COMMENT on A000040, A006562 and A001359 on OEIS

[Nunki]

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

[m_f_h]

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 ?

[Jens K Andersen]

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.

[Nunki]

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

[Jens K Andersen]

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.

[Nunki]

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

[ewmayer]

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."