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 A117563 a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) = 0. 78
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) is the "level" of prime(n). 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). a(n) = 0 only for primes 2, 3 and 7. A118534(n) = prime(n) - g(n) or A000040(n) - A001223(n) if prime(n) - g(n) > g(n), 0 otherwise. LINKS Remi Eismann, Table of n, a(n) for n = 1..10000 Remi Eismann, Java program to decompose a prime as weight*level + gap, or A117078(n)*A117563(n) + A001223(n)). Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT], 2007-2010. EXAMPLE 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. MATHEMATICA a34[n_] := If[n == 1 || n == 2 || n == 4, 0, 2 Prime[n] - Prime[n+1]]; a78[n_] := Block[{a, p = Prime[n], np = Prime[n+1]}, a = Min[Select[ Divisors[2p - np], # > np - p& ]]; If[a == Infinity, 0, a]]; a[n_] := If[a78[n] == 0, 0, a34[n]/a78[n]]; Array[a, 85] (* Jean-François Alcover, Nov 02 2018, after Robert G. Wilson v in A118534  *) CROSSREFS Cf. A117078, A118534. Sequence in context: A102662 A350279 A142048 * A060439 A206283 A135224 Adjacent sequences:  A117560 A117561 A117562 * A117564 A117565 A117566 KEYWORD nonn AUTHOR Rémi Eismann, Apr 29 2006, Feb 14 2008 EXTENSIONS More terms from Robert G. Wilson v, May 05 2006 Edited by N. J. A. Sloane, May 14 2006 STATUS approved

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Last modified September 26 20:57 EDT 2022. Contains 357050 sequences. (Running on oeis4.)