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Re: [PrimeNumbers] Re: consecutive p-smooth integers

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  • Andrey Kulsha
    Hello David, ... as I wrote in http://tech.groups.yahoo.com/group/primenumbers/message/22884, it seems that for fixed p maximal N grows nearly as
    Message 1 of 42 , Aug 7, 2011
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      Hello David,

      > Here is a puzzle about sequences of 5 consecutive integers.
      >
      > Definition: A sequence of consecutive positive
      > integers, beginning with N, has strength
      > s = log(N)/log(p), where p is the largest prime
      > dividing any integer in the sequence.

      as I wrote in http://tech.groups.yahoo.com/group/primenumbers/message/22884,
      it seems that for fixed p maximal N grows nearly as exp(b*sqrt(p)), where b
      is a constant depending on the length of the chain, so it's little sense to
      use log(N)/log(p) as a measure of the "strength". The better would be
      log(N)/sqrt(p), and the largest currently known for 5 integers is

      log(1517)/sqrt(41) = 1.14389...

      Best regards,

      Andrey
    • Mark
      ... Too hard! It was difficult getting just over 2 for the first time with log(287946949)/log(15823). My only consolation is that the above is also good for 15
      Message 42 of 42 , Nov 8, 2011
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        --- In primenumbers@yahoogroups.com, Andrey Kulsha <Andrey_601@...> wrote:
        >
        > >> http://www.primefan.ru/stuff/math/maxs.xls
        > >> http://www.primefan.ru/stuff/math/maxs_plots.gif
        > >
        > > Thanks, Andrey. The gradients are fanning out better now:
        >
        > The files were updated again.
        >
        > Puzzle: find a chain of 13 consecutive p-smooth integers, starting at N,
        > with log(N)/log(p) greater than
        >
        > log(8559986129664)/log(58393) = 2.71328
        >
        > Best regards,
        >
        > Andrey
        >

        Too hard!

        It was difficult getting just over 2 for the first time with
        log(287946949)/log(15823).

        My only consolation is that the above is also good for 15 consecutive 15823-smooth integers.
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