20070430, 18:17  #1 
"R. E."
Apr 2007
France
27_{8} Posts 
COMMENT on A000040, A006562 and A001359 on OEIS
Hi,
The following comments were published on the OnLine 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 ith 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 
20070504, 18:05  #2 
Feb 2007
432_{10} 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 ? 
20070504, 23:01  #3  
Feb 2006
Denmark
2·5·23 Posts 
Quote:
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. 

20070504, 23:49  #4 
"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 
20070505, 01:05  #5 
Feb 2006
Denmark
230_{10} Posts 

20070505, 08:45  #6  
"R. E."
Apr 2007
France
23_{10} Posts 
Quote:
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 

20070702, 18:35  #7 
∂^{2}ω=0
Sep 2002
República de California
2×5×1,163 Posts 
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."
Last fiddled with by ewmayer on 20070702 at 18:37 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Anybody else lose the ability to comment on Youtube?  jasong  jasong  5  20131112 15:34 
10 and strictly prime or composite.Comment.  David John Hill Jr  Miscellaneous Math  7  20100606 12:33 
C230 snfs almost stuck  please comment  Syd  Factoring  9  20090830 19:34 
New Cunningham Tables are ready. Please see sample and comment  garo  Factoring  9  20050802 16:52 
The Rush Limbaugh Comment  eepiccolo  Soap Box  6  20031008 03:10 