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 A117563 a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) = 0. 78

%I

%S 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,

%T 3,5,45,3,49,5,1,3,23,1,59,9,63,27,65,11,1,3,75,45,1,79,21,35,1,1,89,

%U 5,39,93,21,9,3,103,3,3,25,3,115,69,1,39,19,1,75,29,3,3,3,21,139,3,143,61,87

%N a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) = 0.

%C a(n) is the "level" of prime(n).

%C 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 a(n) = 0 only for primes 2, 3 and 7.

%C A118534(n) = prime(n) - g(n) or A000040(n) - A001223(n) if prime(n) - g(n) > g(n), 0 otherwise.

%H Remi Eismann, <a href="/A117563/b117563.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Rémi Eismann, <a href="http://arXiv.org/abs/0711.0865">Decomposition into weight * level + jump and application to a new classification of primes</a>, arXiv:0711.0865 [math.NT], 2007-2010.

%e 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 Cf. A117078, A118534.

%K nonn

%O 1,5

%A _Rémi Eismann_, Apr 29 2006, Feb 14 2008

%E More terms from _Robert G. Wilson v_, May 05 2006

%E Edited by _N. J. A. Sloane_, May 14 2006

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