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Re: [PrimeNumbers] Re: Greatest Prime Factor Sets?

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  • Phil Carmody
    ... p-smooth sets are infinite for all p. Therefore so are sets of numbers with the same gpf. ... Countable and countable. I m not sure I see the difference.
    Message 1 of 7 , Feb 3, 2007
      --- "joel.levenson" <joel.levenson@...> wrote:
      > Hi Phil
      >
      > >The obverse, those which have the same least prime factor, are used
      > > extensively.
      >
      > I must have at some point looked at the Wolfram article on "least
      > prime factor" but I somehow missed the connection. Wolfram says
      > that lpf(n) is standard, so I guess I can use gpf(n).
      >
      > There are two properties that "gpf" sets have that interest me, and
      > "lpf" sets have only one of them, but it took me a while to see that
      > it doesn't have the other.
      >
      > The property they share is that each integer, by definition, has one
      > and only one least- or greatest- prime factor, so you get the mapping.
      >
      > The property they don't share is the one that leads to the finite sets
      > (although it may be there and I'm not seeing it). This is because
      > using the gpf puts an upper limit on the number of integers with the
      > same "not necessarily distinct factors".

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

      > 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.

      Phil

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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 2 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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