2010-01-31, 11:35 | #1 |
"Rémi Eismann"
Apr 2007
France
27_{8} Posts |
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, orthe weight by k(n) = smallest k such that d(n) = a(n) mod k, 0 if no such k exists, orThe 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.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: http://reismann.free.fr/classement_en.php We can do some interesting conjectures about this classification: http://reismann.free.fr/conjecture_en.php I made a big effort of presentation, I updated my web site, my preprint and created a user page on the oeisWiki: http://reismann.free.fr/index_en.html http://arxiv.org/abs/0711.0865 http://oeis.org/wiki/User:R%C3%A9mi_Eismann All comments, criticisms, suggestions and help are welcome. Rémi. 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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