Search: id:A117563
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%I A117563
%S A117563 0,0,1,0,3,1,5,3,1,9,1,3,13,3,1,1,19,5,9,23,1,15,11,9,3,33,11,35,21,3,
               3,
%T A117563 5,45,3,49,5,1,3,23,1,59,9,63,27,65,11,1,3,75,45,1,79,21,35,1,1,89,5,39,
%U A117563 93,21,9,3,103,3,3,25,3,115,69,1,39,19,1,75,29,3,3,3,21,139,3,143,61,87
%N A117563 a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) 
               = 0.
%C A117563 a(n) is the "level" of prime(n).
%C A117563 There is a unique decomposition of the primes: provided the level a(n) 
               is > 0, we have prime(n) = weight * level + gap, or A000040(n)=A117078(n)*a(n)+A001223(n).
%C A117563 a(n) = 0 only for primes 2, 3 and 7.
%C A117563 A118534(n) = prime(n) - g(n) or A000040(n) - A001223(n) if prime(n) - 
               g(n) > g(n), 0 otherwise.
%H A117563 Remi Eismann, <a href="b117563.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A117563 Remi Eismann, <a href="http://reismann.free.fr/download/newAlgo.zip">
               Java program</a> to decompose a prime as weight*level + gap, or A117078(n)*A117563(n) 
               + A001223(n)).
%H A117563 Remi Eismann, <a href="http://arXiv.org/abs/0711.0865">Decomposition 
               of natural numbers into weight * level + jump and application to 
               a new classification of prime numbers</a>
%e A117563 a(7)=15/3=5; a(14)=39/13=3; a(16)=47/47=1; a(18)=55/11=5; a(29)=105/5=11
%Y A117563 Cf. A117078, A118534.
%Y A117563 Sequence in context: A122510 A102662 A142048 this_sequence A060439 A135224 
               A152203
%Y A117563 Adjacent sequences: A117560 A117561 A117562 this_sequence A117564 A117565 
               A117566
%K A117563 nonn
%O A117563 1,5
%A A117563 Remi Eismann (reismann(AT)free.fr), Apr 29 2006, Feb 14 2008
%E A117563 More terms from Robert G. Wilson v, May 05, 2006
%E A117563 Edited by N. J. A. Sloane (njas(AT)research.att.com), May 14 2006

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