35,22,23,1,55,
        26,27,37,47,1,62,1,61,34,35,36,86,1,39,40,83,1,84,1,71,70,
        47,1,119,50,86,52,83,1,115,56,111,58,59,1,158,1,63,94,125,
        66,128,1,107,70,130,1
        Name: Sum of nonprime divisors of n.



        On Mon, 1 Dec 2003, Jack Brennen wrote:

        >
        > 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:
        >
        > 42
        > 1316
        > 131080256
        > 72872313094554244192
        > 37778715690312487141376
        >
        > 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
        > p=(2,3,7,19).
        >
        > 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.
        >
        >
        > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
        > The Prime Pages : http://www.primepages.org/
        >
        >
        >
        > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
        >
        >

        ------------------------------------------------------------
        W. Edwin Clark, Math Dept, University of South Florida,
        http://www.math.usf.edu/~eclark/
        ------------------------------------------------------------
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