Conjecture: The prime numbers classified by level are rarefying among primes.
Edit after Ricardo comment: I use rarefy like in "Théorème de la raréfaction des nombres premiers" but maybe it's not really a mathematical term. The conjecture can be reworded as: the natural density of primes classified by level in relation to the prime numbers is 0.
I would like to know if this conjecture is hard to prove (i think so), if this conjecture is interesting for the repartition of primes (i think so) and the relations of this conjecture with other famous conjecture like RH or Legendre conjecture.
Important precisions : except for $p(1)=2$, $p(2)=3$ and $p(4)=7$, prime numbers are either classified by weight or by level. The prime numbers classified by weight have by definition a jump (first difference) less than $\sqrt{p(n)}$
I put the definitions of decomposition in ${\rm weight} \times {\rm level} + {\rm jump} \,$ on StackExchange so that you don't have to go out but you can find them on the decomposition page on the OEISWiki and in my preprint (arXiv:0711.0865 [math.NT])
Définitions of the decomposition into ${\rm weight} \times {\rm level} + {\rm jump} \,$ :
Let $\scriptstyle {\{a(n)\}}_{n=i_\min}^{\infty} \,$ be a strictly increasing integer sequence of positive integers.
The jump (first difference, gap):
$d(n) := a(n+1) - a(n). \,$
$l(n) := \begin{cases} a(n) - d(n) & \text{if } a(n) - d(n) > d(n), \\ 0 {\rm ~otherwise}. \end{cases} \,$
Alternative definition with $mod$ function: $l(n) := {\rm largest~} l {\rm~such~that~} a(n+1) = a(n) + a(n){\rm~}mod{\rm~}l, {\rm~or~} 0 {\rm~if~no~such~} l {\rm~exists.~}$
The weight:
$k(n) := \begin{cases} {\rm smallest~} k > d(n) {~s.t.~} k|l(n), \\ 0 {\rm ~if~} l(n) = 0. \end{cases} \,$
Alternative definition with $mod$ function: $k(n) := {\rm smallest~} k {\rm~such~that~} a(n+1) = a(n) + a(n){\rm~}mod{\rm~}k, {\rm~or~} 0 {\rm~if~no~such~} k {\rm~exists.~}$
The level:
$L(n) := \begin{cases} \frac{l(n)}{k(n)} & \text{if } k(n) > 0, \\ 0 & \text{if } k(n) = 0. \end{cases} \,$
Decomposition criterion:
The decomposition is possible if and only if $a(n+1) < \frac{3}{2} \times a(n) \,$
A unique decomposition:
The weight is the smallest such that in the Euclidean division of $a(n)$ by its weight the quotient is the level and the remainder is the jump. We have the unique decomposition: $a(n) = k(n) \times L(n) + d(n) = {\rm weight} \times {\rm level} + {\rm jump} \,$
Principles of classification:
If for $a(n)$, $l(n) = k(n) = L(n) = 0 \,$ then $a(n)$ is not classified.
If for $a(n)$, $k(n) > L(n) \,$ then $a(n)$ is classified by level if not $a(n)$ is classified by weight.
For natural numbers, the weight is the smallest prime factor of $n-1$ and the level is the largest proper divisor of $n-1$. The natural numbers classified by weight are the $composites+1$ and the natural numbers classified by level are the $primes+1$.
Edit : minor correction and add explanations on the natural numbers