## COMMENT on A000040, A006562 and A001359 on the OEIS

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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
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

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
• 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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I am not a professional mathematician (just an amateur) but I think that my work

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
• ... 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:

I personally found next to nothing interesting in your post. If I were an

> 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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• 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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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
• 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:

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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• 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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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) :