 mersenneforum.org > Math Decomposition into weight * level + jump
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 Register FAQ Search Today's Posts Mark Forums Read 2010-01-31, 11:35 #1 Nunki   "Rémi Eismann" Apr 2007 France 278 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) byd(n) = a(n+1) - a(n); l(n) byl(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 byk(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 byL(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: 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.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post davieddy Puzzles 71 2013-12-22 07:26 henryzz GMP-ECM 8 2012-09-15 17:00 davieddy Science & Technology 7 2012-08-07 22:59 James Heinrich PrimeNet 11 2011-01-26 20:07 Freightyard PrimeNet 2 2009-11-07 19:48

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