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

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  • Kevin Acres
    Hello David, ... 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,
    Message 1 of 16 , Jun 15, 2011
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      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];}


      Best Regards,

      Kevin.

      [Non-text portions of this message have been removed]
    • 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 2 of 16 , Jun 15, 2011
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        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 3 of 16 , Jun 15, 2011
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          --- 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 4 of 16 , Jun 15, 2011
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            --- 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 5 of 16 , Jun 16, 2011
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              --- 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 6 of 16 , Jun 20, 2011
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                --- 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 7 of 16 , Jun 20, 2011
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                  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 8 of 16 , Jun 23, 2011
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                    --- 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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