 mersenneforum.org COMMENT on A000040, A006562 and A001359 on OEIS
 Register FAQ Search Today's Posts Mark Forums Read 2007-04-30, 18:17 #1 Nunki   "R. E." Apr 2007 France 278 Posts COMMENT on A000040, A006562 and A001359 on OEIS 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/se...sort=0&fmt=0&l... I also realized a Web site to display my work (in french) : http://reismann.free.fr/classement.php Best, Rémi Eismann  2007-05-04, 18:05 #2 m_f_h   Feb 2007 43210 Posts I recently ran across this statement on OEIS prime(n) = weight * level + gap, it was not clear to me, since the definitions seem a bit "recursive" to me. (I understand gap, but I speak of the both others.) I mean, what does "there is a unique decomposition of the primes" mean ? Since, - some are excluded by the fact that w(n)=0 - as to the others, does "unique" mean that (w,L) will be different for each n ? or that there is only 1 couple (w,L) so that this equation holds ? If w(n) is *defined* though some equation, then of course L is given by the relation above, and/or reciprocally, by hoping that this w(n) will divide p(n)-g(n)=p(n)-(p(n+1)-p(n))=2p(n)-p(n+1). I assume that w(n)=0 or 1 in case this is not composite (n=2, 11, 15, 18, 36, 39, 46,... another new sequence...). Is this a "new" theory ? If yes, are there proofs ? If no, are there references ?  2007-05-04, 23:01   #3
Jens K Andersen

Feb 2006
Denmark

2·5·23 Posts Quote:
 Originally Posted by m_f_h I recently ran across this statement on OEIS prime(n) = weight * level + gap,
Eismann has promoted this in many places. It is "unique" by definition.
Let g(n) be the nth prime gap: prime(n+1) = prime(n) + g(n).
The "weight", w(n), is (indirectly) defined as the smallest divisor above g(n) of prime(n) - g(n), or 0 if there is no such divisor (because prime(n) - g(n) <= g(n)).
The "level" is defined as the cofactor: level = (prime - gap)/weight.
The definitions seem arbitrary to me. I cannot think of a use.  2007-05-04, 23:49 #4 Nunki   "R. E." Apr 2007 France 23 Posts Hi, The weight is : - the smallest k such that the gap, g(n) is the remainder of the euclidiean division of p(n) by k, 0 if no such k exists or - the smallest k such that g(n) = p(n) mod k, 0 if no such k exists or - the smallest divisor > g(n) of p(n)-g(n), 0 if no such divisor exists. This decomposition applied to the natural numbers is the sieve of Erathostenes. Not so arbitrary. Rémi  2007-05-05, 01:05   #5
Jens K Andersen

Feb 2006
Denmark

23010 Posts Quote:
 Originally Posted by Nunki This decomposition applied to the natural numbers is the sieve of Erathostenes.
Aha, I have only implemented the sieve of Erathostenes' cousin Eratosthenes. Maybe this has potential for a unique contribution to Miscellaneous Math Threads.  2007-05-05, 08:45   #6
Nunki

"R. E."
Apr 2007
France

2310 Posts Quote:
 Originally Posted by Jens K Andersen Aha, I have only implemented the sieve of Erathostenes' cousin Eratosthenes.
Oups sorry "Eratosthenes". Too funny Jens.

Decomposition of primes :
http://reismann.free.fr/primeSieve.html

Eratosthenes sieve :
http://reismann.free.fr/sieveEra.html

For more explanations (in french) :
http://reismann.free.fr/entiers.php

Rémi  2007-07-02, 18:35   #7
ewmayer
2ω=0

Sep 2002
República de California

2×5×1,163 Posts Quote:
 Originally Posted by Nunki Oups sorry "Eratosthenes". Too funny Jens.
Yes, like most of the rest of us, Jens likes having his time wasted so much, it makes him want to tell a joke or two just by way of "thanks."

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