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Re: [PrimeNumbers] Re: Sphenic tree by factor concatenation on 114

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  • James Merickel
    Yes, suspected my one computer was not adequate in any case. I would have had to do some hands-on measurements to be sure, and probably would not have. So I
    Message 1 of 16 , Jun 13, 2011
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      Yes, suspected my one computer was not adequate in any case. I would have had to do some hands-on measurements to be sure, and probably would not have. So I am glad for this particular response. My one PARI/GP window is done without the dividing line between iterations and has just the first 1.696Million nodes so far (with four 43-digit numbers and current mode at 33 digits with just over 230000).

      On Mon Jun 13th, 2011 6:34 PM EDT djbroadhurst wrote:

      >
      >
      >--- In primenumbers@yahoogroups.com,
      >"djbroadhurst" <d.broadhurst@...> wrote:
      >
      >> 186328 252326 338513 448911 589071
      >
      >Here are my counts for the first 37 iterations:
      >
      >2, 2, 2, 1, 3, 7, 13, 28, 39, 73, 141, 264, 459,
      >757, 1220, 1937, 3054, 4700, 7173, 10766, 15803,
      >23366, 33947, 48816, 69547, 97473, 135489,
      >186328, 252326, 338513, 448911, 589071,
      >766390, 984903, 1253696, 1578502, 1966106,
      >
      >with ratios of successive counts, from the last 10 entries,
      >1.354, 1.342, 1.326, 1.312, 1.301, 1.285, 1.273, 1.259, 1.246,
      >indicating that we are still well below the shrinking point.
      >
      >It is clear that James was very generous in allowing
      >6 possible decimal concatenations of the 3 distinct
      >prime factors of a sphenic number.
      >
      >David
      >
    • djbroadhurst
      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
      Message 2 of 16 , Jun 14, 2011
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        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.

        Example: Here is a chain of length 10:
        1: 114 = 3*2*19
        2: 3219 = 37*29*3
        3: 37293 = 401*31*3
        4: 401313 = 11*3*12161
        5: 11312161 = 7*53*30491
        6: 75330491 = 2887*97*269
        7: 288797269 = 23*187409*67
        8: 2318740967 = 1097*61*34651
        9: 10976134651 = 1163*229*41213
        10: 116322941213 [one may continue this chain]
        which we may conveniently denote by
        [114, 3, 6, 6, 3, 1, 5, 2, 3, 3]
        where 114 is the first number and then we indicate which of
        the permutations {123, 132, 213, 231, 312, 321} were used.

        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];}

        {ischain(s)=local(f,n=s[1],P,t);
        P=[[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]];
        for(k=1,#s,print1(k": ");n=eval(n);f=factor(n)[,1];
        if(n!=f[1]*f[2]*f[3],print("fail");break,
        if(k==#s,print(n" has factors "f~),print1(n " = ");
        n="";for(j=1,3,t=f[P[s[k+1]][j]];print1(t);
        if(j<3,print1("*"),print());n=concat(n,t)))));}

        ischain(sphen80);

        with takes less than a minute to generate the output in
        http://physics.open.ac.uk/~dbroadhu/cert/sphen80.txt
        Sadly, none of the 6 concatenations the 3 primes
        from the 80th member yields a sphenic number.

        Puzzle: Find a sphenic chain with more than 80 members.

        Comment: Please communicate your result in the
        economical format that I used for "sphen80".

        David Broadhurst, 14 June 2011
      • 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 3 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 4 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 5 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 6 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 7 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 8 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 9 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 10 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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