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COMMENT on A000040, A006562 and A001359 on the OEIS

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  • reismann@free.fr
    Hi, The following comments were published on the On-Line Encyclopedia of Integer Sequences : A000040 : prime numbers There is a unique decomposition of the
    Message 1 of 6 , Mar 7, 2007
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      Hi,

      The following comments were published on the On-Line Encyclopedia of
      Integer Sequences :
      A000040 : prime numbers
      There is a unique decomposition of the primes: provided the weight
      A117078(n) is > 0, we have prime(n) = weight * level + gap, or
      A000040(n) = A117078(n) * A117563(n) + A001223(n). - Remi Eismann
      (reismann(AT)free.fr), Feb 16 2007

      A001359 : lesser of twin primes
      Primes for which the weight as defined in A117078 is 3 gives this
      sequence except for the initial 3. - Remi Eismann
      (reismann(AT)free.fr), Feb 15 2007

      A006562 : balanced primes
      Let p(i) denote the i-th prime. If 2 p(n) - p(n+1) is a prime, say p(n-
      i), then we say that p(n) has level(1,i). Sequence gives primes of
      level(1,1). - Remi Eismann (reismann(AT)free.fr), Feb 15 2007

      You can find these comments with this link :
      http://www.research.att.com/~njas/sequences/?q=eismann&sort=0&fmt=0&l

      I also realized a Web site to display my work (in French) :
      http://reismann.free.fr/classement.php

      I wait for comments, for criticisms or for suggestions.

      Best,
      Rémi Eismann
    • reismann@free.fr
      Dear primenumbers Group, Your silence is deafening. I am not a professional mathematician (just an amateur) but I think that my work deserves comments. The
      Message 2 of 6 , Mar 12, 2007
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        Dear primenumbers Group,

        Your silence is deafening.
        I am not a professional mathematician (just an amateur) but I think that my work
        deserves comments.

        The decomposition prime(n)=weight*level+gap is a sieve. If we apply it to the
        natural numbers, we obtain that:
        http://reismann.free.fr/Graph_Entiers_1000000-R.html
        It is a beautiful representation of the sieve of Eratosthene.

        If we apply the decomposition to prime numbers, we obtain that:
        http://reismann.free.fr/Log_L-Log_K_3000000-R.html

        prime(n)=weight*level+gap or prime(n)=weight*level+prime(n+1)-prime(n). It means
        that prime(n+1) is contained in prime(n) or the form of prime(n) determines
        prime(n+1).

        Best,

        Rémi Eismann
      • Phil Carmody
        ... I personally found next to nothing interesting in your post. If I were an editor of OEIS, I d have had serious reservations about your submissions. ... No
        Message 3 of 6 , Mar 12, 2007
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          --- reismann@... wrote:
          > Your silence is deafening.

          I personally found next to nothing interesting in your post. If I were an
          editor of OEIS, I'd have had serious reservations about your submissions.

          > The decomposition prime(n)=weight*level+gap is a sieve.

          No it is not.

          > If we apply it to the
          > natural numbers, we obtain that:
          > http://reismann.free.fr/Graph_Entiers_1000000-R.html
          > It is a beautiful representation of the sieve of Eratosthene.

          No it is not.

          Phil

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        • reismann@free.fr
          Dear Phil and primenumbers Group, Thank you Phil for your answer. Why the decomposition in weight*level+gap is a sieve ?
          Message 4 of 6 , Mar 13, 2007
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            Dear Phil and primenumbers Group,

            Thank you Phil for your answer.

            Why the decomposition in weight*level+gap is a sieve ?

            http://reismann.free.fr/img/sieveNb.jpg
            This graph is the representation of the decomposition of the natural numbers in
            weight * level (or weight * level + gap with gap = 0, the weight is the smallest
            divisor of n). The sets 1, 2, 3 etc... are exactly those whom we obtain by
            making a sieve of Eratosthene on the paper.
            This graph is also the representation of the decomposition of the natural
            numbers in weight * level + gap (with gap = 1, the weight is the smallest
            divisor of n-1). In that case the odd numbers have a weight of 2 (ex: 3 =
            2*1+1).
            The decomposition in weight * level + gap is thus a sieve.

            Applied to prime numbers, we obtain that:
            http://reismann.free.fr/classement.php

            The main differences:
            - for the natural numbers :
            * the weights are prime numbers
            * in the zone weight>level, We have only the numbers of level 1
            - for prime numbers:
            * the weights are odd
            * in the zone weight>level (zone 2), we have several levels (and a limit
            level, very important for the new algo).

            The decomposition in weight * level + gap is a sieve but by factorizing n-gap or
            prime(n)-gap(n). There are "multiples" and "waves" in prime numbers as there is
            among the natural numbers.

            Best,

            Rémi
          • Robin Garcia
            So,we must know p(n+1) to know p(n) and its unique decomposition. Est-ce que vous pensez que cette double et nécessaire connaissance, apportera vraiment un
            Message 5 of 6 , Mar 14, 2007
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              So,we must know p(n+1) to know p(n) and its unique decomposition.

              Est-ce que vous pensez que cette double et nécessaire connaissance, apportera vraiment un air nouveau à notre connaissance des nombres premiers?

              Je ne dis pas que votre travail ne soit pas intéressant, mais je ne vois pas d´algorithmes utilisables.En avez vous?


              reismann@... escribió:
              Dear Phil and primenumbers Group,

              Thank you Phil for your answer.

              Why the decomposition in weight*level+gap is a sieve ?

              http://reismann.free.fr/img/sieveNb.jpg
              This graph is the representation of the decomposition of the natural numbers in
              weight * level (or weight * level + gap with gap = 0, the weight is the smallest
              divisor of n). The sets 1, 2, 3 etc... are exactly those whom we obtain by
              making a sieve of Eratosthene on the paper.
              This graph is also the representation of the decomposition of the natural
              numbers in weight * level + gap (with gap = 1, the weight is the smallest
              divisor of n-1). In that case the odd numbers have a weight of 2 (ex: 3 =
              2*1+1).
              The decomposition in weight * level + gap is thus a sieve.

              Applied to prime numbers, we obtain that:
              http://reismann.free.fr/classement.php

              The main differences:
              - for the natural numbers :
              * the weights are prime numbers
              * in the zone weight>level, We have only the numbers of level 1
              - for prime numbers:
              * the weights are odd
              * in the zone weight>level (zone 2), we have several levels (and a limit
              level, very important for the new algo).

              The decomposition in weight * level + gap is a sieve but by factorizing n-gap or
              prime(n)-gap(n). There are "multiples" and "waves" in prime numbers as there is
              among the natural numbers.

              Best,

              Rémi






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            • reismann@free.fr
              Dear Robin and primenumbers group, Yes it is necessary to know p(n+1) to have the decomposition. I do not propose magic formula. The magic formula would be to
              Message 6 of 6 , Mar 15, 2007
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                Dear Robin and primenumbers group,

                Yes it is necessary to know p(n+1) to have the decomposition. I do not propose
                magic formula. The magic formula would be to have the decomposition without
                knowing p(n+1)...

                Will my vision of prime numbers bring answers to the big questions of the theory
                of the numbers?
                I do not know, I am not sure.
                But to what is of use a classification ? It serves for having a common language.
                It is surprising that in approximately 2500 years nobody did not find the
                sequence of the weights. No?

                >Have you a useful algorithm?
                With my friends Fabien we worked on a program giving the decompostion of prime
                numbers (in assembly for Windows) :
                http://reismann.free.fr/download/class_asm.zip
                There is also a program in Java on the "Download" page :
                http://reismann.free.fr/telechargements.php
                It is a "naive" algorithm. We work at present on a new algorithm closer to the
                sieve of Eratosthène.

                Thank you for your answer Robin

                Best,

                Rémi
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