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Sequences of weights (natural numbers, prime numbers...)

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eism...@gmail.com

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Aug 26, 2007, 5:07:08 AM8/26/07
to
Hi,

An integer a(n) of an increasing integer sequence can be decomposed
into weight*level+jump if a(n+1)<3/2*a(n).

Def :
jump(n) = a(n+1) - a(n),
l(n) = largest l such that jump(n) = MOD(a(n),l), 0 if a such l does
not exist (l(n) = a(n) - jump(n) if a(n) - jump(n) > jump(n), 0
otherwise),
weight(n) = smallest k such that jump(n) = MOD(a(n),k), 0 if a such k
does not exist,
level(n) = l(n) / weight(n), 0 if weight(n) = 0.

Sequences of weights :
Natural numbers (A000027), sieve of Eratosthenes : A020639
http://www.research.att.com/~njas/sequences/A020639
Prime numbers (A000040) : A117078
http://www.research.att.com/~njas/sequences/A117078
Composite numbers (A002808) : A130882
http://www.research.att.com/~njas/sequences/A130882
2-almost primes (A001358) : A130533
http://www.research.att.com/~njas/sequences/A130533
3-almost primes (A014612) : A130650
http://www.research.att.com/~njas/sequences/A130650
Lucky numbers (A000959) : A130889
http://www.research.att.com/~njas/sequences/A130889
Triangular numbers (A000217) : A130703
http://www.research.att.com/~njas/sequences/A130703

Graphs (log(weight);log(level)) (gnuplot) :
Natural numbers, prime numbers, composite numbers and 2-almost
primes :
http://reismann.free.fr/download/graph1.pdf
3-almost primes, prime powers, square free numbers and lucky
numbers :
http://reismann.free.fr/download/graph2.pdf
Triangular numbers (nice graph) :
http://reismann.free.fr/download/graph3.pdf

First application of the decomposition : an experimental
classification of prime numbers :
http://reismann.free.fr/classement.php

Good thoughts,

Rémi Eismann

eism...@gmail.com

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Sep 15, 2007, 4:56:00 AM9/15/07
to
Hi,

Sequences of weights - graphs on the OEIS:
Natural numbers (A000027) : A020639 (sieve of Eratosthenes)
http://www.research.att.com/~njas/sequences/table?a=20639&fmt=5
Prime numbers (A000040) : A117078
http://www.research.att.com/~njas/sequences/table?a=117078&fmt=5
Composite numbers (A002808) : A130882
http://www.research.att.com/~njas/sequences/table?a=130882&fmt=5
2-almost primes (A001358) : A130533
http://www.research.att.com/~njas/sequences/table?a=130533&fmt=5
3-almost primes (A014612) : A130650
http://www.research.att.com/~njas/sequences/table?a=130650&fmt=5
Lucky numbers (A000959) : A130889
http://www.research.att.com/~njas/sequences/table?a=130889&fmt=5
Triangular numbers (A000217) : A130703
http://www.research.att.com/~njas/sequences/table?a=130703&fmt=5
Even numbers (A005843) : A117871
http://www.research.att.com/~njas/sequences/table?a=117871&fmt=5
Odd numbers (A005408) : A090368
http://www.research.att.com/~njas/sequences/table?a=90368&fmt=5

Best,

Rémi Eismann


eism...@gmail.com

unread,
Sep 17, 2007, 10:20:06 AM9/17/07
to
A090369 and A117871 have been merged, so :

Sequences of weights - graphs on the OEIS:

Even numbers (A005843) : A090369
http://www.research.att.com/~njas/sequences/table?a=90369&fmt=5

Rémi Eismann

eism...@gmail.com

unread,
Sep 26, 2007, 4:31:27 AM9/26/07
to
Hi,

Sequence of weights :
Squares numbers (A000290) : A133150
http://www.research.att.com/~njas/sequences/A133150
graph on the OEIS :
http://www.research.att.com/~njas/sequences/table?a=133150&fmt=5

Sequence of weights :
Pentagonal numbers (A000326) : A133151
http://www.research.att.com/~njas/sequences/A133151
graph on the OEIS :
http://www.research.att.com/~njas/sequences/table?a=133151&fmt=5

Rémi Eismann

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