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Re: [PrimeNumbers] Puzzle: Sphenic chain by factor concatenation

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  • Kevin Acres
    ... And then of course, just a few minutes later: {sphen84a = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6, 6, 4, 2, 6, 2, 2, 4, 4, 3, 2,
    Message 1 of 16 , Jun 15, 2011
      At 08:09 PM 15/06/2011, Kevin Acres wrote:
      >Hello David,
      >
      >At 03:19 AM 15/06/2011, djbroadhurst wrote:
      >
      > >Definition: A number is sphenic iff it is the product of 3
      > >distinct primes. A "sphenic chain" is a sequence of sphenic
      > >numbers such that each except the first is one of the 6
      > >decimal concatenations of the primes dividing its predecessor.
      > >
      > >[ snip ]
      > >Here is how generate a chain of length 80, using Pari-GP:
      > >
      > > {sphen80 = [114,
      > > 3, 6, 6, 3, 1, 5, 2, 3, 3, 4, 5, 3, 5, 5, 6, 3, 6, 3, 1, 3,
      > > 3, 4, 5, 3, 4, 4, 6, 3, 1, 1, 1, 1, 2, 1, 1, 5, 6, 6, 2, 5,
      > > 2, 2, 6, 3, 3, 1, 4, 1, 5, 3, 6, 3, 5, 4, 3, 2, 6, 5, 2, 4,
      > > 3, 2, 3, 5, 3, 4, 6, 1, 1, 3, 5, 2, 6, 4, 4, 3, 1, 6, 2];}
      > >
      > >[ snip]
      > >Puzzle: Find a sphenic chain with more than 80 members.
      >
      >Many GHz hours later.....
      >
      > {sphen81 = [114,
      > 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
      > 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
      > 6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
      > 3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4];}

      And then of course, just a few minutes later:

      {sphen84a = [114,
      3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
      6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
      6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
      3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
      4, 3, 4];}

      {sphen84b = [114,
      3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
      6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
      6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
      3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
      4, 3, 3];}



      [Non-text portions of this message have been removed]
    • djbroadhurst
      ... Congrats! I m now in the low 90s, but do not expect to make it to length 100. David
      Message 2 of 16 , Jun 15, 2011
        --- In primenumbers@yahoogroups.com,
        Kevin Acres <research@...> wrote:

        > {sphen84a = [114,
        > 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
        > 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
        > 6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
        > 3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
        > 4, 3, 4];}
        >
        > {sphen84b = [114,
        > 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 3, 6, 2, 2, 1, 1, 3, 4, 2, 6,
        > 6, 4, 2, 6, 2, 2, 4, 4, 3, 2, 1, 3, 5, 6, 2, 4, 5, 5, 1, 4,
        > 6, 4, 6, 4, 5, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 4, 2, 3, 4, 2,
        > 3, 2, 1, 5, 4, 5, 6, 3, 5, 6, 3, 5, 6, 6, 3, 4, 4, 2, 4, 4,
        > 4, 3, 3];}

        Congrats!

        I'm now in the low 90s, but do not expect to make it to length 100.

        David
      • djbroadhurst
        ... In fact, I was able to reach length 105: {sphen105 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6, 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1,
        Message 3 of 16 , Jun 15, 2011
          --- In primenumbers@yahoogroups.com,
          "djbroadhurst" <d.broadhurst@...> wrote:

          > do not expect to make it to length 100

          In fact, I was able to reach length 105:

          {sphen105 = [114,
          3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
          1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
          4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
          5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 3, 6, 4, 2, 4, 5, 2, 6, 1,
          1, 4, 6, 1, 6, 1, 2, 3, 2, 4, 6, 3, 3, 1, 6, 3, 3, 3, 1, 3,
          1, 5, 3, 1];}

          http://physics.open.ac.uk/~dbroadhu/cert/sphen105.gp
          contains 8 helpers, obtained by ECM, and then the chain of
          http://physics.open.ac.uk/~dbroadhu/cert/sphen105.txt
          is generated in less than 4 minutes.

          David
        • djbroadhurst
          ... The best I can do at present is length 108: {sphen108 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6, 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6,
          Message 4 of 16 , Jun 16, 2011
            --- In primenumbers@yahoogroups.com,
            "djbroadhurst" <d.broadhurst@...> wrote:

            > able to reach length 105

            The best I can do at present is length 108:

            {sphen108 = [114,
            3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
            1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
            4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
            5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 3, 6, 4, 2, 4, 5, 2, 6, 1,
            1, 4, 6, 1, 6, 1, 2, 3, 2, 4, 6, 3, 3, 1, 6, 3, 3, 3, 1, 3,
            5, 1, 4, 5, 2, 5, 2];}

            with input and output in
            http://physics.open.ac.uk/~dbroadhu/cert/sphen108.gp
            http://physics.open.ac.uk/~dbroadhu/cert/sphen108.txt

            David
          • djbroadhurst
            ... Length 109 is achieved by {sphen109 = [114, 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6, 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5,
            Message 5 of 16 , Jun 20, 2011
              --- In primenumbers@yahoogroups.com,
              "djbroadhurst" <d.broadhurst@...> wrote:

              > The best I can do at present is length 108

              Length 109 is achieved by

              {sphen109 = [114,
              3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
              1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
              4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
              5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 5, 6, 2, 1, 1, 6, 6, 5, 6,
              6, 4, 6, 6, 2, 4, 4, 6, 6, 2, 5, 3, 1, 2, 2, 1, 2, 2, 4, 3,
              1, 4, 3, 1, 2, 5, 3, 5];}

              with input and output in
              http://physics.open.ac.uk/~dbroadhu/cert/sphen109.gp
              http://physics.open.ac.uk/~dbroadhu/cert/sphen109.txt

              David
            • Kevin Acres
              Hi David, ... Well done for that. Its been windy here and wind = power outage where I live :-) One day I ll get a decent size UPS. Best Regards, Kevin.
              Message 6 of 16 , Jun 20, 2011
                Hi David,

                At 02:20 PM 21/06/2011, djbroadhurst wrote:

                >--- In primenumbers@yahoogroups.com,
                >"djbroadhurst" <d.broadhurst@...> wrote:
                >
                > > The best I can do at present is length 108
                >
                >Length 109 is achieved by
                >
                > {sphen109 = [114,
                > 3, 6, 6, 3, 2, 2, 4, 4, 3, 4, 1, 6, 1, 4, 1, 3, 2, 3, 2, 6,
                > 1, 6, 3, 5, 5, 6, 2, 1, 6, 6, 6, 1, 2, 5, 4, 3, 5, 2, 5, 2,
                > 4, 1, 5, 1, 6, 5, 3, 3, 5, 5, 2, 1, 5, 5, 1, 6, 4, 5, 5, 3,
                > 5, 2, 4, 4, 4, 3, 4, 5, 6, 4, 6, 5, 6, 2, 1, 1, 6, 6, 5, 6,
                > 6, 4, 6, 6, 2, 4, 4, 6, 6, 2, 5, 3, 1, 2, 2, 1, 2, 2, 4, 3,
                > 1, 4, 3, 1, 2, 5, 3, 5];}

                Well done for that. Its been windy here and "wind = power outage"
                where I live :-)

                One day I'll get a decent size UPS.


                Best Regards,

                Kevin.
              • djbroadhurst
                ... The current record is length 112, achieved by {sphen112a = [114, 3, 6, 6, 3, 2, 2, 1, 3, 3, 2, 2, 1, 5, 2, 3, 2, 1, 6, 4, 4, 2, 4, 4, 2, 5, 4, 2, 3, 4, 2,
                Message 7 of 16 , Jun 23, 2011
                  --- In primenumbers@yahoogroups.com,
                  Kevin Acres <research@...> wrote:

                  > > {sphen109 = [114,
                  ....
                  > Well done for that. Its been windy here and "wind = power outage"
                  > where I live :-)

                  The current record is length 112, achieved by

                  {sphen112a = [114,
                  3, 6, 6, 3, 2, 2, 1, 3, 3, 2, 2, 1, 5, 2, 3, 2, 1, 6, 4, 4,
                  2, 4, 4, 2, 5, 4, 2, 3, 4, 2, 1, 3, 3, 6, 5, 4, 3, 1, 5, 4,
                  2, 5, 1, 1, 3, 4, 5, 5, 5, 5, 6, 3, 6, 1, 6, 1, 4, 3, 1, 6,
                  4, 6, 3, 1, 3, 2, 1, 2, 2, 6, 3, 2, 3, 5, 4, 6, 6, 5, 2, 2,
                  2, 3, 4, 6, 5, 2, 3, 3, 2, 6, 5, 1, 2, 6, 1, 4, 3, 5, 3, 3,
                  3, 4, 1, 4, 2, 2, 1, 2, 6, 3, 2];}

                  {sphen112b = [114,
                  3, 6, 6, 3, 2, 2, 1, 3, 3, 2, 2, 1, 5, 2, 3, 2, 1, 6, 4, 4,
                  2, 4, 4, 2, 5, 4, 2, 3, 4, 2, 1, 3, 3, 6, 5, 4, 3, 1, 5, 4,
                  2, 5, 1, 1, 3, 4, 5, 5, 5, 5, 6, 3, 6, 1, 6, 1, 4, 3, 1, 6,
                  4, 6, 3, 1, 3, 2, 1, 2, 2, 6, 3, 2, 3, 5, 4, 6, 6, 5, 2, 2,
                  2, 3, 4, 6, 5, 2, 3, 3, 2, 6, 5, 1, 2, 6, 1, 4, 3, 2, 3, 3,
                  4, 2, 2, 3, 2, 2, 1, 4, 5, 1, 1];}

                  with input and output in
                  http://physics.open.ac.uk/~dbroadhu/cert/sphen112.gp
                  http://physics.open.ac.uk/~dbroadhu/cert/sphen112.txt

                  David
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