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Old 2010-01-31, 11:35   #1
Nunki's Avatar
"R. E."
Apr 2007

278 Posts
Default Decomposition into weight * level + jump

Dear Mersenneforum,

Let be a(n) an integer sequence strictly increasing, we define :
the jump (gap, first difference) of a(n) by
d(n) = a(n+1) - a(n);
l(n) by
l(n) = largest l such that d(n) = a(n) mod l, 0 if no such l exists, or
l(n) = a(n) - d(n) if a(n) - d(n) > d(n), 0 otherwise;
the weight by
k(n) = smallest k such that d(n) = a(n) mod k, 0 if no such k exists, or
k(n) = smallest k greater than d(n), that divides l(n), 0 if l(n)=0;
The level by
L(n) = l(n) / k(n), 0 if k(n) = 0.
In the Euclidean division of a(n) by its weight k(n), the quotient is the level L(n), and the remainder is the jump d(n). We have the unique decomposition a(n) = k(n) * L(n) + d(n) = weight * level + jump when l(n) is different from 0 which can be rewritten when a(n+1) < (3/2) * a(n).
Principles of classification:
If for a(n), l(n) = k(n) = L(n) = 0 then a(n) is not classified.
If for a(n), l(n) <> 0 and k(n) > L(n) then a(n) is classified by level , if not then it is classified by weight.
The decomposition applied to the natural numbers is the sieve of Eratosthenes. Applied to prime numbers, this decomposition is used to obtain a new classification of primes:
We can do some interesting conjectures about this classification:
I made a big effort of presentation, I updated my web site, my preprint and created a user page on the oeisWiki:
All comments, criticisms, suggestions and help are welcome.


PS: please the moderators, don't put this thread in Miscellaneous Math Threads, I don't claim anything just that I invented an original way to see the numbers and I want to share it.
PS2: Th oeisWiki is still under construction. At this time, the main content is derived from the classic OEIS, and all updates should be done there.
PS3: sorry for my poor English.
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