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Re: Greatest Prime Factor Sets?

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  • joel.levenson
    Hi Phil ... numbers with the same gpf. gpf sets are infinite, and their union is the set of all integers greater than one, and their intersection is empty.
    Message 1 of 7 , Feb 3, 2007
      Hi Phil

      > p-smooth sets are infinite for all p. Therefore so are sets of
      numbers with the same gpf.

      gpf sets are infinite, and their union is the set of all integers
      greater than one, and their intersection is empty.

      What interests me about organizing the integers in terms of their gpfs
      is that each one can be uniquely identified by its gpf and its
      position among the rest of the integers with that gpf (when they are
      placed in ascending order, for instance.) But a p-smooth number is
      also p-smooth for every prime greater or equal to p, so it isn't
      uniquely identified. (Payam suggested calling numbers with the same
      gpf "strictly p-smooth numbers" But, I think, from my point of view,
      it is simpler to think in terms of gpfs. Then you don't have to bring
      in any concepts beyond the fundamental theorem of arithmetic and the
      definition of "greatest.")

      I don't know if mapping the primes to the integers is interesting or
      useful, or how many ways it can be done, but this way caught my
      attention.

      > > lpf gpf
      > >
      > > 3 3
      > > 3*2 3*2*2 ... 2*3 3*3
      > > 3*3 3*3*3 ... 2*2*3 2*3*3 3*3*3
      > > 3*5 ...
      > > >
      >> lpf sets are n-dimensional, so to speak. gpf sets are two dimensional.

      > > Countable and countable. I'm not sure I see the difference.
      > > Dense and not dense is the distinction that you could draw.
      > > P-smooth numbers are not dense in the integers, so neither
      > > are sets sharing gpfs.

      (I was using n-dimensional sort of whimsically. I just meant that if
      you try to write out all the lpf numbers on a piece of paper you have
      to go off in all directions.)

      What I am interested in here is the combinatorial structure, involving
      subsets of gpf sets, defined by the number of "not necessarily
      distinct primes." There are an infinite number of these subsets,
      their union is the gpf set, and their intersection is the null set.
      But each one is finite, and the formula for how many elements there
      are in each set, where P(n) is the nth prime and r is the number of
      not necessarily distinct prime factors, is "n choose r with
      repetitions" which is C(n+r-1, n). So you can get a very simple
      structure associating the integers with the primes, and with a little
      manipulation you get these finite sets:

      {2}
      (4,3}
      {8, 6, 9, 5}
      (16, 12, 18, 27, 10, 15, 25, 7}

      This pattern goes on forever. Among other things, the nth set has
      2^(n-1) and contains P(n) and 2^n. The pattern of the other numbers
      has to do with the "not necessarily distinct prime factor" subsets.
      (My second post has enough detail to explain it adequately, I think.)

      I was just surprised to find so much structure, and have been
      exploring it as best I can.

      Thanks for your help! And your time. I'm sorry, I can't help being
      so wordy.
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