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Re: [PrimeNumbers] Re: A074304

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  • Max B
    Interesting, Jack. Thanks. ... A subset of the pseudoperfects... For example, take the number 66. 66 is pseudoperfect because it can be expressed: 66 =
    Message 1 of 7 , Nov 1, 2002
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      Interesting, Jack. Thanks.


      --- In primenumbers@y..., "Max B" <zen_ghost_floating@h...> wrote:
      >
      > "A positive integer n is called unitary nonrepetitive,
      > if, excluding the divisors 1 and n, it is possible to
      > express n-1 as a sum of some or all of the remaining
      > divisors of n using each divisor once and only once. For
      > example, 6 and 20 are unitary nonrepetitive since
      > 5=2+3 and 19=10+5+4. In fact, every perfect number
      > is unitary nonrepetitive." --p.147
      >
      > Are these numbers the same as the pseudoperfects?
      >

      A subset of the pseudoperfects...

      For example, take the number 66.

      66 is pseudoperfect because it can be expressed:

      66 = 11+22+33
      66 = 2+3+6+22+33

      66 is not unitary nonrepetitive because none of the ways of
      expressing 66 as a sum of its distinct divisors include the
      number 1.

      Numbers which are pseudoperfect but not unitary nonrepetitive
      include: 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282,
      318, 348, 354, 366, 372, 402, 414, 426, 438, 444, 474, 490,
      492, 498, 516, 522, 534, 564, 580, 582, 606, 618, 636, 642,
      644, 654, 678, 708, 726, 732, 738, 748, 762, 774, 786.

      It is easy to see that any number of the form 6*p, with
      prime p>=11, is pseudoperfect but not unitary nonrepetitive.

      Similarly for any number of the form 12*p, with prime p>=29.
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