Numbers are nothing

[reismann at free.fr]

Hi seqfans,

The numbers are nothing. Nothing by themselves.
A number can be defined only by the sequence to which it belongs and more
exactly by the jump which it provokes.
The vision of the fundamental theorem of the arithmetic, the analysis of the
numbers by their prime factors is the decomposition in weight*level (jump=0).
It is our vision of the numbers since Erathostènes.
What I propose, it is to generalize the sieve of Erathostènes by the following
formula :
formula of weights: smallest k such that g(n) = N(n) mod k, 0 if a such k does
not exist,
g(n) = N(n+1) - N(n) (jump, gap),
N(n) can belong to the natural numbers, the odd/even numbers, prime numbers...
In fact, the formula can be applied to all the sequences which follow this
condition :
for n sufficiently large, N(n)> 2 * g(n).
Now if we can generalize the sieve of Erathostènes that means that we can
generalize the fundamental theorem of the arithmetic and our definition of a
prime number, our vision of the numbers.
The numbers are nothing, they can be defined only by the sequence to which they
belong.
Ex :
11 among natural numbers and considering jump=0, Erathostènes : 11 is prime, is
of level 1.
11 among natural numbers and considering jump=1 : 11 has a weight of 2 : 11 =
2*5+1.
11 among prime numbers : 11 has a weight of 3 : 11 = 3*3+2.
11 is nothing by itself.

Good thoughts,

Rémi

PS : If these questions were already studied, please indicate to me where I can
find informations. And of course, all the comments are welcome.