  In this paper we introduce an Euclidean decomposition of elements a_n of an
increasing sequence of natural numbers into weight * level + jump which we use
to classify the numbers a_n either by weight or by level. We then show that
this decomposition can be seen as a generalization of the sieve of Eratosthenes
(which is the particular case of the whole sequence of natural numbers). We
apply this decomposition to prime numbers in order to obtain a new
classification of primes, we analyze a few properties of this classification
and we make a series of conjectures based on numerical data. Finally we show
how composite numbers and 2-almost primes behave under the decomposition.
